\documentclass[% %reprint, twocolumn, %superscriptaddress, %groupedaddress, %unsortedaddress, %runinaddress, %frontmatterverbose, % preprint, showpacs, showkeys, preprintnumbers, %nofootinbib, %nobibnotes, %bibnotes, amsmath,amssymb, aps, % prl, pra, % prb, % rmp, %prstab, %prstper, longbibliography, %floatfix, %lengthcheck,% ]{revtex4-1} %\usepackage{cdmtcs-pdf} \usepackage[dvipsnames]{xcolor} \usepackage{mathptmx}% http://ctan.org/pkg/mathptmx \usepackage{amssymb,amsthm,amsmath} \usepackage{tikz} \usetikzlibrary{calc} \usepackage[breaklinks=true,colorlinks=true,anchorcolor=blue,citecolor=blue,filecolor=blue,menucolor=blue,pagecolor=blue,urlcolor=blue,linkcolor=blue]{hyperref} \usepackage{graphicx}% Include figure files \usepackage{url} \usepackage{xcolor} \begin{document} \title{New forms of quantum value indefiniteness suggest that incompatible views on contexts are epistemic} %\title{Routes to value (in)definiteness: quantum contexts as views on states, and states viewed from stitchings of intertwined contexts} %\title{Combining true-implies-false with true-implies-true to obtain true-implies-value indefinite} %Title: %Abstract: Extensions of the Kochen-Specker theorem use quantum logics whose classical interpretation suggest a true-implies-value indefiniteness property. This can be interpreted as an indication that any view of a quantum state beyond a single context is epistemic. \author{Karl Svozil} \email{svozil@tuwien.ac.at} \homepage{http://tph.tuwien.ac.at/~svozil} \affiliation{Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/136, 1040 Vienna, Austria} \date{\today} \begin{abstract} Extensions of the Kochen-Specker theorem use quantum logics whose classical interpretation suggests a true-implies-value indefiniteness property. This can be interpreted as an indication that any view of a quantum state beyond a single context is epistemic. \end{abstract} %A remark by Gleason about the {\it ad hoc} construction of probability measures in Hilbert spaces as a result of the Pythagorean property of vector components is interpreted platonically. Unless there is a total match between preparation and measurement contexts, information about the former from the latter is not ontic but epistemic. This is corroborated by configurations of observables and contexts which a truth-implies-value indefiniteness property. \keywords{Quantum mechanics, Gleason theorem, Kochen-Specker theorem, Born rule} %\preprint{CDMTCS preprint nr. x} \maketitle \section{Quantum contexts as views on states} \label{2018-whycontexts-sec1} Contexts arise naturally in quantum mechanics: they correspond to greatest classical subdomains within the expanse of conceivable quantum propositions. For all empirical matters, every observable within a particular fixed context can be assumed classical with respect and relative to that context. Therefore, according to Gleason~\cite{Gleason}, it appears prudent to assume that classical probabilities should be applicable to such classical mini-universes; and, in particular, when considering observables within a given context. Gleason formalized this in terms of {\em frame functions} and proceeded to show how the quantum probabilities -- in particular, the Born rule -- can be ``stitched together'' from these classical bits and pieces. This paper can be seen as a {\it prolegomenon} to this approach; and as a contribution to the ongoing search for its semantics. Formally, the concept of context can be exposed in two ways: one is in terms of ``largest possible'' sets of orthogonal pure states; that is, in terms of (unit) vectors and their linear spans. Another one is by maximal operators and the perpendicular projection operators in their non-degenerate spectral decomposition. Let us start by supposing that contexts can be represented by orthonormal bases of Hilbert space. Due to the spectral theorem this immediately gives rise to an equivalent conception of context: that as a maximal observable which is formed by some (non-degenerate) spectral sum of the mutually orthogonal perpendicular projection operators corresponding to the basis states. This is just the expression of the dual role of perpendicular projection operators in quantum mechanics: they represent both pure states as well as observable bits; that is, elementary yes-no propositions. For the sake of an elementary example, suppose one is dealing with (lossless) electron spin state (or photon polarization) measurements. As there are two outcomes, the associated Hilbert space is two-dimensional. The two outcomes can be identified with two arbitrary orthogonal normalized vectors therein, forming an orthonormal basis. Suppose, for the sake of further simplicity, that we parametrize this basis to be the standard Cartesian basis in two-dimensional Hilbert space, its two vectors being~\cite[Eq.~(1.8)]{mermin-07} $\vert 0 \rangle = \begin{pmatrix} 1,0 \end{pmatrix}^\intercal$ and $\vert 1 \rangle = \begin{pmatrix}0,1\end{pmatrix}^\intercal$, where the superscript symbol ``$\intercal$'' indicates transposition. Their dyadic products %\begin{equation} %\begin{split} $\textsf{\textbf{E}}_0 =\vert 0 \rangle \langle 0 \vert = \begin{pmatrix}1,0\end{pmatrix}^\intercal \otimes \begin{pmatrix}1,0\end{pmatrix} = \begin{pmatrix}1&0\\0&0\end{pmatrix} ,%\text{ and} %\\ \textsf{\textbf{E}}_1 %=\vert 1 \rangle \langle 1 \vert %= \begin{pmatrix}0,1\end{pmatrix}^\intercal \otimes \begin{pmatrix}0,1\end{pmatrix} = \begin{pmatrix}0&0\\0&1\end{pmatrix} $ %\end{split} %\end{equation} form the corresponding (mutually) orthogonal perpendicular projection operators. These contexts can be either represented in terms of vectors, like ${\cal C} =\left\{ \vert 0 \rangle , \vert 1 \rangle \right\}$, or in terms of perpendicular projection operators, like ${\cal C} =\left\{ \textsf{\textbf{E}}_0 , \textsf{\textbf{E}}_1 \right\}$. Any two distinct numbers $\lambda_0 \neq \lambda_1$ define a maximal operator through the ``weighted'' spectral sum \begin{equation} \textsf{\textbf{A}} = \lambda_0 \textsf{\textbf{E}}_0 + \lambda_1 \textsf{\textbf{E}}_1 = \lambda_0 \vert 0 \rangle \langle 0 \vert + \lambda_1 \vert 1 \rangle \langle 1 \vert = \begin{pmatrix}\lambda_0 & 0\\0 & \lambda_1\end{pmatrix}. \end{equation} The term ``maximal'' refers to the fact that $\textsf{\textbf{A}}$ ``spans'' a ``classical sub-universe'' of mutually commuting operators through variations of $f(\textsf{\textbf{A}}) = f(\lambda_0) \textsf{\textbf{E}}_0 + f(\lambda_1) \textsf{\textbf{E}}_1 $, where $f:\mathbb{R} \mapsto \mathbb{R} $ represents some real valued polynomial or function of a single real argument~\cite[\S~84, Theorems~1{\&}2,p~171]{halmos-vs}. In particular, this includes the context ${\cal C} =\left\{ \textsf{\textbf{E}}_0 , \textsf{\textbf{E}}_1 \right\}$ through the two binary functions $f_i(\lambda_j) = \delta_{ij}$, with $i,j \in \{0,1\}$. \section{Probabilities on contexts in quantum mechanics} \label{2018-whycontexts-sec2} Let us concentrate on probabilities next. As already mentioned, Gleason~\cite{Gleason} observed that classical observables should obey classical probabilities -- this should be the same for Bayesian and frequentist approaches. Can we, therefore, hope for the existence of some {\it ``Realding''} -- that is, some global ontology, some enlarged panorama of ``real physical properties'' -- behind these stitched probabilities? As it turns out, relative to reasonable assumptions and the absence of exotic options, this is futile. Formally this issue can be rephrased by recalling that the main formal entities of quantum mechanics are all based on Hilbert space; that is, on vectors, as well as their relative position and permutations. A pure state represented as a vector $\vert \psi \rangle$ can be conveniently parameterized or encoded by coordinates referring to the respective bases. Because of their convenience one chooses orthonormal bases -- that is, contexts -- for such a parametrization. Why is convenience important? Because, as has been noted earlier, in finite dimensions $D$ any such context ${\cal C} \equiv \{\vert {\bf e}_1 \rangle , \vert {\bf e}_2 \rangle, \ldots ,\vert {\bf e}_D \rangle\}$ can also be interpreted as a maximal set of co-measurable propositions ${\cal C} \equiv \{\textsf{\textbf{E}}_1 , \textsf{\textbf{E}}_2, \ldots ,\textsf{\textbf{E}}_D \}$ with $\textsf{\textbf{E}}_i=\vert {\bf e}_i \rangle \langle {\bf e}_i \vert $, $1 \le i \le D$, as the latter refers to a complete system of orthogonal perpendicular projections which are a resolution of the identity operator $\mathbb{I}_D = \sum_{i=1}^D \textsf{\textbf{E}}_i$. For any such context, classical Kolmogorov probability theory requires the probabilities $P$ to satisfy the following axioms: \begin{itemize} \item[{\bf A1}] -- probabilities are real-valued and non-negative: $P(\textsf{\textbf{E}}_i) \in \mathbb{R}$, and $P(\textsf{\textbf{E}}_i) \ge 0$ for all $\textsf{\textbf{E}}_i \in {\cal C}$, or, equivalently, $1 \le i \le D$; \item[{\bf A2}] -- probabilities of mutually exclusive observables within contexts are additive: $ P\left(\sum_{i=1}^{k\le D} \textsf{\textbf{E}}_i\right) = \sum_{i=1}^{k\le D} P\left(\textsf{\textbf{E}}_i\right) ; $ \item[{\bf A3}] -- probabilities within one context add up to one: $ P(\mathbb{I}_D) = P\left(\sum_{i=1}^D \textsf{\textbf{E}}_i\right) = 1$. %\label{2018-whycontexts-ckpa} \end{itemize} How can probabilities $P_\psi \left( \textsf{\textbf{E}} \right)$ of propositions formalized by perpendicular projection operators (or, more generally, observables whose spectral sums contain such propositions) on given states $\vert \psi \rangle $ be formed which adhere to these axioms? As already Gleason pointed out in the second paragraph of Ref.~\cite[Sect.~1, p.~885]{Gleason}, there is an {\it ad hoc} way to obtain a probability measure on Hilbert spaces: a vector $\vert \psi \rangle $ can be ``viewed'' through a ``probing context'' $\cal C$ as follows: \begin{itemize} \item[(i)] For each closed subspace spanned by the vectors $\vert {\bf e}_i \rangle$ in the context $\cal C$, take the projection $\textsf{\textbf{E}}_i\vert \psi \rangle$ of $\vert \psi \rangle$ onto $\vert {\bf e}_i \rangle$. \item[(ii)] Take the absolute square of the length (norm) of this projection and identify it with the probability $P_{\psi}\left( \textsf{\textbf{E}}_i \right)$ of finding the quantum system which is in state $\vert \psi \rangle $ to be in state $\vert {\bf e}_i \rangle$; that is (the symbol ``$\dagger$'' stands for the Hermitian adjoint): \begin{equation} \begin{split} P_{\psi}\left( \textsf{\textbf{E}}_i \right) = \left(\textsf{\textbf{E}}_i\vert \psi \rangle\right)^\dagger \textsf{\textbf{E}}_i\vert \psi \rangle = \langle \psi \vert \textsf{\textbf{E}}_i^\dagger \textsf{\textbf{E}}_i\vert \psi \rangle \\ = \langle \psi \vert {\bf e}_i \rangle \underbrace{\langle {\bf e}_i \vert {\bf e}_i \rangle}_{=1} \langle {\bf e}_i \vert \psi \rangle %\\ = \langle \psi \vert {\bf e}_i \rangle \langle {\bf e}_i \vert \psi \rangle = \| \langle {\bf e}_i \vert \psi \rangle \|^2 . \end{split} \label{2018-whycontexts-glsqnprob} \end{equation} \end{itemize} Because of the mutual orthogonality of the elements in the context $\cal C$, the Pythagorean theorem enforces the third axiom {\bf A3} as long all vectors involved are normalized; that is, has length (norm)~1. This situation is depicted in Fig.~\ref{2018-whycontexts-f1-basis}. The situation is symmetric in a sense which reflects the duality between observable and state observed: Suppose now that the state $\vert \psi \rangle$ is ``completed'' by other vectors to form an entire context $\cal C'$. Then one could consider this context $\cal C'$, including $\vert \psi \rangle$ to ``probe'' vectors -- now identified as states -- in the original context $\cal C$. Very similarly, probability measures adhering axioms {\bf A1}--{\bf A3} can be constructed by, say, for instance, $P_{\textsf{\textbf{E}}_\psi}\left( \textsf{\textbf{E}}_i \right) $ \begin{figure} \begin{center} \begin{tikzpicture} [scale=0.33] %\tikzstyle{every path}=[line width=2pt] % Draw the axis \draw[line width=2pt, draw=LimeGreen,->] (0,-10cm) %+ (0,-2cm) -- (0,10cm) % -- +(0,2cm) node[above] {$\vert {\bf e}_2 \rangle$}; \draw[line width=2pt,draw=LimeGreen,->] (-10,0cm) %+(-2cm,0) -- (10cm,0) %-- +(2cm,0) node[right] {$\vert {\bf e}_1 \rangle$}; % Draw a circle of radius 1 \draw [draw=black!50,dotted] (0,0) circle[radius=10cm] node[above right] { }; %\draw [line width=0.5pt,color=blue!30,dotted] (0,0) -- ({-10/sqrt(3)},{-10/sqrt(3)}) node[right] {$r=1$} -- ({-10/sqrt(2)},{-10/sqrt(2)}); \draw ({-10/1.3},{-10/1.3}) node[color=black!50,shift={(1.35,0)}] {$r=1$}; % Draw vector x with coordinates \draw [line width=2pt,color=Maroon,->] (-40/5,-30/5) -- (40/5,30/5) node[above right] {$\vert \psi \rangle$}; \draw [line width=0.5pt,color=Maroon] (40/5,0) node[color=LimeGreen,font=\scriptsize,shift={(-0.25,-0.25)}] {$\textsf{\textbf{E}}_1\vert \psi \rangle$} -- (40/5,30/5); \draw [line width=0.5pt,color=Maroon] (0,30/5) node[below left,color=LimeGreen,font=\scriptsize] {$\textsf{\textbf{E}}_2\vert \psi \rangle$} -- (40/5,30/5); \draw [line width=2pt,color=LimeGreen] (40/5,0) circle[radius=7pt]; \draw [line width=2pt,color=Maroon] (40/5,0) circle[radius=2pt]; \draw [line width=2pt,color=LimeGreen] (0,30/5) circle[radius=7pt]; \draw [line width=2pt,color=Maroon] (0,30/5) circle[radius=2pt]; % Draw orthogonal vector y with coordinates \draw [line width=2pt,color=Maroon,dashed,->] (30/5,-40/5) -- (-30/5,40/5) node[above left] {$\vert {\bf \varphi} \rangle$}; \draw [line width=0.5pt,color=Maroon,dashed] (-30/5,0) node[below,color=LimeGreen,font=\scriptsize] {$\textsf{\textbf{E}}_1\vert \varphi \rangle$} -- (-30/5,40/5); \draw [line width=0.5pt,color=Maroon,dashed] (0,40/5) node[color=LimeGreen,font=\scriptsize,shift={(0.95,0)}] {$\textsf{\textbf{E}}_2\vert \varphi \rangle$} -- (-30/5,40/5); \draw [line width=2pt,color=LimeGreen] (-30/5,0) circle[radius=7pt]; \draw [line width=2pt,color=Maroon] (-30/5,0) circle[radius=2pt]; \draw [line width=2pt,color=LimeGreen] (0,40/5) circle[radius=7pt]; \draw [line width=2pt,color=Maroon] (0,40/5) circle[radius=2pt]; \draw [line width=0.5pt,color=LimeGreen] (10,0) -- ({(40/5)*(20/25)},{(30/5)*(20/25)}) node[color=Maroon,font=\tiny,shift={(-0.42,0.1)}] {$\textsf{\textbf{E}}_\psi \vert {\bf e}_1 \rangle$}; \draw [line width=0.5pt,color=LimeGreen] (10,0) -- ({(-30/5)*(-3/5)},{(40/5)*(-3/5)}) node[color=Maroon,font=\scriptsize,shift={(0.65,0)}] {$\textsf{\textbf{E}}_\varphi \vert {\bf e}_1 \rangle$}; \draw [line width=2pt,color=Maroon] ({(40/5)*(20/25)},{(30/5)*(20/25)}) circle[radius=7pt]; \draw [line width=2pt,color=LimeGreen] ({(40/5)*(20/25)},{(30/5)*(20/25)}) circle[radius=2pt]; \draw [line width=2pt,color=Maroon] ({(-30/5)*(-3/5)},{(40/5)*(-3/5)}) circle[radius=7pt]; \draw [line width=2pt,color=LimeGreen] ({(-30/5)*(-3/5)},{(40/5)*(-3/5)}) circle[radius=2pt]; \draw [line width=0.5pt,color=LimeGreen] (0,10) -- ({(40/5)*(3/5)},{(30/5)*(3/5)}) node[color=Maroon,font=\scriptsize,shift={(0.35,-0.3)}] {$\textsf{\textbf{E}}_\psi \vert {\bf e}_2 \rangle$}; \draw [line width=0.5pt,color=LimeGreen] (0,10) -- ({(-30/5)*(20/25)},{(40/5)*(20/25)}) node[color=Maroon,font=\scriptsize,shift={(0.65,0)}] {$\textsf{\textbf{E}}_\varphi \vert {\bf e}_2 \rangle$}; \draw [line width=2pt,color=Maroon] ({(40/5)*(3/5)},{(30/5)*(3/5)}) circle[radius=7pt]; \draw [line width=2pt,color=LimeGreen] ({(40/5)*(3/5)},{(30/5)*(3/5)}) circle[radius=2pt]; \draw [line width=2pt,color=Maroon] ({(-30/5)*(20/25)},{(40/5)*(20/25)}) circle[radius=7pt]; \draw [line width=2pt,color=LimeGreen] ({(-30/5)*(20/25)},{(40/5)*(20/25)}) circle[radius=2pt]; \end{tikzpicture} \end{center} \caption{An orthonormal basis forming a context {\color{LimeGreen} ${\cal C}=\{\vert {\bf e}_1 \rangle , \vert {\bf e}_2 \rangle \}$} represents a frame of reference from which a ``view'' on a state $\vert \psi \rangle$ can be obtained. Formally, if the vectors $\vert \psi \rangle$ and $\vert \varphi \rangle$ are normalized, such that $\langle \psi \vert \psi \rangle = \langle \varphi \vert \varphi \rangle = 1$, then the absolute square of the length (norm) of the projections $\textsf{\textbf{E}}_1\vert \psi \rangle = \vert {\bf e}_1\rangle \langle {\bf e}_1\vert \psi \rangle $ and $\textsf{\textbf{E}}_2\vert \psi \rangle = \vert {\bf e}_2\rangle \langle {\bf e}_2\vert \psi \rangle $ as well as $\textsf{\textbf{E}}_1\vert \varphi \rangle = \vert {\bf e}_1\rangle \langle {\bf e}_1\vert \varphi \rangle $ and $\textsf{\textbf{E}}_2\vert \varphi \rangle = \vert {\bf e}_2\rangle \langle {\bf e}_2\vert \varphi \rangle $ add up to one. Conversely, a second context {\color{Maroon} ${\cal C'}=\{\vert \psi \rangle , \vert \varphi \rangle \}$} grants a frame of reference from which a ``view'' on the first context $\cal C$ can be obtained. } \label{2018-whycontexts-f1-basis} \end{figure} It is important to keep in mind that, although Gleason's {\it Ansatz} is about a single context $\cal C$ it is valid for all contexts; indeed, formally, for a continuum of contexts represented by the continuum of possible orthonormal bases of $D$-dimensional Hilbert space. Every such context entails a particular {\em view} on the state $\vert \psi \rangle $; and there are a {\em continuum} of such views on the state $\vert \psi \rangle $. Furthermore, there is a symmetry between the two contexts $\cal C$ and $\cal C'$ involved. We may call ${\cal C}'$ the ``preparation context'' and ${\cal C}$ the ``measurement context,'' but these denominations are purely conventional. In this sense, it is a matter of convention if we consider ``$\cal C$ probing $\cal C'$'' or ``$\cal C'$ probing $\cal C$.'' There is one ``privileged view'' on the preparation context $\cal C'$: that is the view obtained if both the preparation and measurement contexts coincide: ${\cal C} = {\cal C}'$. Under such circumstances the observables are value definite: their values coincide with those of the preparation. \section{Contexts in partition logics and their probabilities} \label{2018-whycontexts-sec3} This section is a reminder rather than an exposition~\cite{svozil-93,dvur-pul-svo,svozil-ql,svozil-2001-eua,svozil-2008-ql,svozil-2016-s,svozil-2016-pu-book} into partition logics. Suffice it to say that partition logics are probably the most elementary generalization of Boolean algebras: they are the Boolean subalgebras associated with sets of partitions of a given set which are ``pasted'' or ``stitched'' together at their common elements; similar to contexts (blocks, subalgebras) in quantum logic. The main difference is that the latter is a continuous logic based on geometrical entities (vectors), whereas partition logics are discrete, finite algebraic structures based on sets of partitions of a given set. Nevertheless, for empirical purposes, it is always possible to come up with a partition logic mimicking the respective quantum logic~\cite{cheval-or}. Partition logics have two known model realization: automaton logics~\cite{e-f-moore,schaller-95,schaller-96} and generalized urn models~\cite{wright,svozil-2005-ln1e,2010-qchocolate}. Just like classical probabilities on Boolean logics the probabilities on Boolean structures are formed by a convex summation of all two-valued measures~\cite{svozil-2001-cesena,svozil-2016-s,svozil-2016-pu-book} -- corresponding to ball types. Such probabilities will hencefort called {\em (quasi)classical.} \section{Probabilities on pastings or stitchings of contexts} \label{2018-whycontexts-sec4} From dimension $D \ge 3$ onwards, contexts can be non-trivially connected or intertwined~\cite{Gleason} in up to $D-2$ common elements. Such intertwining chains of contexts give rise to various apparently ``non-classical'' logics; and a wealth (some might say a plethora) of publications dealing with ever-increasing ``strange'' or ``magic'' properties of observables hitherto unheard of in classical physics. The following logics have a realization in (mostly three-dimensional if not stated otherwise) Hilbert space. For concrete parametrizations, the reader is either referred to the literature, or to a recent survey~\cite[Chapter~12]{svozil-2016-pu-book}. On such pastings of contexts, (quasi)classical probabilities and their bounds, termed {\em conditions of possible experience} by Boole~\cite[p.~229]{Boole-62}, can be obtained in three steps~\cite{svozil-2001-cesena,svozil-2008-ql,svozil-2016-s,svozil-2016-pu-book}: \begin{itemize} \item[(i)] Enumerate all truth assignments (or two-$\{0,1\}$-valued measures or states) $v_i$. \item[(ii)] The (quasi)classical probabilities are obtained by the formation of the convex sum $\sum_i \lambda_i v_i$ over all such states obtained in (i), with $0 \le \lambda_i \le 1$ and $\sum_i \lambda_i =1$. \item[(ii)] The Bell-type bounds on probabilities and expectations are attained by bundling these truth assignments into vectors, one per two-valued measure, with the coordinates representing the respective values of those states on the atoms (propositions, observables) of the logic; and by subsequently solving the Hull problem for a convex polytope whose vertices are identified with the vectors formed by all truth assignments~\cite{froissart-81,cirelson,pitowsky-86,Pit-94}. \end{itemize} In what follows some such quantum logics will be enumerated whose quantum probabilities co-exist and sometimes violate their (quasi)classical probabilities, if they exist. Such violations can be expected to occur quite regularly, as -- although in both cases the probability axioms {\bf A1}--{\bf A3} are satisfied for mutually compatible observables -- the quantum probabilities are formed very differently from the (quasi)classical ones; that is, not by convex sums as in the (quasi)classical case, but by scalar products among vectors. \subsection{Triangular and square logics in four dimensions} For geometric and algebraic reasons there is no cyclic pasting of three or four contexts in three dimensions, but in four dimensions this is possible; as depicted in Fig.~\ref{2018-whycontexts-triangle4D}. The (quasi)classical probabilities are enumerated in the Appendices~\ref{2018-whycontexts-app-tri} and~\ref{2018-whycontexts-app-square}. \begin{figure} \begin{center} \begin{tabular}{cc} \begin{tikzpicture} [scale=0.22] \newdimen\ms \ms=0.1cm \tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=3*\ms] \tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=2*\ms] \tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=1*\ms] \tikzstyle{every path}=[line width=2pt] % Radius of regular polygons \newdimen\R \R=6cm % outer circle %\r= { \R * sqrt(3) } % inner circle %\newdimen\r %\r= {\R * sqrt(3)/2} % inner circle %\newdimen\K %\K=3cm % Define positions of all observables \path ({90 + 0 * 360 /3}:\R ) coordinate(1) ({90 + 1 * 360 /3}:\R ) coordinate(4) ({90 + 2 * 360 /3}:\R ) coordinate(7) ; % draw contexts %\draw [draw=violet!30] (0,0) circle[radius=\R ] node[above right] { }; \draw [color=orange] (1) -- (4); \draw [color=blue] (4) -- (7); \draw [color=green] (7) -- (1); % %% %% draw atoms %% % \draw (1) coordinate[c3,fill=orange]; % \draw (1) coordinate[c2,fill=green,label=90:\colorbox{white}{\footnotesize $1$}]; % %\draw (1) coordinate[c1,fill=violet!30]; % % \node[c3,fill=orange,label={left:\colorbox{white}{\footnotesize $9$}}] at ( $ (1)!{1/3}!(4) $ ) (2) {}; % % \node[c3,fill=orange,label={left:\colorbox{white}{\footnotesize $8$}}] at ( $ (1)!{2/3}!(4) $ ) (3) {}; % % \draw (4) coordinate[c3,fill=orange]; % \draw (4) coordinate[c2,fill=blue,label=below left:{\footnotesize \colorbox{white}{$7$}}]; % %\draw (4) coordinate[c1,fill=violet!30]; % % \node[c3,fill=blue,label={below:{\footnotesize \colorbox{white}{$6$}}}] at ( $ (4)!{1/3}!(7) $ ) (5) {}; % % \node[c3,fill=blue,label={below:{\footnotesize \colorbox{white}{$5$}}}] at ( $ (4)!{2/3}!(7) $ ) (6) {}; % \draw (7) coordinate[c3,fill=green]; % \draw (7) coordinate[c2,fill=blue,label=below right:{\footnotesize \colorbox{white}{$4$}}]; % %\draw (7) coordinate[c1,fill=violet!30]; % % \node[c3,fill=green,label={right:\colorbox{white}{\footnotesize $3$}}] at ( $ (7)!{1/3}!(1) $ ) (8) {}; % % \node[c3,fill=green,label={right:\colorbox{white}{\footnotesize $2$}}] at ( $ (7)!{2/3}!(1) $ ) (9) {}; % % \end{tikzpicture} % & % \begin{tikzpicture} [scale=0.2] \newdimen\ms \ms=0.1cm \tikzstyle{every path}=[line width=2pt] \tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=3*\ms] \tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=1.5*\ms] \tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=1.1*\ms] % Radius of regular polygons \newdimen\R \R=6cm % outer circle %\r= { \R * sqrt(3) } % inner circle %\newdimen\r %\r= {\R * sqrt(3)/2} % inner circle %\newdimen\K %\K=3cm % Define positions of all observables \path ({45 + 0 * 360 /4}:\R ) coordinate(1) ({45 + 1 * 360 /4}:\R ) coordinate(4) ({45 + 2 * 360 /4}:\R ) coordinate(7) ({45 + 3 * 360 /4}:\R ) coordinate(10) ; % draw contexts \draw [color=orange] (1) -- (4); \draw [color=blue] (4) -- (7); \draw [color=red] (7) -- (10); \draw [color=green] (10) -- (1); % %% %% draw atoms %% % \draw (1) coordinate[c3,fill=orange]; % \draw (1) coordinate[c2,fill=green,label=45:{\footnotesize $1$}]; % %\draw (1) coordinate[c1,fill=violet!30]; % % \node[c3,fill=orange,label={above:{\footnotesize $12$}}] at ( $ (1)!{1/3}!(4) $ ) (2) {}; % % \node[c3,fill=orange,label={above:{\footnotesize $11$}}] at ( $ (1)!{2/3}!(4) $ ) (3) {}; % % \draw (4) coordinate[c3,fill=blue]; % \draw (4) coordinate[c2,fill=orange,label={180-45}:{\footnotesize {$ {10}$}}]; % %\draw (4) coordinate[c1,fill=violet!30]; % % \node[c3,fill=blue,label={left:{\footnotesize {$9$}}}] at ( $ (4)!{1/3}!(7) $ ) (5) {}; % % \node[c3,fill=blue,label={left:{\footnotesize {$8$}}}] at ( $ (4)!{2/3}!(7) $ ) (6) {}; % \draw (7) coordinate[c3,fill=red]; % \draw (7) coordinate[c2,fill=blue,label={180+45}:{\footnotesize {$7$}}]; % %\draw (7) coordinate[c1,fill=violet!30]; % % \node[c3,fill=red,label={below:{\footnotesize $6$}}] at ( $ (7)!{1/3}!(10) $ ) (8) {}; % % \node[c3,fill=red,label={below:{\footnotesize $5$}}] at ( $ (7)!{2/3}!(10) $ ) (9) {}; % % \draw (10) coordinate[c3,fill=green]; % \draw (10) coordinate[c2,fill=red,label= {270+45}:{\footnotesize {$4$}}]; % % \node[c3,fill=green,label={right:{\footnotesize $3$}}] at ( $ (10)!{1/3}!(1) $ ) (11) {}; % % \node[c3,fill=green,label={right:{\footnotesize $2$}}] at ( $ (10)!{2/3}!(1) $ ) (12) {}; % % \end{tikzpicture} \\ (a)&(b) \end{tabular} \end{center} \caption{ Informally, Greechie (or, in another wording, orthogonality) diagrams~\cite{greechie:71} represent contexts by smooth curves such as straight lines or circles. The atoms are represented by circles. Two intertwining contexts are represented by ``broken'' (not smooth) but connected lines. (a) Greechie orthogonality diagram of the triangle logic in four dimensions, realized by (from top) $1: \frac{1}{2} \begin{pmatrix} 1,1,1,1 \end{pmatrix}^\intercal$, $2: \frac{1}{\sqrt{2}} \begin{pmatrix} 1,0,-1,0 \end{pmatrix}^\intercal$, $3: \frac{1}{\sqrt{2}} \begin{pmatrix} 0,1,0,-1 \end{pmatrix}^\intercal$, $4: \frac{1}{2} \begin{pmatrix} -1,1,-1,1 \end{pmatrix}^\intercal$, $5: \frac{1}{\sqrt{2}} \begin{pmatrix} 0,1,1,0 \end{pmatrix}^\intercal$, $6: \frac{1}{\sqrt{2}} \begin{pmatrix} 1,0,0,1 \end{pmatrix}^\intercal$, $7: \frac{1}{2} \begin{pmatrix} 1,1,-1,-1 \end{pmatrix}^\intercal$, $8: \frac{1}{\sqrt{2}} \begin{pmatrix} 0,0,1,-1 \end{pmatrix}^\intercal$, and $9: \frac{1}{\sqrt{2}} \begin{pmatrix} 1,-1,0,0 \end{pmatrix}^\intercal$. (b) Greechie orthogonality diagram of the square logic in four dimensions, realized by (from top right) $1: \begin{pmatrix} 1,0,0,0 \end{pmatrix}^\intercal$, $2: \frac{1}{\sqrt{2}} \begin{pmatrix} 0,1,0,1 \end{pmatrix}^\intercal$, $3: \frac{1}{\sqrt{2}} \begin{pmatrix} 0,1,0,-1 \end{pmatrix}^\intercal$, $4: \begin{pmatrix} 0,0,1,0 \end{pmatrix}^\intercal$, $5: \frac{1}{\sqrt{2}} \begin{pmatrix} 1,1,0,0 \end{pmatrix}^\intercal$, $6: \frac{1}{\sqrt{2}} \begin{pmatrix} 1,-1,0,0 \end{pmatrix}^\intercal$, $7: \begin{pmatrix} 0,0,0,1 \end{pmatrix}^\intercal$, and $8: \frac{1}{\sqrt{2}} \begin{pmatrix} 1,0,1,0 \end{pmatrix}^\intercal$, $9: \frac{1}{\sqrt{2}} \begin{pmatrix} 1,0,-1,0 \end{pmatrix}^\intercal$, $10: \begin{pmatrix} 0,1,0,0 \end{pmatrix}^\intercal$, $11: \frac{1}{\sqrt{2}} \begin{pmatrix} 0,0,1,1 \end{pmatrix}^\intercal$, $12: \frac{1}{\sqrt{2}} \begin{pmatrix} 0,0,1,-1 \end{pmatrix}^\intercal$. The associated (quasi)classical probabilities are obtained from a convex summation over all truth assignments, and listed in the Appendices~\ref{2018-whycontexts-app-tri} and~\ref{2018-whycontexts-app-square}. } \label{2018-whycontexts-triangle4D} \end{figure} Summation of the (quasi)classical probabilities on the intertwining atoms of the triangle logic yields $ p_1+p_4+p_7= \lambda_1 + \lambda_2 + \lambda_7 + \lambda_{12} + \lambda_{13} + \lambda_{14} \le 1 $. However, the axioms of probability theory are too restrictive to allow for quantum violations of these probabilities: after all, these adjacent vertices are mutually orthogonal, and thus are in the same context (augmented with the fourth atom of that context). Other inequalities, such as $p_1+p_2=\lambda_1 + \lambda_2 \le p_5+p_6= (\lambda_1 + \lambda_3 + \lambda_4 + \lambda_8 + \lambda_9) + (\lambda_2 + \lambda_5 + \lambda_6 + \lambda_{10} + \lambda_{11})$, compare vertices with the adjacent ``inner'' atoms; but again, due to the probability axiom {\bf A3}, the quantum probabilities must obey these inequalities as well. Komei Fukuda's {\tt cddlib} package~\cite{cdd-pck} can be employed for a calculation of the hull problem, yielding all Bell-type inequalities associated with the convex polytope whose vertices are associated with the 14 or 34 truth assignments (two-valued measures) on the respective triangle and square logics. It turns out that all of them are expressions of the axioms {\bf A1}--{\bf A3} which are mandatory also for the quantum probabilities within contexts. \subsection{Pentagon (pentagram) logic} \label{2018-whycontexts-pentagons} The pentagon (graph theoretically equivalent to a pentagram) logic is a cyclic stitching or pasting of five contexts~\cite{wright:pent,kalmbach-83,Beltrametti-1995,Klyachko-2008,Bub-2009,Bub-2010,Badziag-2011} as depicted in Fig.~\ref{2018-whycontexts-pentagon}. The (quasi)classical probabilities~\cite[p.~289, Fig.~11.8]{svozil-2016-s} can be obtained by taking the convex sum of all the 11 two-valued measures~\cite{wright:pent}, as listed in Appendix~\ref{2018-whycontexts-app-pent}. Because of the convex sum of all $\lambda$'s add up to one, the sum of the (quasi)classical probabilities enumerated in Eq.~(\ref{2018-whycontexts-ppent}), taken merely on the 5 intertwining observables, yields \begin{equation} \begin{aligned} &p_1+p_3+p_5+p_7+p_9\\ & = \lambda_1 + \lambda_4 + \lambda_7 + \lambda_9 + \lambda_{10} + 2\left( \lambda_2 + \lambda_3 + \lambda_5 + \lambda_6 + \lambda_8\right) \\ &\le 2 \sum_{i=1}^{11}\lambda_i =2 . \end{aligned} \end{equation} This inequality is in violation of quantum predictions~\cite{Bub-2009,Badziag-2011} of $\sqrt{5}>2$. Note that, in order to obtain the probabilities on the five intertwining observables (vertices) all of them need to be determined. However, only adjacent pairs share a common context. So at least three incompatible measurement types are necessary. \begin{figure} \begin{center} \begin{tikzpicture} [scale=0.2] \newdimen\ms \ms=0.1cm \tikzstyle{every path}=[line width=2pt] \tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=3*\ms] \tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=1.5*\ms] \tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=1.1*\ms] % Radius of regular polygons \newdimen\R \R=6cm % outer circle %\r= { \R * sqrt(3) } % inner circle %\newdimen\r %\r= {\R * sqrt(3)/2} % inner circle %\newdimen\K %\K=3cm % Define positions of all observables \path ({90 + 0 * 360 /5}:\R ) coordinate(1) ({90 + 36 + 0 * 360 /5}:{\R * sqrt((25+10*sqrt(5))/(50+10*sqrt(5)))} ) coordinate(2) ({90 + 1 * 360 /5}:\R ) coordinate(3) ({90 + 36 + 1 * 360 /5}:{\R * sqrt((25+10*sqrt(5))/(50+10*sqrt(5)))} ) coordinate(4) ({90 + 2 * 360 /5}:\R ) coordinate(5) ({90 + 36 + 2 * 360 /5}:{\R * sqrt((25+10*sqrt(5))/(50+10*sqrt(5)))} ) coordinate(6) ({90 + 3 * 360 /5}:\R ) coordinate(7) ({90 + 36 + 3 * 360 /5}:{\R * sqrt((25+10*sqrt(5))/(50+10*sqrt(5)))} ) coordinate(8) ({90 + 4 * 360 /5}:\R ) coordinate(9) ({90 + 36 + 4 * 360 /5}:{\R * sqrt((25+10*sqrt(5))/(50+10*sqrt(5)))} ) coordinate(10) ; % draw contexts \draw [color=orange] (1) -- (2) -- (3); \draw [color=red] (3) -- (4) -- (5); \draw [color=green] (5) -- (6) -- (7); \draw [color=blue] (7) -- (8) -- (9); \draw [color=magenta] (9) -- (10) -- (1); % % %% %% draw atoms %% % \draw (1) coordinate[c3,fill=orange,label=90:{\footnotesize $1$}]; % \draw (1) coordinate[c2,fill=magenta]; % % \draw (2) coordinate[c3,fill=orange,label={above left:\footnotesize $10$}]; % % \draw (3) coordinate[c3,fill=red,label={left:\footnotesize $9$}]; % \draw (3) coordinate[c2,fill=orange]; % % \draw (4) coordinate[c3,fill=red,label={left:\footnotesize $8$}]; % % \draw (5) coordinate[c3,fill=green,label={below left:\footnotesize $7$}]; % \draw (5) coordinate[c2,fill=red]; % % \draw (6) coordinate[c3,fill=green,label={below:\footnotesize $6$}]; % \draw (7) coordinate[c3,fill=blue,label={below right:\footnotesize $5$}]; % \draw (7) coordinate[c2,fill=green]; % % \draw (8) coordinate[c3,fill=blue,label={right:\footnotesize $4$}]; % % \draw (9) coordinate[c3,fill=magenta,label={right:\footnotesize $3$}]; \draw (9) coordinate[c2,fill=blue]; % % \draw (10) coordinate[c3,fill=magenta,label={above right:\footnotesize $2$}]; % % % \end{tikzpicture} \end{center} \caption{Greechie orthogonality diagram of the pentagon (pentagram) logic. The associated (quasi)classical probabilities are obtained from a convex summation over all truth assignments, and listed in the Appendix~\ref{2018-whycontexts-app-pent}. } \label{2018-whycontexts-pentagon} \end{figure} \subsection{Specker bug logic with the true--implies--false property} \label{2018-whycontexts-Specker-bug} A pasting of two pentagon logics, the ``Specker bug'' logic, has been introduced~\cite[Fig.~1, p.~182]{kochen2} and used~\cite[$\Gamma_1$, p.~68]{kochen1} by Kochen and Specker and discussed by many researchers~\cite{redhead,Pitowsky2003395,pitowsky-06}; see also Refs.~\cite[Fig.~B.l. p.~64]{Belinfante-73}, \cite[p.~588-589]{stairs83}, \cite[Sects.~IV, Fig.~2]{clifton-93} and \cite[p.~39, Fig.~2.4.6]{pulmannova-91}. It is a pasting~\cite{kalmbach-83,nav:91} of seven contexts in such a tight way [cf. Fig.~\ref{2018-whycontexts-fbugcombo}(a)] that preparation of a (quasi)classical system in state $ {\bf a} $ entails non-occurrence of observable $ {\bf b}$. As has been observed by Stairs~\cite[p.~588-589]{stairs83} and Clifton~\cite[Sects.~II,III, Fig.~1]{clifton-93,Johansen-1994,Vermaas-1994}, this is no longer the case for quantum states and quantum observables. Therefore, if one prepares a system in a state $\vert {\bf a} \rangle$ and measures $\textsf{\textbf{E}}_b = \vert {\bf b}\rangle \langle {\bf b} \vert$, associated with state $\vert {\bf b} \rangle$, then the mere occurrence of $\vert {\bf b} \rangle$ implies the non-classicality of the quantized system. Again the (quasi)classical probabilities~\cite[p.~286, Fig.~11.5(iii)]{svozil-2016-s} enumerated in Appendix~\ref{2018-whycontexts-app-sbcombo} can be obtained by taking the convex sum of all the 14 two-valued measures~\cite[p.~579, Table~7]{svozil-2008-ql}. Pt{\'{a}}k and Pulmannov{\'{a}}~\cite[p.~39, Fig.~2.4.6]{pulmannova-91} as well as Pitowsky in Refs.~\cite[p.~402, Fig.~2]{Pitowsky2003395} and~\cite[pp.~224,225, Fig.~10.2]{pitowsky-06} noted that, for (quasi)classical probabilities -- including ones on partition logics -- the sum of the probabilities on $\vert {\bf a} \rangle$ and $\vert {\bf b} \rangle$ must not exceed $\frac{3}{2}$. Therefore both cannot be true at the same time, because this would result in their sum being $2$. This might be called a {\em true--implies--false property}~\cite{2018-minimalYIYS} ({\it aka} one-zero rule~\cite{svozil-2006-omni}) on the atoms ${\bf a}$ and ${\bf b}$. Actually, this classical bound can be tightened by explicity summing the (quasi)classical probabilities of ${\bf a}$ and ${\bf b}$ enumerated in Eq.~(\ref{2018-whycontexts-pspeckerbug}) Because of the convex sum of all $\lambda$'s add up to one, this yields yields \begin{equation} %\begin{aligned} p_{\bf a}+p_{\bf b} = \lambda_1+\lambda_2+\lambda_3+\lambda_6+\lambda_{13}+\lambda_{14} \le \sum_{i=1}^{14}\lambda_i =1 . %\end{aligned} \end{equation} This inequality is in violation of quantum predictions for a system prepared in state $\vert {\bf a} \rangle$; in this case~\cite{cabello-1994}, $\frac{10}{9}>1$. Indeed, Cabello~\cite{cabello-1994} (see also his dissertation~\cite[pp.~55,56]{Cabello-1996-diss}) pointed out that in three dimensions, $\vert {\bf a} \rangle$ and $\vert {\bf b} \rangle$ must be at least an angle $\angle ({\bf a},{\bf b}) \ge \textrm{arcsec} (3) = \textrm{arccos}\left(\frac{1}{3}\right)= \frac{\pi}{2} - \textrm{arccot}\left(2 \sqrt{2}\right) = \textrm{arctan}\left(2 \sqrt{2}\right)$ apart. Therefore, the probability to find a state prepared along $\vert {\bf a} \rangle \equiv \begin{pmatrix} 1,0,0 \end{pmatrix}^\intercal$ in a state $\vert {\bf b} \rangle \equiv \begin{pmatrix} \cos \angle ({\bf a},{\bf b}) ,\sin \angle ({\bf a},{\bf b}) ,0 \end{pmatrix}^\intercal$ cannot exceed $\vert \langle {\bf b} \vert {\bf a} \rangle \vert^2= 1/9$. Thus in at most one-ninth of all cases will quantum mechanical probabilities violate the classical ones, as the classical prediction demands zero probability to measure ${\bf b}$, given ${\bf a}$. (This prediction is relative to the assumption of non-contextuality, such that the truth assignment is independent of the particular context.) For a concrete ``optimal'' realization~\cite[p.~206, Fig.~1]{tkadlec-96} (see also~\cite[Fig.~4, p.~5387]{svozil-tkadlec}), take $\vert {\bf a}\rangle = \frac{1}{\sqrt{3}}\begin{pmatrix} 1,\sqrt{2},0 \end{pmatrix}^\intercal $ and $\vert {\bf b}\rangle = \frac{1}{\sqrt{3}}\begin{pmatrix} -1,\sqrt{2},0 \end{pmatrix}^\intercal $ which yield $\vert \langle {\bf b} \vert {\bf a}\rangle \vert = \frac{1}{3}$. %\begin{figure} %\begin{center} % %\end{center} %\caption{Greechie orthogonality diagram of the Specker bug logic. %A proof that, if the system is prepared in state ${\bf a}$, then classical (non-contextual) truth assignments require ${\bf b}$ not to occur, proceeds as follows: %In such a truth assignment, as {\it per} axiom {\bf A3}, there is only one true atom per context; all the others have to be false. %In a proof by contradiction, suppose that both ${\bf a}$ and ${\bf b}$ are true. %Then all atoms connected to them (2,4,7,9) must be false. %This in turn requires that the observables (3,8) connecting them must both be true. %Alas, those two observables (3,8) are connected by a ``middle'' context {\color{cyan}$\{3,11,8\}$} %are both true, which is forbidden by axiom {\bf A3}. %The only consistent alternative is to disallow ${\bf b}$ to be true if ${\bf a}$ is assumed to be true; or, conversely, % to disallow ${\bf a}$ to be true if ${\bf b}$ is assumed to be true. %} %\label{2018-whycontexts-fbug} %\end{figure} Another true-implies-false configuration depicted in Fig.~\ref{2018-whycontexts-acsl1}(a) has an immediate quantum realization~\cite[Table.~1, p.~102201-7]{2015-AnalyticKS} for $\vert \langle {\bf a} \vert {\bf b}\rangle \vert^2=\frac{1}{2}$, and can be constructively (i.e., algorithmically computable) extended to arbitrary angles between non-collinear and non-orthogonal vectors. \subsection{Combo of Specker bug logic with the true--implies--true as well as inseparability properties} This non-classical behaviour can be ``boosted'' by an extension of the Specker bug logic~\cite[$\Gamma_1$, p.~68]{kochen1}, including two additional contexts {\color{violet} $\left\{ {\bf a}, {\bf c} , {\bf b}' \right\}$} as well as {\color{brown} $\left\{ {\bf b}, {\bf c} , {\bf a}' \right\}$}, as depicted in Fig.~\ref{2018-whycontexts-fbugcombo}(b). It implements a {\em true-implies-true property}~\cite{2018-minimalYIYS} ({\it aka} one-one rule~\cite{svozil-2006-omni}) for ${\bf a}$ and ${\bf a}'$. Cabello's bound on the angle $\angle ({\bf a},{\bf b})$ between ${\bf a}$ and ${\bf b}$ mentioned earlier results in bounds between ${\bf a}$ and ${\bf a}'$ as well as ${\bf b}$ and ${\bf b}'$: since ${\bf a}$ and ${\bf b}'$ as well as ${\bf b}$ and ${\bf a}'$ are orthogonal, that is, $ \angle ({\bf a},{\bf b}') = \angle ({\bf b},{\bf a}') = \frac{\pi}{2} $, it follows for planar configurations that %\begin{equation} %\begin{split} $\angle ({\bf a},{\bf a}') = \angle ({\bf b},{\bf a}') - \angle ({\bf a},{\bf b}) %\\ \le \frac{\pi}{2} - \textrm{arccos} \left(\frac{1}{3} \right)= \textrm{arccot} \left( 2\sqrt{2} \right)= \textrm{arccsc} \left( {3} \right)= \textrm{arcsin} \left(\frac{1}{3} \right) $. %\end{split} %\end{equation} For symmetry reasons, the same estimate holds for planar configurations between ${\bf b}$ and ${\bf b}'$. For non-planar configurations the angles must be even less than for planar ones. True-implies-true properties have also been studied by Stairs~\cite[p.~588-589, note added in proof]{stairs83}; Clifton~\cite[Sects.~II,III, Fig.~1]{clifton-93,Johansen-1994,Vermaas-1994} presents a similar argument, based upon another true-implies-true logic inspired by Bell~\cite[Fig.~C.l. p.~67]{Belinfante-73} (cf. also Pitowsky~\cite[p.~394]{Pitowsky-1982-subs}), on the Specker bug logic~\cite[Sects.~IV, Fig.~2]{clifton-93}. More recently Hardy~\cite{Hardy-92,Hardy-93,hardy-97} as well as Cabello and Garc{\'{i}}a-Alcaine and others~\cite{Cabello-1995-ppks,cabello-96,cabello-97-nhvp,Badziag-2011,Cabello-2013-HP,Cabello-2013-Hardylike} have discussed such scenarios. Another true-implies-true configuration depicted in Fig.~\ref{2018-whycontexts-acsl1}(b) has an immediate quantum realization~\cite[Table.~1, p.~102201-7]{2015-AnalyticKS} for $\vert \langle {\bf a} \vert {\bf b}\rangle \vert^2=\frac{1}{2}$, and can be extended to arbitrary angles between non-collinear and non-orthogonal vectors. A combo of Specker bug logics renders a non-separable set of two-valued states~\cite[$\Gamma_3$, p.~70]{kochen1}: in the logic depicted in Fig.~\ref{2018-whycontexts-fbugcombo}(c), ${\bf a}$ and ${\bf a}'$ as well as ${\bf b}$ and ${\bf b}'$ cannot be ``separated'' from one another by any non-contextual (quasi)classical truth assignment enumerated in Appendix~\ref{2018-whycontexts-app-sbcombo}. Kochen and Specker~\cite[Theorem~0, p.~67]{kochen1} pointed out that this type of inseparability is a necessary and sufficient condition for a logic to be not embeddable in any classical Boolean algebra. Therefore, whereas both the Specker bug logic as well as its extension true-implies-true logic can be represented by a partition logic, the combo Specker bug logic cannot. \begin{figure} \begin{center} \begin{tabular}{c} \begin{tikzpicture} [scale=0.25] \newdimen\ms \ms=0.1cm \tikzstyle{every path}=[line width=2pt] \tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=3*\ms] \tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=1.5*\ms] \tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=1.1*\ms] % Radius of regular polygons \newdimen\R \R=6cm % outer circle %\r= { \R * sqrt(3) } % inner circle %\newdimen\r %\r= {\R * sqrt(3)/2} % inner circle %\newdimen\K %\K=3cm % Define positions of all observables \path ({90 + 0 * 360 /6}:\R ) coordinate(1) ({90 + 30 + 0 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(2) ({90 + 1 * 360 /6}:\R ) coordinate(3) ({90 + 30 + 1 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(4) ({90 + 2 * 360 /6}:\R ) coordinate(5) ({90 + 30 + 2 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(6) ({90 + 3 * 360 /6}:\R ) coordinate(7) ({90 + 30 + 3 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(8) ({90 + 4 * 360 /6}:\R ) coordinate(9) ({90 + 30 + 4 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(10) ({90 + 5 * 360 /6}:\R ) coordinate(11) ({90 + 30 + 5 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(12) (0,0) coordinate(13) ; % draw contexts \draw [color=orange] (1) -- (2) -- (3); \draw [color=red] (3) -- (4) -- (5); \draw [color=green] (5) -- (6) -- (7); \draw [color=blue] (7) -- (8) -- (9); \draw [color=gray] (9) -- (10) -- (11); % \draw [color=magenta] (11) -- (12) -- (1); % \draw [color=cyan] (4) -- (10); % %% %% draw atoms %% % \draw (1) coordinate[c3,fill=magenta,label=90:\LARGE ${\bf a}$]; % \draw (1) coordinate[c2,fill=orange]; % % \draw (2) coordinate[c3,fill=orange,label=140:{\footnotesize 10}]; % % \draw (3) coordinate[c3,fill=red,label=180:{\footnotesize 9}]; % \draw (3) coordinate[c2,fill=orange]; % % \draw (4) coordinate[c3,fill=red,label=180:{\footnotesize 8}]; % \draw (4) coordinate[c2,fill=cyan]; % % \draw (5) coordinate[c3,fill=green,label=180:{\footnotesize 7}]; % \draw (5) coordinate[c2,fill=red]; % % \draw (6) coordinate[c3,fill=green,label=220:{\footnotesize 6}]; % \draw (7) coordinate[c3,fill=blue,label=270:\LARGE ${\bf b}$]; % \draw (7) coordinate[c2,fill=green]; % % \draw (8) coordinate[c3,fill=blue,label=320:{\footnotesize 5}]; % % \draw (9) coordinate[c3,fill=gray,label=0:{\footnotesize 4}]; \draw (9) coordinate[c2,fill=blue]; % % \draw (10) coordinate[c3,fill=gray,label=0:{\footnotesize 3}]; % \draw (10) coordinate[c2,fill=cyan]; % % \draw (11) coordinate[c3,fill=magenta,label=0:{\footnotesize 2}]; % \draw (11) coordinate[c2,fill=gray]; % % \draw (12) coordinate[c3,fill=magenta,label=40:{\footnotesize 1}]; % % \draw (13) coordinate[c3,fill=cyan,label=90:{\footnotesize 11}]; % \end{tikzpicture} \\ (a) \\ \begin{tikzpicture} [scale=0.25] \newdimen\ms \ms=0.1cm \tikzstyle{every path}=[line width=2pt] \tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=3*\ms] \tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=1.8*\ms] \tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=1.1*\ms] % Radius of regular polygons \newdimen\R \R=6cm % outer circle %\r= { \R * sqrt(3) } % inner circle %\newdimen\r %\r= {\R * sqrt(3)/2} % inner circle \newdimen\K \K=18cm % Define positions of all observables \path ({90 + 0 * 360 /6}:\R ) coordinate(1) ({90 + 30 + 0 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(2) ({90 + 1 * 360 /6}:\R ) coordinate(3) ({90 + 30 + 1 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(4) ({90 + 2 * 360 /6}:\R ) coordinate(5) ({90 + 30 + 2 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(6) ({90 + 3 * 360 /6}:\R ) coordinate(7) ({90 + 30 + 3 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(8) ({90 + 4 * 360 /6}:\R ) coordinate(9) ({90 + 30 + 4 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(10) ({90 + 5 * 360 /6}:\R ) coordinate(11) ({90 + 30 + 5 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(12) (0,0) coordinate(13) ({90 + 0 * 360 /6}:\R ) + (\K,0) coordinate(21) ({90 + 30 + 0 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(22) ({90 + 1 * 360 /6}:\R ) + (\K,0) coordinate(23) ({90 + 30 + 1 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(24) ({90 + 2 * 360 /6}:\R ) + (\K,0) coordinate(25) ({90 + 30 + 2 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(26) ({90 + 3 * 360 /6}:\R ) + (\K,0) coordinate(27) ({90 + 30 + 3 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(28) ({90 + 4 * 360 /6}:\R ) + (\K,0) coordinate(29) ({90 + 30 + 4 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(210) ({90 + 5 * 360 /6}:\R ) + (\K,0) coordinate(211) ({90 + 30 + 5 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(212) (0,0) + (\K,0) coordinate(213) (0,0) + ({\K/2},0) coordinate(40) ; % draw contexts \draw [color=orange] (1) -- (2) -- (3); \draw [color=red] (3) -- (4) -- (5); \draw [color=green] (5) -- (6) -- (7); \draw [color=blue] (7) -- (8) -- (9); \draw [color=gray] (9) -- (10) -- (11); % \draw [color=magenta] (11) -- (12) -- (1); % \draw [color=cyan] (4) -- (10); \draw [color=violet] (7) to [out=0,in=270] (40) to [out=90,in=180] (21); \draw [color=brown] (27) to [out=180,in=270] (40) to [out=90,in=0] (1); % %% %% draw atoms %% % \draw (1) coordinate[c3,fill=magenta,label=90:\LARGE ${\bf a}$]; % \draw (1) coordinate[c2,fill=orange]; % \draw (1) coordinate[c1,fill=brown]; % % \draw (2) coordinate[c3,fill=orange,label={[label distance=-2]140:{\footnotesize 10}}]; % % \draw (3) coordinate[c3,fill=red,label={[label distance=-2]180:{\footnotesize 9}}]; % \draw (3) coordinate[c2,fill=orange]; % % \draw (4) coordinate[c3,fill=red,label={[label distance=-2]180:{\footnotesize 8}}]; % \draw (4) coordinate[c2,fill=cyan]; % % \draw (5) coordinate[c3,fill=green,label={[label distance=-2]180:{\footnotesize 7}}]; % \draw (5) coordinate[c2,fill=red]; % % \draw (6) coordinate[c3,fill=green,label={[label distance=-2]220:{\footnotesize 6}}]; % \draw (7) coordinate[c3,fill=blue,label=270:\LARGE ${\bf b}$]; % \draw (7) coordinate[c2,fill=green]; % \draw (7) coordinate[c1,fill=violet]; % % \draw (8) coordinate[c3,fill=blue,label={[label distance=-2]90:{\footnotesize 5}}]; % % \draw (9) coordinate[c3,fill=gray,label={[label distance=-2]0:{\footnotesize 4}}]; \draw (9) coordinate[c2,fill=blue]; % % \draw (10) coordinate[c3,fill=gray,label={[label distance=-2]0:{\footnotesize 3}}]; % \draw (10) coordinate[c2,fill=cyan]; % % \draw (11) coordinate[c3,fill=magenta,label={[label distance=-2]0:{\footnotesize 2}}]; % \draw (11) coordinate[c2,fill=gray]; % % \draw (12) coordinate[c3,fill=magenta,label={[label distance=-2]270:{\footnotesize 1}}]; % % \draw (13) coordinate[c3,fill=cyan,label={[label distance=-2]90:{\footnotesize 11}}]; % \draw (21) coordinate[c3,fill=violet,label=90:\LARGE ${\bf a}'$]; % \draw (27) coordinate[c3,fill=brown,label=270:\LARGE ${\bf b}'$]; % \draw (40) coordinate[c3,fill=brown,label={[label distance=15]90:\LARGE ${\bf c}$}]; % \draw (40) coordinate[c2,fill=violet]; % \end{tikzpicture} \\ (b) \\ \begin{tikzpicture} [scale=0.25] \newdimen\ms \ms=0.1cm \tikzstyle{every path}=[line width=2pt] \tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=3*\ms] \tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=1.8*\ms] \tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=1.1*\ms] % Radius of regular polygons \newdimen\R \R=6cm % outer circle %\r= { \R * sqrt(3) } % inner circle %\newdimen\r %\r= {\R * sqrt(3)/2} % inner circle \newdimen\K \K=18cm % Define positions of all observables \path ({90 + 0 * 360 /6}:\R ) coordinate(1) ({90 + 30 + 0 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(2) ({90 + 1 * 360 /6}:\R ) coordinate(3) ({90 + 30 + 1 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(4) ({90 + 2 * 360 /6}:\R ) coordinate(5) ({90 + 30 + 2 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(6) ({90 + 3 * 360 /6}:\R ) coordinate(7) ({90 + 30 + 3 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(8) ({90 + 4 * 360 /6}:\R ) coordinate(9) ({90 + 30 + 4 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(10) ({90 + 5 * 360 /6}:\R ) coordinate(11) ({90 + 30 + 5 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(12) (0,0) coordinate(13) ({90 + 0 * 360 /6}:\R ) + (\K,0) coordinate(21) ({90 + 30 + 0 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(22) ({90 + 1 * 360 /6}:\R ) + (\K,0) coordinate(23) ({90 + 30 + 1 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(24) ({90 + 2 * 360 /6}:\R ) + (\K,0) coordinate(25) ({90 + 30 + 2 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(26) ({90 + 3 * 360 /6}:\R ) + (\K,0) coordinate(27) ({90 + 30 + 3 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(28) ({90 + 4 * 360 /6}:\R ) + (\K,0) coordinate(29) ({90 + 30 + 4 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(210) ({90 + 5 * 360 /6}:\R ) + (\K,0) coordinate(211) ({90 + 30 + 5 * 360 /6}:{\R * sqrt(3)/2} ) + (\K,0) coordinate(212) (0,0) + (\K,0) coordinate(213) (0,0) + ({\K/2},0) coordinate(40) ; % draw contexts \draw [color=orange] (1) -- (2) -- (3); \draw [color=red] (3) -- (4) -- (5); \draw [color=green] (5) -- (6) -- (7); \draw [color=blue] (7) -- (8) -- (9); \draw [color=gray] (9) -- (10) -- (11); % \draw [color=magenta] (11) -- (12) -- (1); % \draw [color=cyan] (4) -- (10); \draw [color=orange!30!white] (21) -- (22) -- (23); \draw [color=red!30!white] (23) -- (24) -- (25); \draw [color=green!30!white] (25) -- (26) -- (27); \draw [color=blue!30!white] (27) -- (28) -- (29); \draw [color=gray!30!white] (29) -- (210) -- (211); % \draw [color=magenta!30!white] (211) -- (212) -- (21); % \draw [color=cyan!30!white] (24) -- (210); \draw [color=violet] (7) to [out=0,in=270] (40) to [out=90,in=180] (21); \draw [color=brown] (27) to [out=180,in=270] (40) to [out=90,in=0] (1); % %% %% draw atoms %% % \draw (1) coordinate[c3,fill=magenta,label=90:\LARGE ${\bf a}$]; % \draw (1) coordinate[c2,fill=orange]; % \draw (1) coordinate[c1,fill=brown]; % % \draw (2) coordinate[c3,fill=orange,label={[label distance=-2]140:{\footnotesize 10}}]; % % \draw (3) coordinate[c3,fill=red,label={[label distance=-2]180:{\footnotesize 9}}]; % \draw (3) coordinate[c2,fill=orange]; % % \draw (4) coordinate[c3,fill=red,label={[label distance=-2]180:{\footnotesize 8}}]; % \draw (4) coordinate[c2,fill=cyan]; % % \draw (5) coordinate[c3,fill=green,label={[label distance=-2]180:{\footnotesize 7}}]; % \draw (5) coordinate[c2,fill=red]; % % \draw (6) coordinate[c3,fill=green,label={[label distance=-2]220:{\footnotesize 6}}]; % \draw (7) coordinate[c3,fill=blue,label=270:\LARGE ${\bf b}$]; % \draw (7) coordinate[c2,fill=green]; % \draw (7) coordinate[c1,fill=violet]; % % \draw (8) coordinate[c3,fill=blue,label={[label distance=-2]90:{\footnotesize 5}}]; % % \draw (9) coordinate[c3,fill=gray,label={[label distance=-2]0:{\footnotesize 4}}]; \draw (9) coordinate[c2,fill=blue]; % % \draw (10) coordinate[c3,fill=gray,label={[label distance=-2]0:{\footnotesize 3}}]; % \draw (10) coordinate[c2,fill=cyan]; % % \draw (11) coordinate[c3,fill=magenta,label={[label distance=-2]0:{\footnotesize 2}}]; % \draw (11) coordinate[c2,fill=gray]; % % \draw (12) coordinate[c3,fill=magenta,label={[label distance=-2]270:{\footnotesize 1}}]; % % \draw (13) coordinate[c3,fill=cyan,label={[label distance=-2]90:{\footnotesize 11}}]; % \draw (21) coordinate[c3,fill=magenta!30!white,label=90:\LARGE ${\bf a}'$]; % \draw (21) coordinate[c2,fill=orange!30!white]; % \draw (21) coordinate[c1,fill=violet]; % % \draw (22) coordinate[c3,fill=orange!30!white,label={[label distance=-2]270:{\footnotesize 10'}}]; % % \draw (23) coordinate[c3,fill=red!30!white,label={[label distance=-2]180:{\footnotesize 9'}}]; % \draw (23) coordinate[c2,fill=orange!30!white]; % % \draw (24) coordinate[c3,fill=red!30!white,label={[label distance=-2]180:{\footnotesize 8'}}]; % \draw (24) coordinate[c2,fill=cyan!30!white]; % % \draw (25) coordinate[c3,fill=green!30!white,label={[label distance=-2]180:{\footnotesize 7'}}]; % \draw (25) coordinate[c2,fill=red!30!white]; % % \draw (26) coordinate[c3,fill=green!30!white,label={[label distance=-2]90:{\footnotesize 6'}}]; % \draw (27) coordinate[c3,fill=blue!30!white,label=270:\LARGE ${\bf b}'$]; % \draw (27) coordinate[c2,fill=green!30!white]; % \draw (27) coordinate[c1,fill=brown]; % % \draw (28) coordinate[c3,fill=blue!30!white,label={[label distance=-2]320:{\footnotesize 5'}}]; % % \draw (29) coordinate[c3,fill=gray!30!white,label={[label distance=-2]0:{\footnotesize 4'}}]; \draw (29) coordinate[c2,fill=blue!30!white]; % % \draw (210) coordinate[c3,fill=gray!30!white,label={[label distance=-2]0:{\footnotesize 3'}}]; % \draw (210) coordinate[c2,fill=cyan!30!white]; % % \draw (211) coordinate[c3,fill=magenta!30!white,label={[label distance=-2]0:{\footnotesize 2'}}]; % \draw (211) coordinate[c2,fill=gray!30!white]; % % \draw (212) coordinate[c3,fill=magenta!30!white,label={[label distance=-2]40:{\footnotesize 1'}}]; % % \draw (213) coordinate[c3,fill=cyan!30!white,label={[label distance=-2]90:{\footnotesize 11'}}]; % \draw (40) coordinate[c3,fill=brown,label={[label distance=15]90:\LARGE ${\bf c}$}]; % \draw (40) coordinate[c2,fill=violet]; % \end{tikzpicture} \\ (c) \end{tabular} \end{center} \caption{Greechie orthogonality diagram of (a) the Specker bug logic~\cite[Fig.~1, p.~182]{kochen2}. A proof that, if the system is prepared in state ${\bf a}$, then classical (non-contextual) truth assignments require ${\bf b}$ not to occur, proceeds as follows: In such a truth assignment, as {\it per} axiom {\bf A3}, there is only one true atom per context; all the others have to be false. In a proof by contradiction, suppose that both ${\bf a}$ and ${\bf b}$ are true. Then all atoms connected to them (2,4,7,9) must be false. This in turn requires that the observables (3,8) connecting them must both be true. Alas, those two observables (3,8) are connected by a ``middle'' context {\color{cyan}$\{3,11,8\}$}. But the occurrence of two true observables within the same context is forbidden by axiom {\bf A3}. The only consistent alternative is to disallow ${\bf b}$ to be true if ${\bf a}$ is assumed to be true; or, conversely, to disallow ${\bf a}$ to be true if ${\bf b}$ is assumed to be true. (b)~Greechie orthogonality diagram of a Specker bug logic extended by two contexts which has the true-implies-true property on ${\bf a'}$, given ${\bf a}$ to be true~\cite[$\Gamma_1$, p.~68]{kochen1}. (c)~Greechie orthogonality diagram of a combo of two Specker bug logics~\cite[$\Gamma_3$, p.~70]{kochen1}. If ${\bf a}$ is assumed to be true then the remaining atoms in the context {\color{violet} $\left\{ {\bf a}, {\bf c}, {\bf b}' \right\}$} connecting ${\bf a}$ with ${\bf b'}$, and, in particular, ${\bf c}$, have to be false. Also if ${\bf a}$ is true then ${\bf b}$ is false. Therefore, ${\bf a}'$ needs to be true if ${\bf b}$ and ${\bf c}$ both are false, because they form the context {\color{brown} $\left\{ {\bf b}, {\bf c}, {\bf a}' \right\}$}. This argument is valid even in the absence of a second Specker bug logic. Introduction of a second Specker bug logic ensures the converse: whenever ${\bf a}'$ is true, ${\bf a}$ must be true as well. Therefore ${\bf a}$ and ${\bf a}'$ (and by symmetry also ${\bf b}$ and ${\bf b}'$) cannot be separated by any truth assignment.} \label{2018-whycontexts-fbugcombo} \end{figure} \subsection{Logics inducing partial value (in)definiteness} Probably the strongest forms on value indefiniteness~\cite{pitowsky:218,hru-pit-2003} are theorems~\cite{2012-incomput-proofsCJ,PhysRevA.89.032109,2015-AnalyticKS} stating that relative to reasonable (admissibility, non-contextuality) assumptions, if a quantized system is prepared in some pure state $\vert {\bf a} \rangle$, then {\em any} observable which is not identical or orthogonal to $\vert {\bf a} \rangle$ is undefined. That is, there exist finite systems of quantum contexts whose pasting are demanding that any pure state $\vert {\bf b} \rangle$ not belonging to some context with $\vert {\bf a} \rangle$ can {\em neither be true nor false}; else a complete contradiction would follow from the assumption of classically pre-existent truth values on some pasting of contexts such as the Specker bug logic. What does ``strong'' mean here? Suppose one prepares the system in a particular context ${\cal C}$ such that a single vector $\vert {\bf a} \rangle \in {\cal C}$ is true; that is, $\vert {\bf a} \rangle$ has probability measure $1$ when measured along ${\cal C}$. Then, if one measures a complementary variable $\vert {\bf b} \rangle$, and $\vert {\bf b} \rangle$ is sufficiently separated from $\vert {\bf a} \rangle$ (more precisely, at least an angle $\textrm{arccos} \left( \frac{1}{3} \right)$ apart for the Specker bug logic), then intertwined quantum propositional structures (such as the Specker bug logic) exist which, interpreted (quasi)classically, demand that $\vert {\bf b} \rangle$ can never occur (cannot be true) -- and yet quantum system allow $\vert {\bf b} \rangle$ to occur. Likewise, other intertwined contexts which correspond to true-implies-true configurations of quantum observables (termed Hardy-like~\cite{Hardy-92,Hardy-93,hardy-97} by Cabello~\cite{Cabello-2013-Hardylike}) (quasi)classically imply that some endpoint $\vert {\bf b}' \rangle$ must always occur, given $\vert {\bf a} \rangle$ is true. And yet, quantum mechanically, since $\vert {\bf a} \rangle$ and $\vert {\bf b}' \rangle$ are not collinear, quantum mechanics predicts that occasionally $\vert {\bf b}' \rangle$ does {\em not} occur. In the ``strongest'' form~\cite{2012-incomput-proofsCJ,PhysRevA.89.032109,2015-AnalyticKS} of classical ``do's and don'ts'' there are no possibilities whatsoever for an observable proposition to be either true or false. That is, even if the Specker bug simultaneously allows some $\vert {\bf a} \rangle$ to be true and $\vert {\bf b} \rangle$ to be false (although disallowing the latter to be true), there is another, supposedly more sophisticated finite configuration of intertwined quantum contexts, which can be constructively enumerated and which disallows $\vert {\bf b} \rangle$ even to be false (it cannot be true either). For the sake of an explicit example take the logic~\cite[Fig.~2, p.~102201-8]{2015-AnalyticKS} depicted in Fig.~\ref{2018-whycontexts-acsl1}(c). It is the composite of two logics depicted in Figs.~\ref{2018-whycontexts-acsl1}(a),(b), which perform very differently at ${\bf b}$ given ${\bf a}$ to be true: whereas (a) implements a true-implies-false property, (b) has a true-implies-true property for the atoms ${\bf a}$ and ${\bf b}$, respectively. Both (a) and (b) are proper subsets (lacking 2 contexts) of the logic in Fig.~\ref{2018-whycontexts-acsl1}(c); and, apart from their difference in 4 contexts, are identical. More precisely, as explicated in Appendix~\ref{2018-whycontexts-app-ACS}, both of these logics (a) and (b) allow 13 truth assignments (two-valued states), but only a single one allows ${\bf a}$ to be true on either of them. (This uniqueness is not essential to the argument.) The logic in (c) allows for 8 truth assignments, but all of them assign falsity to ${\bf a}$. By combining the logics (a) and (b) one obtains (c) which, if ${\bf a}$ is assumed to be true, implies that ${\bf b}$ can neither be true -- this would contradict the true-implies-false property of (a) -- nor can it be false -- because this would contradict the true-implies-true property of (b). Hence we are left with the only consistent alternative, (relative to the assumptions): that a system prepared in state ${\bf a}$ must be value indefinite for observable ${\bf b}$. Thereby, as the truth assignment on ${\bf b}$ is not defined, it must be partial on the entire logic depicted in Fig.~\ref{2018-whycontexts-acsl1}(c). \newif\iflabel \labelfalse \begin{figure} \begin{center} \begin{tabular}{ccc} \begin{tikzpicture} [scale=0.3] \tikzstyle{every path}=[line width=1.5pt] \tikzstyle{c1}=[color=green,circle,inner sep=2.5] \tikzstyle{s1}=[color=red,rectangle,inner sep=3.5] \tikzstyle{l1}=[draw=none,circle,minimum size=4] % Define positions of all observables \draw [color=orange] (4,0) coordinate[c1,fill,label=225:{\color{black}\LARGE ${\bf b}$}] (b) -- (13,0) coordinate[c1,fill,label=270:{\iflabel \tiny $P_2$\fi}] (2) -- (22,0) coordinate[s1,fill,label=315:{\iflabel \tiny $P_3$\fi}] (3); \draw [color=blue, ] (3) -- (26,12) coordinate[c1,fill,pos=0.8,label=0:{\iflabel \tiny $P_{21}$\fi}] (21) coordinate[c1,fill,label=0:{\iflabel \tiny $P_{23}$\fi}] (23); \draw [color=white] (23) -- (22,18.5) coordinate[c1,fill,pos=0.4,color=white,label=0:{\iflabel \tiny $P_{29}$\fi}] (29) coordinate[c1,fill,label=45:{\iflabel \tiny $P_5$\fi}] (5); \draw [color=magenta,] (5)-- (13,18.5)coordinate[s1,fill,label=90:{\color{black}\LARGE ${\bf a}$}] (a) -- (4,18.5) coordinate[c1,fill,label=135:{\iflabel \tiny $P_4$\fi}] (4); \draw [color=CadetBlue, ] (4) -- (0,12) coordinate[c1,fill,pos=0.6,label=180:{\iflabel \tiny $P_{10}$\fi}] (10) coordinate[s1,fill,label=180:{\iflabel \tiny $P_7$\fi}] (7); \draw [color=brown, ](7) -- (b) coordinate[c1,fill,pos=0.2,label=180:{\iflabel \tiny $P_6$\fi}] (6); \draw [color=gray] (a) -- (2) coordinate[c1,fill,pos=0.5,label=315:{\iflabel \tiny $P_1$\fi}] (1); \draw [color=violet] (5) -- (22,6) coordinate[s1,fill,pos=0.4,label=0:{\iflabel \tiny $P_{11}$\fi}] (11) coordinate[c1,fill,label=0:{\iflabel \tiny $P_9$\fi}] (9); \draw [color=Apricot] (9) -- (b) coordinate[s1,fill,pos=0.3,label=280:{\iflabel \tiny $P_8$\fi}] (8); \draw [color=TealBlue] (4) -- (4,6) coordinate[s1,fill,pos=0.4,label=180:{\iflabel \tiny $P_{28}$\fi}] (28) coordinate[c1,fill,label=180:{\iflabel \tiny $P_{22}$\fi}] (22); \draw [color=YellowGreen] (22) -- (3) coordinate[c1,fill,pos=0.2,label=260:{\iflabel \tiny $P_{19}$\fi}] (19); \coordinate (25) at ([xshift=-4cm]1); \coordinate (27) at ([xshift=4cm]1); \draw [color=MidnightBlue] (22) -- (25) coordinate[c1,fill,pos=0.5,label=115:{\iflabel \tiny $P_{24}$\fi}] (24) coordinate[s1,fill,label=270:{\iflabel \tiny $P_{25}$\fi}] (25); \draw [color=Mulberry] (25) -- (9) coordinate[c1,fill,pos=0.8,label=90:{\iflabel \tiny $P_{35}$\fi}] (35); \draw [color=BrickRed] (7) -- (27) coordinate[c1,fill,pos=0.5,label=90:{\iflabel \tiny $P_{34}$\fi}] (34) coordinate[c1,fill,label=90:{\iflabel \tiny $P_{27}$\fi}] (27); \draw [color=Emerald] (27) -- (23) coordinate[s1,fill,pos=0.25,label=270:{\iflabel \tiny $P_{26}$\fi}] (26); \draw [color=BlueGreen] (10) -- (15.5,17.5) coordinate[c1,fill,pos=0.5,label=90:{\iflabel \tiny $P_{12}$\fi}] (12) coordinate[s1,fill,label=15:{\iflabel \tiny $P_{13}$\fi}] (13); %\draw [color=Tan] (13) -- (29) coordinate[c1,fill,pos=0.4,label=90:{\iflabel \tiny $P_{31}$\fi}] (31); \draw [color=RawSienna] (28) -- (10.5,15) coordinate[c1,fill,pos=0.5,label=90:{\iflabel \tiny $P_{30}$\fi}] (30) coordinate[c1,fill,label=90:{\iflabel \tiny $P_{15}$\fi}] (15); \draw [color=SpringGreen] (15) -- (11) coordinate[c1,fill,pos=0.6,label=90:{\iflabel \tiny $P_{14}$\fi}] (14); \draw [color=Salmon] (15) -- (1) coordinate[s1,fill,pos=0.2,label=15:{\iflabel \tiny $P_{17}$\fi}] (17); \draw [color=Fuchsia] (1)-- (13) coordinate[c1,fill,pos=0.3,label=0:{\iflabel \tiny $P_{16}$\fi}] (16); \draw [color=CornflowerBlue] (19) -- (16) coordinate[s1,fill,pos=0.3,label=180:{\iflabel \tiny $P_{18}$\fi}] (18); \draw [color=pink] (16) -- (8) coordinate[c1,fill,pos=0.7,label=180:{\iflabel \tiny $P_{32}$\fi}] (32); \draw [color=PineGreen] (6) -- (17) coordinate[c1,fill,pos=0.7,label=90:{\iflabel \tiny $P_{33}$\fi}] (33); \draw [color=DarkOrchid] (17) -- (21) coordinate[c1,fill,pos=0.4,label=90:{\iflabel \tiny $P_{20}$\fi}] (20); \draw [color=black] (25) -- (1) -- (27); %\coordinate (ContextLabel) at ([shift=({-2cm,-3mm})]1); %\draw (ContextLabel) coordinate[l1,label=90:{\iflabel \tiny $C_{26}$\fi}]; \end{tikzpicture} \\ (a) \\ \begin{tikzpicture} [scale=0.3] \tikzstyle{every path}=[line width=1.5pt] \tikzstyle{c1}=[color=green,circle,inner sep=2.5] \tikzstyle{s1}=[color=red,rectangle,inner sep=3.5] \tikzstyle{l1}=[draw=none,circle,minimum size=4] % Define positions of all observables \draw [color=orange] (4,0) coordinate[s1,fill,label=225:{\color{black}\LARGE ${\bf b}$}] (b) -- (13,0) coordinate[c1,fill,label=270:{\iflabel \tiny $P_2$\fi}] (2) -- (22,0) coordinate[c1,fill,label=315:{\iflabel \tiny $P_3$\fi}] (3); \draw [color=blue, ] (3) -- (26,12) coordinate[c1,fill,pos=0.8,label=0:{\iflabel \tiny $P_{21}$\fi}] (21) coordinate[s1,fill,label=0:{\iflabel \tiny $P_{23}$\fi}] (23); \draw [color=green] (23) -- (22,18.5) coordinate[c1,fill,pos=0.4,label=0:{\iflabel \tiny $P_{29}$\fi}] (29) coordinate[c1,fill,label=45:{\iflabel \tiny $P_5$\fi}] (5); \draw [color=magenta,] (5)-- (13,18.5)coordinate[s1,fill,label=90:{\color{black}\LARGE ${\bf a}$}] (a) -- (4,18.5) coordinate[c1,fill,label=135:{\iflabel \tiny $P_4$\fi}] (4); \draw [color=white] (4) -- (0,12) coordinate[c1,color=white,fill,pos=0.6,label=180:{\iflabel \tiny $P_{10}$\fi}] (10) coordinate[c1,fill,label=180:{\iflabel \tiny $P_7$\fi}] (7); \draw [color=brown, ] (7) -- (b) coordinate[c1,fill,pos=0.2,label=180:{\iflabel \tiny $P_6$\fi}] (6); \draw [color=gray] (a) -- (2) coordinate[c1,fill,pos=0.5,label=315:{\iflabel \tiny $P_1$\fi}] (1); \draw [color=violet] (5) -- (22,6) coordinate[s1,fill,pos=0.4,label=0:{\iflabel \tiny $P_{11}$\fi}] (11) coordinate[c1,fill,label=0:{\iflabel \tiny $P_9$\fi}] (9); \draw [color=Apricot] (9) -- (b) coordinate[c1,fill,pos=0.3,label=280:{\iflabel \tiny $P_8$\fi}] (8); \draw [color=TealBlue] (4) -- (4,6) coordinate[s1,fill,pos=0.4,label=180:{\iflabel \tiny $P_{28}$\fi}] (28) coordinate[c1,fill,label=180:{\iflabel \tiny $P_{22}$\fi}] (22); \draw [color=YellowGreen] (22) -- (3) coordinate[s1,fill,pos=0.2,label=260:{\iflabel \tiny $P_{19}$\fi}] (19); \coordinate (25) at ([xshift=-4cm]1); \coordinate (27) at ([xshift=4cm]1); \draw [color=MidnightBlue] (22) -- (25) coordinate[c1,fill,pos=0.5,label=115:{\iflabel \tiny $P_{24}$\fi}] (24) coordinate[s1,fill,label=270:{\iflabel \tiny $P_{25}$\fi}] (25); \draw [color=Mulberry] (25) -- (9) coordinate[c1,fill,pos=0.8,label=90:{\iflabel \tiny $P_{35}$\fi}] (35); %\draw [color=BrickRed] (7) -- (27) coordinate[c1,fill,pos=0.5,label=90:{\iflabel \tiny $P_{34}$\fi}] (34) coordinate[c1,fill,label=90:{\iflabel \tiny $P_{27}$\fi}] (27); \draw [color=BrickRed] (7) -- (27) coordinate[s1,fill,pos=0.5,label=90:{\iflabel \tiny $P_{34}$\fi}] (34) coordinate[c1,fill,label=90:{\iflabel \tiny $P_{27}$\fi}] (27); \draw [color=Emerald] (27) -- (23) coordinate[c1,fill,pos=0.25,label=270:{\iflabel \tiny $P_{26}$\fi}] (26); \draw [color=white] (10) -- (15.5,17.5) coordinate[c1,color=white,fill,pos=0.5,label=90:{\iflabel \tiny $P_{12}$\fi}] (12) coordinate[s1,fill,label=15:{\iflabel \tiny $P_{13}$\fi}] (13); \draw [color=Tan] (13) -- (29) coordinate[c1,fill,pos=0.4,label=90:{\iflabel \tiny $P_{31}$\fi}] (31); \draw [color=RawSienna] (28) -- (10.5,15) coordinate[c1,fill,pos=0.5,label=90:{\iflabel \tiny $P_{30}$\fi}] (30) coordinate[c1,fill,label=90:{\iflabel \tiny $P_{15}$\fi}] (15); \draw [color=SpringGreen] (15) -- (11) coordinate[c1,fill,pos=0.6,label=90:{\iflabel \tiny $P_{14}$\fi}] (14); \draw [color=Salmon] (15) -- (1) coordinate[s1,fill,pos=0.2,label=15:{\iflabel \tiny $P_{17}$\fi}] (17); \draw [color=Fuchsia] (1)-- (13) coordinate[c1,fill,pos=0.3,label=0:{\iflabel \tiny $P_{16}$\fi}] (16); \draw [color=CornflowerBlue] (19) -- (16) coordinate[c1,fill,pos=0.3,label=180:{\iflabel \tiny $P_{18}$\fi}] (18); \draw [color=pink] (16) -- (8) coordinate[s1,fill,pos=0.7,label=180:{\iflabel \tiny $P_{32}$\fi}] (32); \draw [color=PineGreen] (6) -- (17) coordinate[c1,fill,pos=0.7,label=90:{\iflabel \tiny $P_{33}$\fi}] (33); \draw [color=DarkOrchid] (17) -- (21) coordinate[c1,fill,pos=0.4,label=90:{\iflabel \tiny $P_{20}$\fi}] (20); \draw [color=black] (25) -- (1) -- (27); \coordinate (ContextLabel) at ([shift=({-2cm,-3mm})]1); \draw (ContextLabel) coordinate[l1,label=90:{\iflabel \tiny $C_{26}$\fi}]; \end{tikzpicture} \\ (b) \\ \begin{tikzpicture} [scale=0.3] \labeltrue \tikzstyle{every path}=[line width=1.5pt] %\tikzstyle{c1}=[circle,fill,inner sep=4] %\tikzstyle{c2}=[circle,fill,inner sep=2.7] %\tikzstyle{c3}=[circle,fill,inner sep=1] \tikzstyle{c1}=[circle,fill,inner sep=3] \tikzstyle{c2}=[circle,fill,inner sep=2] \tikzstyle{c3}=[circle,fill,inner sep=1] \tikzstyle{s1}=[color=red,rectangle,minimum size=8,inner sep=6] \tikzstyle{d1}=[draw=none,circle,minimum size=4] \tikzstyle{e1}=[color=gray,rectangle,minimum size=8,inner sep=6] % Define positions of all observables \draw [color=orange] (4,0) coordinate[c1,fill,label=225:{\color{black}\LARGE ${\bf b}$}] (b) -- (13,0) coordinate[c1,fill,label={[label distance=-1]270:{\iflabel \footnotesize \color{black} $2$\fi}}] (2) -- (22,0) coordinate[c1,fill,label={[label distance=-1]315:{\iflabel \footnotesize \color{black} $3$\fi}}] (3); \draw [color=blue] (3) -- (26,12) coordinate[c1,fill,pos=0.8,label={[label distance=-1]0:{\iflabel \footnotesize \color{black} ${21}$\fi}}] (21) coordinate[c1,fill,label={[label distance=-3]0:{\iflabel \footnotesize \color{black} ${23}$\fi}}] (23); \draw [color=green] (23) -- (22,18.5) coordinate[c1,fill,pos=0.4,label={[label distance=-1]0:{\iflabel \footnotesize \color{black} ${29}$\fi}}] (29) coordinate[c1,fill,label={[label distance=-1]45:{\iflabel \footnotesize \color{black} $5$\fi}}] (5); \draw [color=magenta] (5)-- (13,18.5)coordinate[c1,fill,label=90:{\color{black}\LARGE ${\bf a}$}] (a) -- (4,18.5) coordinate[c1,fill,label={[label distance=-1]135:{\iflabel \footnotesize \color{black} $4$\fi}}] (4); \draw [color=CadetBlue] (4) -- (0,12) coordinate[c1,fill,pos=0.6,label={[label distance=-1]180:{\iflabel \footnotesize \color{black} ${10}$\fi}}] (10) coordinate[c1,fill,label={[label distance=-1]180:{\iflabel \footnotesize \color{black} $7$\fi}}] (7); \draw [color=brown] (7) -- (b) coordinate[c1,fill,pos=0.2,label={[label distance=-1]180:{\iflabel \footnotesize \color{black} $6$\fi}}] (6); \draw [color=gray] (a) -- (2) coordinate[c1,fill,pos=0.52,label={[label distance=-1, yshift=2]357.5:{\iflabel \footnotesize \color{black} $1$\fi}}] (1); \draw [color=violet] (5) -- (22,6) coordinate[c1,fill,pos=0.4,label={[label distance=-1]0:{\iflabel \footnotesize \color{black} ${11}$\fi}}] (11) coordinate[c1,fill,label={[label distance=-1]0:{\iflabel \footnotesize \color{black} $9$\fi}}] (9); \draw [color=Apricot] (9) -- (b) coordinate[c1,fill,pos=0.3,label={[label distance=-1]280:{\iflabel \footnotesize \color{black} $8$\fi}}] (8); \draw [color=TealBlue] (4) -- (4,6) coordinate[c1,fill,pos=0.4,label={[label distance=-1]180:{\iflabel \footnotesize \color{black} ${28}$\fi}}] (28) coordinate[c1,fill,label={[label distance=-3]180:{\iflabel \footnotesize \color{black} ${22}$\fi}}] (22); \draw [color=YellowGreen] (22) -- (3) coordinate[c1,fill,pos=0.2,label={[label distance=-1]260:{\iflabel \footnotesize \color{black} ${19}$\fi}}] (19); \coordinate (25) at ([xshift=-4cm]1); \coordinate (27) at ([xshift=4cm]1); \draw [color=MidnightBlue] (22) -- (25) coordinate[c1,fill,pos=0.5,label={[label distance=-1]180:{\iflabel \footnotesize \color{black} ${24}$\fi}}] (24) coordinate[c1,fill,label={[label distance=-1]180:{\iflabel \footnotesize \color{black} ${25}$\fi}}] (25); \draw [color=Mulberry] (25) -- (9) coordinate[c1,fill,pos=0.8,label={[label distance=-1]90:{\iflabel \footnotesize \color{black} ${35}$\fi}}] (35); \draw [color=BrickRed] (7) -- (27) coordinate[c1,fill,pos=0.5,label={[label distance=-3]90:{\iflabel \footnotesize \color{black} ${34}$\fi}}] (34) coordinate[c1,fill,label={[label distance=-1]90:{\iflabel \footnotesize \color{black} ${27}$\fi}}] (27); \draw [color=Emerald] (27) -- (23) coordinate[c1,fill,pos=0.25,label={[label distance=-1]320:{\iflabel \footnotesize \color{black} ${26}$\fi}}] (26); \draw [color=BlueGreen] (10) -- (15.5,17.5) coordinate[c1,fill,pos=0.5,label={[label distance=-1]90:{\iflabel \footnotesize \color{black} ${12}$\fi}}] (12) coordinate[c1,fill,label={[label distance=-1,xshift=5]270:{\iflabel \footnotesize \color{black} ${13}$\fi}}] (13); \draw [color=Tan] (13) -- (29) coordinate[c1,fill,pos=0.4,label={[label distance=-1]90:{\iflabel \footnotesize \color{black} ${31}$\fi}}] (31); \draw [color=RawSienna] (28) -- (10.5,15) coordinate[c1,fill,pos=0.5,label={[label distance=-3, yshift=-3]160:{\iflabel \footnotesize \color{black} ${30}$\fi}}] (30) coordinate[c1,fill,label={[label distance=-5]45:{\iflabel \footnotesize \color{black} ${15}$\fi}}] (15); \draw [color=SpringGreen] (15) -- (11) coordinate[c1,fill,pos=0.6,label={[label distance=-1]90:{\iflabel \footnotesize \color{black} ${14}$\fi}}] (14); \draw [color=Salmon] (15) -- (1) coordinate[c1,fill,pos=0.2,label={[label distance=-1, yshift=2]180:{\iflabel \footnotesize \color{black} ${17}$\fi}}] (17); \draw [color=Fuchsia] (1)-- (13) coordinate[c1,fill,pos=0.3,label={[label distance=-1]0:{\iflabel \footnotesize \color{black} ${16}$\fi}}] (16); \draw [color=CornflowerBlue] (19) -- (16) coordinate[c1,fill,pos=0.3,label={[label distance=-1]180:{\iflabel \footnotesize \color{black} ${18}$\fi}}] (18); \draw [color=pink] (16) -- (8) coordinate[c1,fill,pos=0.7,label={[label distance=-1]180:{\iflabel \footnotesize \color{black} ${32}$\fi}}] (32); \draw [color=PineGreen] (6) -- (17) coordinate[c1,fill,pos=0.7,label={[label distance=-1, yshift=2]180:{\iflabel \footnotesize \color{black} ${33}$\fi}}] (33); \draw [color=DarkOrchid] (17) -- (21) coordinate[c1,fill,pos=0.4,label={[label distance=-3]20:{\iflabel \footnotesize \color{black} ${20}$\fi}}] (20); \draw [color=black] (25) -- (1) -- (27); \draw (a) coordinate[c2,fill=gray]; \draw (b) coordinate[c2,fill=brown]; \draw (b) coordinate[c3,fill=Apricot]; \draw (4) coordinate[c2,fill=CadetBlue]; \draw (4) coordinate[c3,fill=TealBlue]; \draw (5) coordinate[c2,fill=magenta]; \draw (5) coordinate[c3,fill=violet]; \draw (7) coordinate[c2,BrickRed]; \draw (7) coordinate[c3,fill=brown]; \draw (23) coordinate[c2,fill=green]; \draw (23) coordinate[c3,fill=Emerald]; \draw (29) coordinate[c2,fill=Tan]; \draw (21) coordinate[c2,fill=DarkOrchid]; \draw (3) coordinate[c2,fill=blue]; \draw (3) coordinate[c3,fill=YellowGreen]; \draw (11) coordinate[c2,fill=SpringGreen]; \draw (9) coordinate[c2,fill=Apricot]; \draw (9) coordinate[c3,fill=Mulberry]; \draw (27) coordinate[c2,fill=Emerald]; \draw (27) coordinate[c3,fill=black]; \draw (13) coordinate[c2,fill=Tan]; \draw (13) coordinate[c3,fill=Fuchsia]; \draw (7) coordinate[c2,BrickRed]; \draw (7) coordinate[c3,fill=brown]; \draw (8) coordinate[c2,fill=pink]; \draw (19) coordinate[c2,fill=CornflowerBlue]; \draw (16) coordinate[c2,fill=CornflowerBlue]; \draw (16) coordinate[c3,fill=pink]; \draw (1) coordinate[c2,fill=Salmon]; \draw (1) coordinate[c3,fill=Fuchsia]; \draw (17) coordinate[c2,fill=PineGreen]; \draw (17) coordinate[c3,fill=DarkOrchid]; \draw (15) coordinate[c2,fill=SpringGreen]; \draw (15) coordinate[c3,fill=Salmon]; \draw (25) coordinate[c2,fill=Mulberry]; \draw (25) coordinate[c3,fill=black]; \draw (22) coordinate[c2,fill=YellowGreen]; \draw (22) coordinate[c3,fill=MidnightBlue]; \draw (28) coordinate[c2,fill=RawSienna]; \draw (10) coordinate[c2,fill=BlueGreen]; \draw (6) coordinate[c2,fill=PineGreen]; \draw (2) coordinate[c2,fill=gray]; \end{tikzpicture} \\ (c) \end{tabular} \end{center} \caption{Greechie orthogonality diagram of a logic~\cite[Fig.~2, p.~102201-8]{2015-AnalyticKS} realizable in $\mathbb{R}^3$ (a) with the true--implies--false property; (b) with the true--implies--true property; (c) with the true--implies--value indefiniteness (neither true nor false) property on the atoms ${\bf a}$ and ${\bf b}$, respectively. (a) and (b) contain the single (out of 13) value assignment which is possible, and for which ${\bf a}$ is true. All 8 value assignments of the logic depicted in (c) require ${\bf a}$ to be false. } \label{2018-whycontexts-acsl1} \end{figure} The scheme of the proof is as follows: \begin{itemize} \item[(i)] Find a logic (collection of intertwined contexts of observables) exhibiting a true-implies-false property on the two atoms ${\bf a}$ and ${\bf b}$. \item[(ii)] Find another logic exhibiting a true-implies-true property on the same two atoms ${\bf a}$ and ${\bf b}$. \item[(iii)] Then join (paste) these logics into a larger logic, which, given ${\bf a}$, neither allows ${\bf b}$ to be true nor false. Consequently ${\bf b}$ must be value indefinite. \end{itemize} The most suggestive candidate for such a pasting is, however, unavailable: it is the combination of a Specker bug logic and another, extended Specker bug logic, as depicted in Fig.~\ref{2018-whycontexts-impossible}. Such logic cannot be realized in three dimensions, as the angles cannot be chosen consistently; that is, obeying the Cabello bounds on the relative angles, respectively. \begin{figure} \begin{center} \begin{tikzpicture} [xscale=0.25, yscale=0.25, rotate=0] \newdimen\ms \ms=0.1cm \tikzstyle{every path}=[line width=2pt] \tikzstyle{c5}=[circle,inner sep={\ms/8},minimum size=5*\ms] \tikzstyle{c4}=[circle,inner sep={\ms/8},minimum size=4*\ms] \tikzstyle{c3}=[circle,inner sep={\ms/8},minimum size=3*\ms] \tikzstyle{c2}=[circle,inner sep={\ms/8},minimum size=1.8*\ms] \tikzstyle{c1}=[circle,inner sep={\ms/8},minimum size=1.1*\ms] % Radius of regular polygons \newdimen\R \R=6cm % outer circle %\r= { \R * sqrt(3) } % inner circle %\newdimen\r %\r= {\R * sqrt(3)/2} % inner circle \newdimen\K \K=12cm \newdimen\KK \KK=11cm \newdimen\KKK \KKK=11.5cm \newdimen\RR \RR=-1cm \newdimen\RRR \RRR=-0.5cm % Define positions of all observables \path ({90 + 0 * 360 /6}:\R ) coordinate(1) ({90 + 30 + 0 * 360 /6}:{\R * sqrt(3)/2} ) + (0,\RRR) coordinate(2) ({90 + 1 * 360 /6}:\R ) + (0,\RR) coordinate(3) ({90 + 30 + 1 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(4) ({90 + 2 * 360 /6}:\R ) coordinate(5) ({90 + 30 + 2 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(6) ({90 + 3 * 360 /6}:\R ) coordinate(7) ({90 + 30 + 3 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(8) ({90 + 4 * 360 /6}:\R ) coordinate(9) ({90 + 30 + 4 * 360 /6}:{\R * sqrt(3)/2} ) coordinate(10) ({90 + 5 * 360 /6}:\R ) + (0,\RR) coordinate(11) ({90 + 30 + 5 * 360 /6}:{\R * sqrt(3)/2} ) + (0,\RRR) coordinate(12) (0,0) coordinate(13) ({90 + 0 * 360 /6}:\R ) + (0,\K) coordinate(21) ({90 + 30 + 0 * 360 /6}:{\R * sqrt(3)/2} ) + (0,\K) coordinate(22) ({90 + 1 * 360 /6}:\R ) + (0,\K) coordinate(23) ({90 + 30 + 1 * 360 /6}:{\R * sqrt(3)/2} ) + (0,\K) coordinate(24) ({90 + 2 * 360 /6}:\R ) + (0,\KK) coordinate(25) ({90 + 30 + 2 * 360 /6}:{\R * sqrt(3)/2} ) + (0,\KKK) coordinate(26) ({90 + 3 * 360 /6}:\R ) + (0,\K) coordinate(27) ({90 + 30 + 3 * 360 /6}:{\R * sqrt(3)/2} ) + (0,\KKK) coordinate(28) ({90 + 4 * 360 /6}:\R ) + (0,\KK) coordinate(29) ({90 + 30 + 4 * 360 /6}:{\R * sqrt(3)/2} ) + (0,\K) coordinate(210) ({90 + 5 * 360 /6}:\R ) + (0,\K) coordinate(211) ({90 + 30 + 5 * 360 /6}:{\R * sqrt(3)/2} ) + (0,\K) coordinate(212) (0,0) + (0,\K) coordinate(213) (0,0) + (14,{\K/2}) coordinate(40) (0,0) + (7,{\K/2}) coordinate(41) ; % draw contexts \draw [color=orange] (1) -- (2) -- (3); \draw [color=red] (3) -- (4) -- (5); \draw [color=green] (5) -- (6) -- (7); \draw [color=blue] (7) -- (8) -- (9); \draw [color=gray] (9) -- (10) -- (11); % \draw [color=magenta] (11) -- (12) -- (1); % \draw [color=cyan] (4) -- (10); \draw [color=orange!30!white] (21) -- (22) -- (23); \draw [color=red!30!white] (23) -- (24) -- (25); \draw [color=green!30!white] (25) -- (26) -- (27); \draw [color=blue!30!white] (27) -- (28) -- (29); \draw [color=gray!30!white] (29) -- (210) -- (211); % \draw [color=magenta!30!white] (211) -- (212) -- (21); % \draw [color=cyan!30!white] (24) -- (210); \draw [color=violet] (7) to [out=0,in=270] (40) to [out=90,in=0] (21); \draw [color=brown] (27) -- (40); % %% %% draw atoms %% % \draw (1) coordinate[c5,fill=magenta,label=90:\LARGE ${\bf a}$]; % \draw (1) coordinate[c4,fill=orange]; % \draw (1) coordinate[c1,fill=brown]; % % \draw (2) coordinate[c3,fill=orange]; % % \draw (3) coordinate[c3,fill=red]; % \draw (3) coordinate[c2,fill=orange]; % % \draw (4) coordinate[c3,fill=red]; % \draw (4) coordinate[c2,fill=cyan]; % % \draw (5) coordinate[c3,fill=green]; % \draw (5) coordinate[c2,fill=red]; % % \draw (6) coordinate[c3,fill=green]; % \draw (7) coordinate[c3,fill=blue,label=270:\LARGE ${\bf b}$]; % \draw (7) coordinate[c2,fill=green]; % \draw (7) coordinate[c1,fill=violet]; % % \draw (8) coordinate[c3,fill=blue]; % % \draw (9) coordinate[c3,fill=gray]; \draw (9) coordinate[c2,fill=blue]; % % \draw (10) coordinate[c3,fill=gray]; % \draw (10) coordinate[c2,fill=cyan]; % % \draw (11) coordinate[c3,fill=magenta]; % \draw (11) coordinate[c2,fill=gray]; % % \draw (12) coordinate[c3,fill=magenta]; % % \draw (13) coordinate[c3,fill=cyan]; % \draw (21) coordinate[c3,fill=magenta!30!white,label=90:\LARGE ${\bf c}$]; % {\bf a}' \draw (21) coordinate[c2,fill=orange!30!white]; % \draw (21) coordinate[c1,fill=violet]; % % \draw (22) coordinate[c3,fill=orange!30!white]; % % \draw (23) coordinate[c3,fill=red!30!white]; % \draw (23) coordinate[c2,fill=orange!30!white]; % % \draw (24) coordinate[c3,fill=red!30!white]; % \draw (24) coordinate[c2,fill=cyan!30!white]; % % \draw (25) coordinate[c3,fill=green!30!white]; % \draw (25) coordinate[c2,fill=red!30!white]; % % \draw (26) coordinate[c3,fill=green!30!white]; % \draw (27) coordinate[c3,fill=blue!30!white,label=270:{}]; % \draw (27) coordinate[c2,fill=green!30!white]; % \draw (27) coordinate[c1,fill=brown]; % % \draw (28) coordinate[c3,fill=blue!30!white]; % % \draw (29) coordinate[c3,fill=gray!30!white]; \draw (29) coordinate[c2,fill=blue!30!white]; % % \draw (210) coordinate[c3,fill=gray!30!white]; % \draw (210) coordinate[c2,fill=cyan!30!white]; % % \draw (211) coordinate[c3,fill=magenta!30!white]; % \draw (211) coordinate[c2,fill=gray!30!white]; % % \draw (212) coordinate[c3,fill=magenta!30!white]; % % \draw (213) coordinate[c3,fill=cyan!30!white]; % \draw (40) coordinate[c3,fill=brown,label=0:\LARGE $$]; % {\bf c} \draw (40) coordinate[c2,fill=violet]; % \draw (41) coordinate[c3,fill=brown,label=290:\LARGE $$]; % {\bf b}' \end{tikzpicture} \end{center} \caption{Greechie orthogonality diagram of a logic which is value indefinite on ${\bf b}$ (as well as on ${\bf c}$ for symmetry reasons), given ${\bf a}$ is true; alas such a logic has no realization in three dimensional Hilbert space, as the angles $\angle ({\bf a},{\bf b})$ between ${\bf a}$ and ${\bf b}$ should simultaneously obey $ 1.2 \approx \textrm{arcsec} (3) \le \angle ({\bf a},{\bf b}) \le \textrm{arccsc} (3)\approx 0.3$.} \label{2018-whycontexts-impossible} \end{figure} The latter result about the partiality of the truth assignment has already been discussed by Pitowsky~\cite{pitowsky:218}, and later by Hrushovski and Pitowsky~\cite{hru-pit-2003}. It should also be mentioned that the logic (c) has been realized with a particular configuration in three-dimensional real Hilbert space~\cite[Tables~I,II, p.~102201-7]{2015-AnalyticKS} which are an angle $\angle ({\bf a} , {\bf b})=\textrm{arccos}\left(\frac{1}{\sqrt{2}}\right)$ apart, but, as has been mentioned earlier, this kind of value indefiniteness on any particular state ${\bf b}$, given that the system has been prepared in state ${\bf a}$, can be constructively obtained by an extension of the above configuration whenever ${\bf a}$ and ${\bf b}$ are neither collinear (in this case ${\bf b}$ would be true) nor orthogonal (in this case ${\bf b}$ would be false). So basically all states not identical (or orthogonal) to the state prepared must be value indefinite. All three logics in Fig.~\ref{2018-whycontexts-impossible}(a)-(c) have another non-classical feature: they are non-unital~\cite{tkadlec-96}, meaning that the truth assignments on some of their atoms can only acquire the value false, regardless of the preparation. That is, in this ``state-independend'' form, whenever a proposition corresponding to such an atom is measured to be true, this can be interpreted as indication of non-classicality. (Note that one can always rotate the entire set of rays so that this particular atom coincides with some observable measured.) \section{Propositional logic does not uniquely determine probabilities} \label{2018-whycontexts-sec5} By now it should be clear that the propositional structure does in general {\em not} uniquely determine its probabilities. The Specker bug in Fig.~\ref{2018-whycontexts-fbugcombo}(a) serves as a good example for that: it supports (quasi)classical probabilities, explicitly enumerated in Refs.~\cite[p.~286, Fig.~11.5(iii)]{svozil-2016-s} and~\cite[p.~91, Fig.~12.10]{svozil-2016-pu-book} which are formed by convex combinations of all two-valued states on them. Other propositional structures such as the pentagon logic support ``exotic'' probability measures~\cite{wright:pent} which do not vanish at their interlink observables and are equally weighted with value $\frac{1}{2}$ there. This measure is neither realized in the (quasi)classical partition logic setup explicitly discussed in Refs.~\cite[p.~289, Fig.~11.8]{svozil-2016-s} and~\cite[p.~88, Fig.~12.8]{svozil-2016-pu-book}, nor in quantum mechanics. It remains to be seen if a more general theory of probability measures based on the axioms {\bf A1}-{\bf A3} can be found. \section{Some Platonist afterthoughts} \label{2018-whycontexts-summary} The author's not-so-humble reading of all these aforementioned ``mind-boggling'' non-classical quantum predictions is a rather sober one: in view of the numerous indications that classical value definiteness cannot be extended to more than a single context, the most plausible supposition is that, besides exotic possibilities~\cite{pitowsky-83,meyer:99}, ontologically there is only one such {\it ``Realding''} -- indeed a rather obvious candidate suggesting itself as ontology: a single vector, or rather a single context. Quantized systems can be completely and exhaustively characterized by a unique context, and a ``true'' proposition within this context. Suppose for a moment that this hypothesis is correct, and that there is no ontology, no {\it ``Realding,''} beyond a single context. There is one preferred view -- namely the context identical to the context in which the system has been prepared -- and all but one epistemic views. And yet, a confusing experience is the apparent ease by which an experimenter appears to measure, without any difficulty, a context or (maximal) observable not (or only partly through intertwines) matching the preparation context. In such a situation one may assume that the measurement grants an ``imperfect'' view on the preparation context. In this process, information -- in particular, the relative locatedness of the measurement context with respect to the preparation context -- is {\em augmented} by properties of the measurement device, thereby effectively generating entanglement~\cite{schrodinger,london-Bauer-1983} {\it via} context translation~\cite{svozil-2003-garda}. Frames of reference which do not coincide with the {\it ``Realding''} or preparation context necessarily include stochastic elements which are not caused or determined by any property of the formerly individual {\it ``Realding.''} One may conclude~\cite{Howard2004-HOWWIT} with Bohr's 1972 Como lecture~\cite[p.~580]{bohr:28} that {\em ``any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected. Accordingly, an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation.''} That is, any interaction between the previously separated individual object and the measurement device results in a joint physical state which is no longer determined by the states of the (previously) individual constituents~\cite{v-neumann-49,london-Bauer-1983}. Instead, the joint state exhibits what Schr\"odinger later called entanglement~\cite{schrodinger}. Entanglement characterized by a value definite {\em relational}~\cite{zeil-99} or {\em collective} (re-)encoding of information with respect to the constituent parts, thereby (since the unitary quantum evolution is injective) resulting in a value indefiniteness of the previously individual and separate parts. As a result, knowledge about observations obtained by different contexts than the preparation context are necessarily (at least partially in the sense of the augmented information from the measurement device) epistemic. Another possible source of perplexity might be the various {\em types} of algabaic or logical structures involved. Classically, empirical logics are Boolean algebras. Then, in a first step towards non-classicality, there are partition logics which are not Boolean any longer (they feature complementarity through non-distributivity) but nevertheless still allow for a certain type of (quasi)classicality; that is, a separating and unital set of two-valued states. And then, further on this road, there are (finite) quantum logics which do not allow any definite state at all. One might be puzzled by the fact that there exist ``intermediate'' logics, such as the Specker bug or the pentagon (pentagram) logic discussed in Sects.~\ref{2018-whycontexts-Specker-bug} and~\ref{2018-whycontexts-pentagons}, which still allow (even classical) simultaneous value indefiniteness, although they contain observables which are mutually complementary (non-collinear and non-orthogonal). However, this apparent paradox should rather be interpreted epistemically, as means (configuration) relative~\cite{Myrvold2011237}: in the case of the pentagon we have decided to concentrate on 10 observables in a cyclic pasting of 5 contexts, but we have thereby implicitly chosen to ``look the other way'' and disregard the abundance of other observables which impose much more stringent conditions on the value definiteness of the observables in the pentagon logic than the pentagon logic itself. So, properties such as the true-implies-false, the true-implies-true properties, as well as inseparability and even value indefiniteness, are means relative and valid only if one restricts or broadens one's attention to sometimes very specific, limited sublogics of the realm of all conceivable quantum logics, which are structures formed by perpendicular projection operators in Hilbert spaces of dimension larger than two. Pointedly stated, sets of intertwining contexts connecting two (or more) relevant complementary observables ${\bf a}$ and ${\bf b}$ should be considered as totally arbitrary when it comes to the inclusion or exclusion of particular contexts interconnecting them: there is neither a necessity nor even a compelling reason to take into account one such structue and disregard another, or favour one over the other. Indeed, in an extreme, {\it sui generis}, form of the argument, suppose a single quantum is prepared in some state ${\bf a}$. Then {\em every single} outcome of a measurement of {\em every} complementary (non-collinear and non-orthogonal relative to the state prepared) quantum observable may be considered as ``proof'' or ``certification of non-classicality'' (or, in another terminology, ``contextuality''). Because those observable can be identified with the ``endpoint'' ${\bf b}$ of either some true-implies-false, or alternatively true-implies-true configuration [say the one sketched in Figs.~\ref{2018-whycontexts-impossible}(a),(b)], depending on whether the classical false or true predictions need to contradict the particular outcome, respectively. For quantum logics with a unital set of two-valued states -- such as the logics depicted by Tkadlec~\cite[p.~207, Fig.~2]{tkadlec-96} or the ones in Figs.~\ref{2018-whycontexts-impossible}(a),(b) -- one could even get rid of the state preparation if ${\bf b}$ occurs and is identified with an observable which, according to the classical predictions associated with that logic, cannot occur. There is no principle which could prevent us from arguing that way if we insist on the simultaneous existence of multiple contexts encountered in quantum mechanics. Indeed, are intertwining contexts not scholastic~\cite{specker-60} sophisms in desperate need of deconstruction? An interesting historical question arises: Kochen and Specker, in a succession of papers on partial algebras~\cite{kochen2,kochen3,kochen1} have insisted that logical operations should only be defined within contexts and must not be applied to propositions outside of it. And yet they have considered extended counterfactual structures of pasted context, ending up in a holistic argument involving complementary observables. Of course, an immediate reply might be that without intertwined contexts there cannot be any non-trivial (non-classical, non-Boolean) configuration of observables which is of any interest. For the reasons mentioned earlier, the emphasis should not be on ``completing'' quantum mechanics by some sort of hidden parameter theory -- such as, for instance, Valentini~\cite{Valentini-2018} envisioning a theory which is to quantum mechanics as statistical physics is to thermodynamics -- but just the opposite: the challenge is to acknowledge the scarcity of resources -- the {\it ``Realding''} or physical state as mere vector -- despite the continuum of possible views on it, resulting in an illusory over-abundance and over-determination. In this line of thought the question of what might be the reason behind the futility to co-define non-commuting quantum observables (from two or more different contexts) simultaneously should be answered in terms of a serious lack of a proper perspective of what one is dealing with: Metaphorically speaking, it is almost as if one pretends to take a $360^\circ$ panorama of what lies in the outside world while actually merely takes photos from some sort of echo chamber, or house of mirrors -- partly reflecting what is in it, and partly reproducing the observer (photographer) in almost endless reflections. Stitching together photos from these reflections yields a panorama of one and the same object in seemingly endless varieties. In this way one might end up with a horribly distorted image of this situation; and with the inside turned outside. This is not dissimilar to what Plato outlined in the {\em Republic's} {\em cave metaphor}~\cite[Book~7, 515c, p.~221]{plato-republic}: {\em ``what people in this situation would take for truth would be nothing more than the shadows of the manufactured objects.''} In the quantum transcription of this metaphor, the vectors are the objects, and the shadows taken for truth are the views on these objects, mediated or translated~\cite{svozil-2003-garda} by arbitrary mismatching contexts. \begin{acknowledgments} Federico Holik has constantly inspired me to (re-)think probabilities and invited me to Argentina. I kindly thank Ad\'an Cabello and Jos\'{e} R. Portillo for numerous explanations and discussions during an ongoing collaboration. I am also deeply indebted to Alastair A. Abbott and Cristian S. Calude for their contributions and collaboration on the localization of value indefiniteness. Josef Tkadlec has kindly provided a {\it Pascal} program to compute two-valued states on logics, given the contexts (blocks). Christoph Lemell and the Nonlinear Dynamics group at the Vienna University of Technology has provided the computational framework for the Hull calculations performed. \end{acknowledgments} %\clearpage \appendix \section{Two-valued states, (quasi)classical probabilities on the triangular logic in 4 dimensions} \label{2018-whycontexts-app-tri} The two-valued states ({\it aka} truth tables) have been enumerated by Josef Tkadlec's Pascal program {\tt 2states}~\cite{tkadlec-2states-2017}. Implicitly, the convex sums over the respective probabilities encode the truth tables, as, on any particular atom, the $i$'th truth table entry is 1 if $\lambda_i$ appears in the listing of the classical probability $p_i$. Otherwise, the $i$'th truth table entry is zero. The bounds for classical probabilities have been obtained by Komei Fukuda's {\tt cddlib} package~\cite{cdd-pck}. %\subsection{Classical truth assignments} %\begin{table} %\begin{tabular}{c|cccccccccccccccccccccccccc} %\hline\hline % &1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ %\hline % 1 &1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ % 2 &1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ % 3 &0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ % 4 &0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ % 5 &0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ % 6 &0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ % 7 &0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ % 8 &0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ % 9 &0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ %10 &0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ %11 &0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ %12 &0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ %13 &0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ %14 &0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ %\hline\hline %\end{tabular} %\caption{Truth table (two-valued measures) on the triangle logic in four dimensions.} %\end{table} % %\subsection{Classical truth assignments and probabilities} There are 9 propositions forming three contexts $\{1, 2, 3, 4\}$, $\{4, 5, 6, 7\}$, and $\{7, 8, 9, 1\}$ allowing 14 (separating, unital) two-valued states whose convex sum yield the following (quasi)classical probabilities: \begin{equation} \begin{aligned} p_{ 1} =&\lambda_1 + \lambda_2 ,\\ p_{ 2} =&\lambda_3 + \lambda_4 + \lambda_5 + \lambda_6 + \lambda_7 ,\\ p_{ 3} =&\lambda_8 + \lambda_9 + \lambda_{10} + \lambda_{11}+ \lambda_{12} ,\\ p_{ 4} =&\lambda_{13} + \lambda_{14} ,\\ p_{ 5} =&\lambda_1 + \lambda_3 + \lambda_4 +\lambda_8 + \lambda_9 ,\\ p_{ 6} =&\lambda_2 + \lambda_5 + \lambda_6 + \lambda_{10} + \lambda_{11} ,\\ p_{ 7} =&\lambda_7 + \lambda_{12} ,\\ p_{ 8} =&\lambda_3 + \lambda_5 + \lambda_8 + \lambda_{10} + \lambda_{13} ,\\ p_{ 9} =&\lambda_4 + \lambda_6 + \lambda_{9} + \lambda_{11} + \lambda_{14} . \end{aligned} \end{equation} %\subsection{Boole-Bell type inequalities} % %\begin{equation} %\begin{split} % 0 \le p_{1}, p_{2}, p_{3}, p_{5}, p_{6}, p_{8} ,\\ % 0 \le -p_{1} +p_{5} +p_{6} ,\\ % -1 \le -2p_{1} -p_{2} -p_{3} +p_{5} +p_{6} -p_{8} ,\\ % 0 \le p_{1} +p_{2} +p_{3} -p_{5} -p_{6} ,\\ % -1 \le -p_{1} -p_{2} -p_{3} ,\\ % 1 \le p_{1} +p_{2} +p_{3} +p_{4} ,\\ % 0 \le -p_{1} -p_{2} -p_{3} +p_{5} +p_{6} +p_{7} ,\\ % 1 \le 2p_{1} +p_{2} +p_{3} -p_{5} -p_{6} +p_{8} +p_{9} %\end{split} %\end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Truth assignments, (quasi)classical probabilities on the square logic in 4 dimensions} \label{2018-whycontexts-app-square} %\subsection{Classical truth assignments} %\begin{table} %\begin{tabular}{c|cccccccccccccccccccccccccc} %\hline\hline % &1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ %\hline % 1 & 1 &0 &0 &0 &1 &0 &0 &1 &0 &0 &0 &0 \\ % 2 & 1 &0 &0 &0 &1 &0 &0 &0 &1 &0 &0 &0 \\ % 3 & 1 &0 &0 &0 &0 &1 &0 &1 &0 &0 &0 &0 \\ % 4 & 1 &0 &0 &0 &0 &1 &0 &0 &1 &0 &0 &0 \\ % 5 & 1 &0 &0 &0 &0 &0 &1 &0 &0 &0 &0 &0 \\ % 6 & 0 &1 &0 &0 &1 &0 &0 &1 &0 &0 &1 &0 \\ % 7 & 0 &1 &0 &0 &1 &0 &0 &1 &0 &0 &0 &1 \\ % 8 & 0 &1 &0 &0 &1 &0 &0 &0 &1 &0 &1 &0 \\ % 9 & 0 &1 &0 &0 &1 &0 &0 &0 &1 &0 &0 &1 \\ %10 & 0 &1 &0 &0 &1 &0 &0 &0 &0 &1 &0 &0 \\ %11 & 0 &1 &0 &0 &0 &1 &0 &1 &0 &0 &1 &0 \\ %12 & 0 &1 &0 &0 &0 &1 &0 &1 &0 &0 &0 &1 \\ %13 & 0 &1 &0 &0 &0 &1 &0 &0 &1 &0 &1 &0 \\ %14 & 0 &1 &0 &0 &0 &1 &0 &0 &1 &0 &0 &1 \\ %15 & 0 &1 &0 &0 &0 &1 &0 &0 &0 &1 &0 &0 \\ %16 & 0 &1 &0 &0 &0 &0 &1 &0 &0 &0 &1 &0 \\ %17 & 0 &1 &0 &0 &0 &0 &1 &0 &0 &0 &0 &1 \\ %18 & 0 &0 &1 &0 &1 &0 &0 &1 &0 &0 &1 &0 \\ %19 & 0 &0 &1 &0 &1 &0 &0 &1 &0 &0 &0 &1 \\ %20 & 0 &0 &1 &0 &1 &0 &0 &0 &1 &0 &1 &0 \\ %21 & 0 &0 &1 &0 &1 &0 &0 &0 &1 &0 &0 &1 \\ %22 & 0 &0 &1 &0 &1 &0 &0 &0 &0 &1 &0 &0 \\ %23 & 0 &0 &1 &0 &0 &1 &0 &1 &0 &0 &1 &0 \\ %24 & 0 &0 &1 &0 &0 &1 &0 &1 &0 &0 &0 &1 \\ %25 & 0 &0 &1 &0 &0 &1 &0 &0 &1 &0 &1 &0 \\ %26 & 0 &0 &1 &0 &0 &1 &0 &0 &1 &0 &0 &1 \\ %27 & 0 &0 &1 &0 &0 &1 &0 &0 &0 &1 &0 &0 \\ %28 & 0 &0 &1 &0 &0 &0 &1 &0 &0 &0 &1 &0 \\ %29 & 0 &0 &1 &0 &0 &0 &1 &0 &0 &0 &0 &1 \\ %30 & 0 &0 &0 &1 &0 &0 &0 &1 &0 &0 &1 &0 \\ %31 & 0 &0 &0 &1 &0 &0 &0 &1 &0 &0 &0 &1 \\ %32 & 0 &0 &0 &1 &0 &0 &0 &0 &1 &0 &1 &0 \\ %33 & 0 &0 &0 &1 &0 &0 &0 &0 &1 &0 &0 &1 \\ %34 & 0 &0 &0 &1 &0 &0 &0 &0 &0 &1 &0 &0 \\ %\hline\hline %\end{tabular} %\caption{Truth table (two-valued measures) on the square logic in four dimensions.} %\end{table} %\subsection{Classical truth assignments and probabilities} There are 12 propositions forming four contexts $\{1, 2, 3, 4\}$, $\{4, 5, 6, 7\}$, $\{7, 8, 9, 10\}$, and $\{10, 11, 12, 1\}$ allowing 34 (separating, unital) two-valued states whose convex sum yield the following (quasi)classical probabilities: \begin{equation} \begin{aligned} p_{ 1} =&\lambda_1 + \lambda_2+ \lambda_3+ \lambda_4+ \lambda_5,\\ p_{ 2} =&\lambda_6 + \lambda_7 + \lambda_8+ \lambda_9 + \lambda_{10}+ \lambda_{11} \\& + \lambda_{12}+ \lambda_{13}+ \lambda_{14}+ \lambda_{15}+ \lambda_{16}+ \lambda_{17},\\ p_{ 3} =&\lambda_{18}+ \lambda_{19}+ \lambda_{20}+ \lambda_{21}+ \lambda_{22}+ \lambda_{23} \\&+ \lambda_{24}+ \lambda_{25}+ \lambda_{26}+ \lambda_{24}+ \lambda_{28}+ \lambda_{29},\\ p_{ 4} =&\lambda_{30}+ \lambda_{31}+ \lambda_{32}+ \lambda_{33}+ \lambda_{34},\\ p_{ 5} =&\lambda_{1}+ \lambda_{2}+ \lambda_{6}+ \lambda_{7}+ \lambda_{8}+ \lambda_{9} \\& + \lambda_{10}+ \lambda_{18}+ \lambda_{19}+ \lambda_{20}+ \lambda_{21}+ \lambda_{22},\\ p_{ 6} =&\lambda_{3}+ \lambda_{4}+ \lambda_{11}+ \lambda_{12}+ \lambda_{13}+ \lambda_{14} \\& + \lambda_{15}+ \lambda_{23}+ \lambda_{24}+ \lambda_{25}+ \lambda_{26}+ \lambda_{27},\\ p_{ 7} =&\lambda_{5}+ \lambda_{16}+ \lambda_{17}+ \lambda_{28}+ \lambda_{29},\\ p_{ 8} =&\lambda_{1}+ \lambda_{3}+ \lambda_{6}+ \lambda_{7}+ \lambda_{11}+ \lambda_{12} \\&+ \lambda_{18}+ \lambda_{19}+ \lambda_{23}+ \lambda_{24}+ \lambda_{30}+ \lambda_{31},\\ p_{ 9} =&\lambda_{2}+ \lambda_{4}+ \lambda_{8}+ \lambda_{9}+ \lambda_{13}+ \lambda_{14} \\&+ \lambda_{20}+ \lambda_{21}+ \lambda_{25}+ \lambda_{26}+ \lambda_{32}+ \lambda_{33},\\ p_{10} =&\lambda_{10}+ \lambda_{15}+ \lambda_{22}+ \lambda_{27}+ \lambda_{34},\\ p_{11} =&\lambda_{6}+ \lambda_{8}+ \lambda_{11}+ \lambda_{13}+ \lambda_{16}+ \lambda_{18} \\&+ \lambda_{20}+ \lambda_{23}+ \lambda_{25}+ \lambda_{28}+ \lambda_{30}+ \lambda_{32},\\ p_{12} =&\lambda_{7}+ \lambda_{9}+ \lambda_{12}+ \lambda_{14}+ \lambda_{17}+ \lambda_{19} \\&+ \lambda_{21}+ \lambda_{24}+ \lambda_{26}+ \lambda_{29}+ \lambda_{31}+ \lambda_{33}. \end{aligned} \end{equation} %\subsection{Boole-Bell type inequalities} % %\begin{equation} %\begin{split} % 0 \le p_1, p_2, p_{3}, p_5, p_6, p_8, p_{10}, p_{11} ,\\ % 0 \le p_1 + p_2 + p_3 -p_5 -p_6 ,\\ % 0 \le p_{2} +p_{3} -p_{5} -p_{6} +p_{8} +p_{9} -p_{11} ,\\ %-1 \le -p_{1} -p_{2} -p_{3} +p_{5} +p_{6} -p_{8} -p_{9} ,\\ % 1 \le -p_{1} -p_{2} -p_{3} ,\; % 1 \le p_{1} +p_{2} +p_{3} +p_{4} ,\\ % 0 \le -p_{1} -p_{2} -p_{3} +p_{5} +p_{6} +p_{7} ,\\ % 1 \le p_{1} +p_{2} +p_{3} -p_{5} -p_{6} +p_{8} +p_{9} +p_{10} ,\\ % 0 \le -p_{2} -p_{3} +p_{5} +p_{6} -p_{8} -p_{9} +p_{11} +p_{12} . % \end{split} %\end{equation} \section{Two-valued states, (quasi)classical probabilities on the pentagon (pentagram) logic in 3 dimensions} \label{2018-whycontexts-app-pent} %\subsection{Classical truth assignments and probabilities} There are 5 contexts $\{1, 2, 3\}$, $\{3, 4, 5\}$, $\{5, 6, 7\}$, $\{7, 8, 9\}$, and $\{9, 10, 1\}$ allowing 11 (separating, unital) two-valued states~\cite{wright:pent} whose convex sum yield the following (quasi)classical probabilities: \begin{equation} \begin{aligned} p_1=&\lambda_1 + \lambda_2 + \lambda_3 ,\\ p_2=&\lambda_4 + \lambda_5 + \lambda_7 + \lambda_9 + \lambda_{11} ,\\ p_3=&\lambda_6 + \lambda_8 + \lambda_{10} ,\\ p_4=&\lambda_1 + \lambda_2 + \lambda_4 + \lambda_7 + \lambda_{11} ,\\ p_5=&\lambda_3 + \lambda_5 + \lambda_9 ,\\ p_6=&\lambda_1 + \lambda_4 + \lambda_6 + \lambda_{10} + \lambda_{11} ,\\ p_7=&\lambda_2 + \lambda_7 + \lambda_8 ,\\ p_8=&\lambda_1 + \lambda_3 + \lambda_9 + \lambda_{10} + \lambda_{11} ,\\ p_9=&\lambda_4 + \lambda_5 + \lambda_6 ,\\ p_{10}=&\lambda_7 + \lambda_8 + \lambda_9 + \lambda_{10} + \lambda_{11} . \end{aligned} \label{2018-whycontexts-ppent} \end{equation} %\subsection{Boole-Bell type inequalities} % %\begin{equation} %\begin{split} % 0 \le p_{1}, p_{2}, p_{4}, p_{6}, p_{8} ,\\ % 0 \le - p_{1} + p_{4} + p_{8} ,\\ % -1 \le - p_{1} - p_{2} + p_{4} - p_{6} ,\\ % -1 \le -2p_{1} - p_{2} + p_{4} - p_{6} + p_{8} ,\\ % 0 \le p_{1} + p_{2} - p_{4} ,\\ % 0 \le p_{1} + p_{2} - p_{4} + p_{6} - p_{8} ,\\ % -1 \le - p_{1} - p_{2} ,\\ % 1 \le p_{1} + p_{2} + p_{3} ,\\ % 0 \le - p_{1} - p_{2} + p_{4} + p_{5} ,\\ % 1 \le p_{1} + p_{2} - p_{4} + p_{6} + p_{7} ,\\ % 0 \le - p_{1} - p_{2} + p_{4} - p_{6} + p_{8} + p_{9} ,\\ % 1 \le 2p_{1} + p_{2} - p_{4} + p_{6} - p_{8} + p_{10} . % \end{split} %\end{equation} \section{Truth assignments, (quasi)classical probabilities on the Specker bug combo logic} \label{2018-whycontexts-app-sbcombo} The logic depicted in Fig.~\ref{2018-whycontexts-fbugcombo}(c) contains 27 propositions forming 16 contexts $\{ {\bf a},1,2 \}$, $\{ 2,3,4 \}$, $\{ 4,5,{\bf b} \}$, $\{ {\bf b},6,7 \}$, $\{ 7,8,9 \}$, $\{ 9,10,{\bf a} \}$, $\{ 3 , 8 , 11 \}$, {\color{brown}$\{ {\bf a},{\bf c},{\bf b'} \}$}, {\color{violet}$\{ {\bf b},{\bf c},{\bf a}' \}$}, $\{ {\bf a}',1',2' \}$, $\{ 2',3',4' \}$, $\{ 4',5',{\bf b}' \}$, $\{ {\bf b}',6',7' \}$, $\{ 7',8',9' \}$, $\{ 9',10',{\bf a}' \}$, and $\{ 3' , 8' , 11' \}$, allowing 82 non-separating on ${\bf a}$/${\bf a'}$ and ${\bf b}$/${\bf b'}$, unital two-valued states (not enumerated here because of volume). 9 and 9 of these permit ${\bf a}$ as well as ${\bf a'}$, and ${\bf b}$ as well as ${\bf b'}$ to be true, respectively. The logic depicted in Fig.~\ref{2018-whycontexts-fbugcombo}(b) contains 16 propositions forming 9 contexts $\{ {\bf a},1,2 \}$, $\{ 2,3,4 \}$, $\{ 4,5,{\bf b} \}$, $\{ {\bf b},6,7 \}$, $\{ 7,8,9 \}$, $\{ 9,10,{\bf a} \}$, $\{ 3 , 8 , 11 \}$, {\color{brown}$\{ {\bf a},{\bf c},{\bf b'} \}$}, {\color{violet}$\{ {\bf b},{\bf c},{\bf a}' \}$}, allowing 22 (separating and unital) two-valued states which, through their convex summation, yield the (quasi-)classical probabilities: \begin{equation} \begin{aligned} p_{{\bf a}}=&\lambda_1+\lambda_2+\lambda_3 ,\\ p_{{\bf b}}=&\lambda_8+\lambda_{21}+\lambda_{22} ,\\ p_{{\bf a}'}=&\lambda_1+\lambda_2+\lambda_3+\lambda_5+\lambda_7+\lambda_{10} \\&+\lambda_{12}+\lambda_{14}+\lambda_{16}+\lambda_{18}+\lambda_{20} ,\\ p_{{\bf b}'}=&\lambda_5+\lambda_7+\lambda_8+\lambda_{10}+\lambda_{12}+\lambda_{14} \\& +\lambda_{16}+\lambda_{18}+\lambda_{20}+\lambda_{21}+\lambda_{22} , \\ p_{{\bf c}}=&\lambda_4+\lambda_6+\lambda_9+\lambda_{11}+\lambda_{13}+\lambda_{15}+\lambda_{17}+\lambda_{19} ,\\ p_{ 1}=&\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_9 \\& +\lambda_{10}+\lambda_{11}+\lambda_{12}+\lambda_{13}+\lambda_{14} ,\\ p_{ 2}=&\lambda_{15}+\lambda_{16}+\lambda_{17}+\lambda_{18}+\lambda_{19}+\lambda_{20}+\lambda_{21}+\lambda_{22} ,\\ p_{ 3}=&\lambda_1+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_8 ,\\ p_{ 4}=&\lambda_2+\lambda_3+\lambda_9+\lambda_{10}+\lambda_{11} +\lambda_{12}+\lambda_{13}+\lambda_{14} ,\\ p_{ 5}=&\lambda_1+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_{15} \\& +\lambda_{16}+\lambda_{17}+\lambda_{18}+\lambda_{19}+\lambda_{20} ,\\ p_{ 6}=&\lambda_2+\lambda_4+\lambda_5+\lambda_9+\lambda_{10}+\lambda_{11} \\& +\lambda_{12}+\lambda_{15}+\lambda_{16}+\lambda_{17}+\lambda_{18} ,\\ p_{ 7}=&\lambda_1+\lambda_3+\lambda_6+\lambda_7+\lambda_{13}+\lambda_{14}+\lambda_{19}+\lambda_{20} ,\\ p_{ 8}=&\lambda_2+\lambda_9+\lambda_{10}+\lambda_{15}+\lambda_{16}+\lambda_{21} ,\\ p_{ 9}=&\lambda_4+\lambda_5+\lambda_8+\lambda_{11}+\lambda_{12} +\lambda_{17}+\lambda_{18}+\lambda_{22} ,\\ p_{10}=&\lambda_6+\lambda_7+\lambda_9+\lambda_{10}+\lambda_{13} +\lambda_{14} \\& +\lambda_{15}+\lambda_{16}+\lambda_{19}+\lambda_{20}+\lambda_{21} ,\\ p_{11}=&\lambda_3+\lambda_{11}+\lambda_{12}+\lambda_{13}+\lambda_{14} +\lambda_{17} \\& +\lambda_{18}+\lambda_{19}+\lambda_{20}+\lambda_{22} . \end{aligned} \end{equation} Note that, for all configurations, $p_{{\bf a}} =\lambda_1+\lambda_2+\lambda_3 \le p_{{\bf a}'}$, implying that, whenever ${\bf a}$ is true ${\bf a}'$ must be true as well. The Specker bug logic depicted in Fig.~\ref{2018-whycontexts-fbugcombo}(a) contains 13 propositions forming 7 contexts $\{ {\bf a},1,2 \}$, $\{ 2,3,4 \}$, $\{ 4,5,{\bf b} \}$, $\{ {\bf b},6,7 \}$, $\{ 7,8,9 \}$, $\{ 9,10,{\bf a} \}$, $\{ 3 , 8 , 11 \}$, allowing 14 (separating and unital) two-valued states: \begin{equation} \begin{aligned} p_{{\bf a}}=&\lambda_1+\lambda_2+\lambda_3 ,\\ p_{{\bf b}}=&\lambda_6+\lambda_{13}+\lambda_{14} ,\\ p_{ 1}=&\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_9 ,\\ p_{ 2}=&\lambda_{10}+\lambda_{11}+\lambda_{12}+\lambda_{13}+\lambda_{14} ,\\ p_{ 3}=&\lambda_1+\lambda_4+\lambda_5+\lambda_6 ,\\ p_{ 4}=&\lambda_2+\lambda_3+\lambda_7+\lambda_8+\lambda_9 ,\\ p_{ 5}=&\lambda_1+\lambda_4+\lambda_5+\lambda_{10}+\lambda_{11}+\lambda_{12} ,\\ p_{ 6}=&\lambda_2+\lambda_4+\lambda_7+\lambda_8+\lambda_{10}+\lambda_{11} ,\\ p_{ 7}=&\lambda_1+\lambda_3+\lambda_5+\lambda_9+\lambda_{12} ,\\ p_{ 8}=&\lambda_2+\lambda_7+\lambda_{10}+\lambda_{13} ,\\ p_{ 9}=&\lambda_4+\lambda_6+\lambda_8+\lambda_{11}+\lambda_{14} ,\\ p_{10}=&\lambda_5+\lambda_7+\lambda_9+\lambda_{10}+\lambda_{12}+\lambda_{13} ,\\ p_{11}=&\lambda_3+\lambda_8+\lambda_9+\lambda_{11}+\lambda_{12}+\lambda_{14} . \end{aligned} \label{2018-whycontexts-pspeckerbug} \end{equation} Note that, for all configurations, whenever ${\bf a}$ is true {\bf b} is false, and {\it vice versa}. \section{Truth assignments, (quasi)classical probabilities on truth-implies-value indefiniteness logic in 3 dimensions} \label{2018-whycontexts-app-ACS} Fig.~\ref{2018-whycontexts-impossible}(c) depicts 37 propositions $\{{\bf a}, {\bf b}, 1, 2, 3, \ldots , 35\}$ in 26 contexts $\{ {\bf a} ,1 ,2 \}$, $\{ {\bf b} ,2 ,3 \}$, $\{ 4 ,{\bf a} ,5 \}$, $\{ {\bf b} ,6 ,7 \}$, {\color{magenta}$[\{ 7 ,10 , 4 \}]_\textrm{(a),(c)}$}, {\color{magenta}$[\{ 10 , 12 , 13 \}]_\textrm{(a),(c)}$}, {\color{blue}$[\{ 5 ,29 , 23 \}]_\textrm{(b),(c)}$}, {\color{blue}$[\{ 13 , 31 , 29 \}]_\textrm{(b),(c)}$}, $\{ 3 ,21 , 23 \}$, $\{ 4 ,28 , 22 \}$, $\{ 22 , 19 , 3 \}$, $\{ {\bf b} ,8 ,9 \}$, $\{ 9 ,11 , 5 \}$, $\{ 28 , 30 , 15 \}$, $\{ 15 , 14 , 11 \}$, $\{ 6 ,33 , 17 \}$, $\{ 17 , 20 , 21 \}$, $\{ 7 ,34 , 27 \}$, $\{ 27 , 26 , 23 \}$, $\{ 22 , 24 , 25 \}$, $\{ 25 , 35 , 9 \}$, $\{ 15 , 17 , 1 \}$, $\{ 13 , 16 , 1 \}$, $\{ 16 , 18 , 19 \}$, $\{ 16 , 32 , 8 \}$, $\{ 25 , 1 , 27 \}$, allowing 8 (non-separating, non-unital on ${\bf a}$, 2, 13, 15, 16, 17, 25, 27) two-valued states whose convex sum yield the following weights: \begin{equation} \begin{aligned} p_{{\bf a}}=&p_{ 2}= p_{13}= p_{15}= p_{16}=p_{17}=p_{25}= p_{27}= 0 , \\ p_{{\bf b}}=& \lambda_1 +\lambda_2 +\lambda_3 +\lambda_4 ,\\ p_{ 1}=& \lambda_1 +\lambda_2 +\lambda_3 +\lambda_4 +\lambda_5 +\lambda_6 +\lambda_7 +\lambda_8 = 1 , \\ p_{ 3}=& +\lambda_5 +\lambda_6 +\lambda_7 +\lambda_8 , \\ p_{ 4}=& \lambda_1 +\lambda_2 +\lambda_5 +\lambda_6 , \\ p_{ 5}=& \lambda_3 +\lambda_4 +\lambda_7 +\lambda_8 , \\ p_{ 6}=& \lambda_5 +\lambda_6 +\lambda_7 , \\ p_{ 7}=&\lambda_8 , p_{ 9}= \lambda_6 , p_{22}= \lambda_4, p_{23}= \lambda_2 , \\ p_{ 8}=& \lambda_5 +\lambda_7 +\lambda_8 , \\ p_{10}=& \lambda_3 +\lambda_4 +\lambda_7 , \\ p_{11}=& \lambda_1 +\lambda_2 +\lambda_5 , \\ p_{12}=& \lambda_1 +\lambda_2 +\lambda_5 +\lambda_6 +\lambda_8 , \\ p_{14}=& \lambda_3 +\lambda_4 +\lambda_6 +\lambda_7 +\lambda_8 , \\ p_{18}=& \lambda_4 +\lambda_5 +\lambda_6 +\lambda_7 +\lambda_8 , \\ p_{19}=& \lambda_1 +\lambda_2 +\lambda_3 , \\ p_{20}=& +\lambda_2 +\lambda_5 +\lambda_6 +\lambda_7 +\lambda_8 , \\ p_{21}=& \lambda_1 +\lambda_3 +\lambda_4 , \\ p_{24}=& \lambda_1 +\lambda_2 +\lambda_3 +\lambda_5 +\lambda_6 +\lambda_7 +\lambda_8 , \\ p_{26}=& \lambda_1 +\lambda_3 +\lambda_4 +\lambda_5 +\lambda_6 +\lambda_7 +\lambda_8 , \\ p_{28}=& \lambda_3 +\lambda_7 +\lambda_8 , \\ p_{29}=& \lambda_1 + \lambda_5 +\lambda_6 , \\ p_{30}=& \lambda_1 +\lambda_2 +\lambda_4 +\lambda_5 +\lambda_6 , \\ p_{31}=& \lambda_2 +\lambda_3 +\lambda_4 +\lambda_7 +\lambda_8 , \\ p_{32}=& \lambda_1 +\lambda_2 +\lambda_3 +\lambda_4 +\lambda_6 , \\ p_{33}=& \lambda_1 +\lambda_2 +\lambda_3 +\lambda_4 +\lambda_8 , \\ p_{34}=& \lambda_1 +\lambda_2 +\lambda_3 +\lambda_4 +\lambda_5 +\lambda_6 +\lambda_7 , \\ p_{35}=& \lambda_1 +\lambda_2 +\lambda_3 +\lambda_4 +\lambda_5 +\lambda_7 +\lambda_8 . \end{aligned} \end{equation} The logics in Figs.~\ref{2018-whycontexts-impossible}(a) and~\ref{2018-whycontexts-impossible}(b) contain 35 observables in 24 contexts which are the same as before in Fig.~\ref{2018-whycontexts-impossible}(c) lacking two contexts {\color{blue}$[\{ 5 ,29 , 23 \}]_\textrm{(b),(c)}$} and {\color{blue}$[\{ 13 , 31 , 29 \}]_\textrm{(b),(c)}$}, as well as {\color{magenta}$[\{ 7 ,10 , 4 \}]_\textrm{(a),(c)}$} and {\color{magenta}$[\{ 10 , 12 , 13 \}]_\textrm{(a),(c)}$}, respectively. The logic in Figs.~\ref{2018-whycontexts-impossible}(a) allows 13 (non-unital on 16) two-valued states whose convex sum yield the following weights: \begin{equation} \begin{aligned} p_{{\bf a}}=&\lambda_1 ,\\ p_{{\bf b}}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7 ,\\ p_{16}=& 0,\\ p_{ 1}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{ 2}=&\lambda_{12}+\lambda_{13} ,\\ p_{ 3}=&\lambda_1+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{ 4}=&\lambda_2+\lambda_3+\lambda_8+\lambda_9+\lambda_{12} ,\\ p_{ 5}=&\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_{10}+\lambda_{11}+\lambda_{13} ,\\ p_{ 6}=&\lambda_8+\lambda_9+\lambda_{10}+\lambda_{12} ,\\ p_{ 7}=&\lambda_1+\lambda_{11}+\lambda_{13} ,\\ p_{ 8}=&\lambda_1+\lambda_8+\lambda_{10}+\lambda_{11}+\lambda_{13} ,\\ p_{ 9}=&\lambda_9+\lambda_{12} ,\\ p_{10}=&\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_{10} ,\\ p_{11}=&\lambda_1+\lambda_2+\lambda_3+\lambda_8 ,\\ p_{12}=&\lambda_2+\lambda_3+\lambda_8+\lambda_9+\lambda_{11} ,\\ p_{13}=&\lambda_1+\lambda_{12}+\lambda_{13} ,\\ p_{14}=&\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_9+\lambda_{10}+\lambda_{11}+\lambda_{13} ,\\ p_{15}=&\lambda_{12} ,\\ p_{17}=&\lambda_1+\lambda_{13} ,\\ p_{18}=&\lambda_1+\lambda_5+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{19}=&\lambda_2+\lambda_3+\lambda_4+\lambda_6+\lambda_{12}+\lambda_{13} ,\\ p_{20}=&\lambda_3+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{21}=&\lambda_2+\lambda_4+\lambda_5+\lambda_{12} ,\\ p_{22}=&\lambda_5+\lambda_7 ,\\ p_{23}=&\lambda_3+\lambda_6+\lambda_7+\lambda_{13} ,\\ p_{24}=&\lambda_2+\lambda_3+\lambda_4+\lambda_6+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11}+\lambda_{12} ,\\ p_{25}=&\lambda_1+\lambda_{13} ,\\ p_{26}=&\lambda_1+\lambda_2+\lambda_4+\lambda_5+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{27}=&\lambda_{12} ,\\ p_{28}=&\lambda_1+\lambda_4+\lambda_6+\lambda_{10}+\lambda_{11}+\lambda_{13} ,\\ p_{30}=&\lambda_2+\lambda_3+\lambda_5+\lambda_7+\lambda_8+\lambda_9 ,\\ p_{32}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_9+\lambda_{12} ,\\ p_{33}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_{11} ,\\ p_{34}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10} ,\\ p_{35}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_{10}+\lambda_{11} . \end{aligned} \end{equation} Therefore, whenever ${\bf a}$ is true, that is, $p_{{\bf a}}=\lambda_1 =1$, ${\bf b}$ has to be false, because $p_{{\bf b}}=\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7 =0$. Conversely, the logic in Figs.~\ref{2018-whycontexts-impossible}(b) allows 13 (non-separating on 15/27 and non-unital on 16) two-valued states whose convex sum yield the following weights: \begin{equation} \begin{aligned} p_{{\bf a}}=&\lambda_1,\\ p_{{\bf b}}=&\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5,\\ p_{16}=& 0,\\ p_{ 1}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{ 2}=&\lambda_{12}+\lambda_{13} ,\\ p_{ 3}=&\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{ 4}=&\lambda_2+\lambda_3+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{12} ,\\ p_{ 5}=&\lambda_4+\lambda_5+\lambda_{10}+\lambda_{11}+\lambda_{13} ,\\ p_{ 6}=&\lambda_6+\lambda_7+\lambda_{10}+\lambda_{13} ,\\ p_{ 7}=&\lambda_8+\lambda_9+\lambda_{11}+\lambda_{12} ,\\ p_{ 8}=&\lambda_6+\lambda_8+\lambda_{10}+\lambda_{11}+\lambda_{12}+\lambda_{13} ,\\ p_{ 9}=&\lambda_7+\lambda_9,\\ p_{11}=&\lambda_1+\lambda_2+\lambda_3+\lambda_6+\lambda_8+\lambda_{12} ,\\ p_{13}=&\lambda_1+\lambda_{12}+\lambda_{13} ,\\ p_{14}=&\lambda_4+\lambda_5+\lambda_7+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{15}=& p_{27}=\lambda_{13} ,\\ p_{17}=&\lambda_1+\lambda_{12} ,\\ p_{18}=&\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11}+\lambda_{13} ,\\ p_{19}=&\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_{12} ,\\ p_{20}=&\lambda_3+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{21}=&\lambda_2+\lambda_4+\lambda_5+\lambda_{13} ,\\ p_{22}=&\lambda_5+\lambda_{13} ,\\ p_{23}=&\lambda_1+\lambda_3+\lambda_{12} ,\\ p_{24}=&\lambda_2+\lambda_3+\lambda_4+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{25}=&\lambda_1+\lambda_{12} ,\\ p_{26}=&\lambda_2+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{10}+\lambda_{11} ,\\ p_{28}=&\lambda_1+\lambda_4+\lambda_{10}+\lambda_{11} ,\\ p_{29}=&\lambda_2+\lambda_6+\lambda_7+\lambda_8+\lambda_9 ,\\ p_{30}=&\lambda_2+\lambda_3+\lambda_5+\lambda_6+\lambda_7+\lambda_8+\lambda_9+\lambda_{12} ,\\ p_{31}=&\lambda_3+\lambda_4+\lambda_5+\lambda_{10}+\lambda_{11} ,\\ p_{32}=&\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_7+\lambda_9,\\ p_{33}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_8+\lambda_9+\lambda_{11} ,\\ p_{34}=&\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_7+\lambda_{10} ,\\ p_{35}=&\lambda_2+\lambda_3+\lambda_4+\lambda_5+\lambda_6+\lambda_8+\lambda_{10}+\lambda_{11}+\lambda_{13} . \end{aligned} \end{equation} Therefore, whenever ${\bf a}$ is true, that is, $p_{{\bf a}}=\lambda_1 =1$, ${\bf b}$ has to be true, because $p_{{\bf b}}=\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5 =\lambda_1=1$. \clearpage %\medskip %\bibliography{svozil} %merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked %Control: key (0) %Control: author (0) dotless jnrlst %Control: editor 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[{\citenamefont {Kochen}\ and\ \citenamefont {Specker}(1967)}]{kochen1}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Simon}\ \bibnamefont {Kochen}}\ and\ \bibinfo {author} {\bibfnamefont {Ernst~P.}\ \bibnamefont {Specker}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The problem of hidden variables in quantum mechanics},}\ }\href {\doibase 10.1512/iumj.1968.17.17004} {\bibfield {journal} {\bibinfo {journal} {Journal of Mathematics and Mechanics (now Indiana University Mathematics Journal)}\ }\textbf {\bibinfo {volume} {17}},\ \bibinfo {pages} {59--87} (\bibinfo {year} {1967})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Redhead}(1990)}]{redhead}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Michael}\ \bibnamefont {Redhead}},\ }\href@noop {} {\emph {\bibinfo {title} {Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics}}}\ (\bibinfo {publisher} {Clarendon Press},\ \bibinfo {address} {Oxford},\ \bibinfo {year} {1990})\BibitemShut {NoStop}% \bibitem [{\citenamefont {Pitowsky}(2003)}]{Pitowsky2003395}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Itamar}\ \bibnamefont {Pitowsky}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Betting on the outcomes of measurements: a bayesian theory of quantum probability},}\ }\href {\doibase 10.1016/S1355-2198(03)00035-2} {\bibfield {journal} {\bibinfo {journal} {Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics}\ }\textbf {\bibinfo {volume} {34}},\ \bibinfo {pages} {395--414} (\bibinfo {year} {2003})},\ \bibinfo {note} {quantum Information and Computation},\ \Eprint {http://arxiv.org/abs/arXiv:quant-ph/0208121} {arXiv:quant-ph/0208121} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Pitowsky}(2006)}]{pitowsky-06}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Itamar}\ \bibnamefont {Pitowsky}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum mechanics as a theory of probability},}\ }in\ \href {\doibase 10.1007/1-4020-4876-9_10} {\emph {\bibinfo {booktitle} {Physical Theory and its Interpretation}}},\ \bibinfo {series} {The Western Ontario Series in Philosophy of Science}, Vol.~\bibinfo {volume} {72},\ \bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont {William}\ \bibnamefont {Demopoulos}}\ and\ \bibinfo {editor} {\bibfnamefont {Itamar}\ \bibnamefont {Pitowsky}}}\ (\bibinfo {publisher} {Springer Netherlands},\ \bibinfo {year} {2006})\ pp.\ \bibinfo {pages} {213--240},\ \Eprint {http://arxiv.org/abs/arXiv:quant-ph/0510095} {arXiv:quant-ph/0510095} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Belinfante}(1973)}]{Belinfante-73}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Frederik~Jozef}\ \bibnamefont {Belinfante}},\ }\href {\doibase 10.1016/B978-0-08-017032-9.50001-7} {\emph {\bibinfo {title} {A Survey of Hidden-Variables Theories}}},\ \bibinfo {series} 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\bibinfo {pages} {443--447} (\bibinfo {year} {1993})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Pt{\'{a}}k}\ and\ \citenamefont {Pulmannov{\'{a}}}(1991)}]{pulmannova-91}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Pavel}\ \bibnamefont {Pt{\'{a}}k}}\ and\ \bibinfo {author} {\bibfnamefont {Sylvia}\ \bibnamefont {Pulmannov{\'{a}}}},\ }\href@noop {} {\emph {\bibinfo {title} {Orthomodular Structures as Quantum Logics. {I}ntrinsic Properties, State Space and Probabilistic Topics}}},\ \bibinfo {series} {Fundamental Theories of Physics}, Vol.~\bibinfo {volume} {44}\ (\bibinfo {publisher} {Kluwer Academic Publishers, Springer Netherlands},\ \bibinfo {address} {Dordrecht},\ \bibinfo {year} {1991})\BibitemShut {NoStop}% \bibitem [{\citenamefont {Navara}\ and\ \citenamefont {Rogalewicz}(1991)}]{nav:91}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Mirko}\ \bibnamefont {Navara}}\ and\ \bibinfo {author} {\bibfnamefont {Vladim\'ir}\ \bibnamefont 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{Abbott}, \citenamefont {Calude},\ and\ \citenamefont {Svozil}}]{2015-AnalyticKS}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alastair~A.}\ \bibnamefont {Abbott}}, \bibinfo {author} {\bibfnamefont {Cristian~S.}\ \bibnamefont {Calude}}, \ and\ \bibinfo {author} {\bibfnamefont {Karl}\ \bibnamefont {Svozil}},\ }\bibfield {title} {\enquote {\bibinfo {title} {A variant of the {K}ochen-{S}pecker theorem localising value indefiniteness},}\ }\href {\doibase 10.1063/1.4931658} {\bibfield {journal} {\bibinfo {journal} {Journal of Mathematical Physics}\ }\textbf {\bibinfo {volume} {56}},\ \bibinfo {eid} {102201} (\bibinfo {year} {2015})},\ \Eprint {http://arxiv.org/abs/arXiv:1503.01985} {arXiv:1503.01985} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Pitowsky}(1982)}]{Pitowsky-1982-subs}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Itamar}\ \bibnamefont {Pitowsky}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Substitution and truth in quantum 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{79}},\ \bibinfo {pages} {2755--2758} (\bibinfo {year} {1997})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Cabello}\ and\ \citenamefont {Garc{\'{i}}a-Alcaine}(1995)}]{Cabello-1995-ppks}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ad{\'{a}}n}\ \bibnamefont {Cabello}}\ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Garc{\'{i}}a-Alcaine}},\ }\bibfield {title} {\enquote {\bibinfo {title} {A hidden-variables versus quantum mechanics experiment},}\ }\href {\doibase 10.1088/0305-4470/28/13/016} {\bibfield {journal} {\bibinfo {journal} {Journal of Physics A: Mathematical and General Physics}\ }\textbf {\bibinfo {volume} {28}} (\bibinfo {year} {1995}),\ 10.1088/0305-4470/28/13/016}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Cabello}\ \emph {et~al.}(1996)\citenamefont {Cabello}, \citenamefont {Estebaranz},\ and\ \citenamefont {Garc{\'{i}}a-Alcaine}}]{cabello-96}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ad{\'{a}}n}\ \bibnamefont {Cabello}}, \bibinfo {author} {\bibfnamefont {Jos{\'{e}}~M.}\ \bibnamefont {Estebaranz}}, \ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Garc{\'{i}}a-Alcaine}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{B}ell-{K}ochen-{S}pecker theorem: A proof with 18 vectors},}\ }\href {\doibase 10.1016/0375-9601(96)00134-X} {\bibfield {journal} {\bibinfo {journal} {Physics Letters A}\ }\textbf {\bibinfo {volume} {212}},\ \bibinfo {pages} {183--187} (\bibinfo {year} {1996})},\ \Eprint {http://arxiv.org/abs/arXiv:quant-ph/9706009} {arXiv:quant-ph/9706009} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Cabello}(1997)}]{cabello-97-nhvp}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ad\'an}\ \bibnamefont {Cabello}},\ }\bibfield {title} {\enquote {\bibinfo {title} {No-hidden-variables proof for two spin- particles preselected and postselected in unentangled states},}\ }\href {\doibase 10.1103/PhysRevA.55.4109} {\bibfield {journal} {\bibinfo {journal} {Physical Review A}\ }\textbf {\bibinfo {volume} {55}},\ \bibinfo {pages} {4109--4111} (\bibinfo {year} {1997})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Chen}\ \emph {et~al.}(2013)\citenamefont {Chen}, \citenamefont {Cabello}, \citenamefont {Xu}, \citenamefont {Su}, \citenamefont {Wu},\ and\ \citenamefont {Kwek}}]{Cabello-2013-HP}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Jing-Ling}\ \bibnamefont {Chen}}, \bibinfo {author} {\bibfnamefont {Ad\'an}\ \bibnamefont {Cabello}}, \bibinfo {author} {\bibfnamefont {Zhen-Peng}\ \bibnamefont {Xu}}, \bibinfo {author} {\bibfnamefont {Hong-Yi}\ \bibnamefont {Su}}, \bibinfo {author} {\bibfnamefont {Chunfeng}\ \bibnamefont {Wu}}, \ and\ \bibinfo {author} {\bibfnamefont {L.~C.}\ \bibnamefont {Kwek}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Hardy's paradox for high-dimensional systems},}\ }\href {\doibase 10.1103/PhysRevA.88.062116} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {88}},\ \bibinfo {pages} {062116} (\bibinfo {year} {2013})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Cabello}\ \emph {et~al.}(2013{\natexlab{b}})\citenamefont {Cabello}, \citenamefont {Badziag}, \citenamefont {Terra~Cunha},\ and\ \citenamefont {Bourennane}}]{Cabello-2013-Hardylike}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ad\'an}\ \bibnamefont {Cabello}}, \bibinfo {author} {\bibfnamefont {Piotr}\ \bibnamefont {Badziag}}, \bibinfo {author} {\bibfnamefont {Marcelo}\ \bibnamefont {Terra~Cunha}}, \ and\ \bibinfo {author} {\bibfnamefont {Mohamed}\ \bibnamefont {Bourennane}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Simple {H}ardy-like proof of quantum contextuality},}\ }\href {\doibase 10.1103/PhysRevLett.111.180404} {\bibfield {journal} {\bibinfo {journal} {Physical Review Letters}\ }\textbf {\bibinfo {volume} {111}},\ \bibinfo {pages} {180404} (\bibinfo {year} {2013}{\natexlab{b}})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Pitowsky}(1998)}]{pitowsky:218}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Itamar}\ \bibnamefont {Pitowsky}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Infinite and finite {G}leason's theorems and the logic of indeterminacy},}\ }\href {\doibase 10.1063/1.532334} {\bibfield {journal} {\bibinfo {journal} {Journal of Mathematical Physics}\ }\textbf {\bibinfo {volume} {39}},\ \bibinfo {pages} {218--228} (\bibinfo {year} {1998})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Hrushovski}\ and\ \citenamefont {Pitowsky}(2004)}]{hru-pit-2003}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ehud}\ \bibnamefont {Hrushovski}}\ and\ \bibinfo {author} {\bibfnamefont {Itamar}\ \bibnamefont {Pitowsky}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Generalizations of {K}ochen and {S}pecker's theorem and the effectiveness of {G}leason's theorem},}\ }\href {\doibase 10.1016/j.shpsb.2003.10.002} {\bibfield {journal} {\bibinfo {journal} {Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics}\ }\textbf {\bibinfo {volume} {35}},\ \bibinfo {pages} {177­194} (\bibinfo {year} {2004})},\ \Eprint {http://arxiv.org/abs/arXiv:quant-ph/0307139} {arXiv:quant-ph/0307139} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Abbott}\ \emph {et~al.}(2012)\citenamefont {Abbott}, \citenamefont {Calude}, \citenamefont {Conder},\ and\ \citenamefont {Svozil}}]{2012-incomput-proofsCJ}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alastair~A.}\ \bibnamefont {Abbott}}, \bibinfo {author} {\bibfnamefont {Cristian~S.}\ \bibnamefont {Calude}}, \bibinfo {author} {\bibfnamefont {Jonathan}\ \bibnamefont {Conder}}, \ and\ \bibinfo {author} {\bibfnamefont {Karl}\ \bibnamefont {Svozil}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Strong {K}ochen-{S}pecker theorem and incomputability of quantum randomness},}\ }\href {\doibase 10.1103/PhysRevA.86.062109} {\bibfield {journal} {\bibinfo {journal} {Physical Review A}\ }\textbf {\bibinfo {volume} {86}},\ \bibinfo {pages} {062109} (\bibinfo {year} {2012})},\ \Eprint {http://arxiv.org/abs/arXiv:1207.2029} {arXiv:1207.2029} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Abbott}\ \emph {et~al.}(2014)\citenamefont {Abbott}, \citenamefont {Calude},\ and\ \citenamefont {Svozil}}]{PhysRevA.89.032109}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alastair~A.}\ \bibnamefont {Abbott}}, \bibinfo {author} {\bibfnamefont {Cristian~S.}\ \bibnamefont {Calude}}, \ and\ \bibinfo {author} {\bibfnamefont {Karl}\ \bibnamefont {Svozil}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Value-indefinite observables are almost everywhere},}\ }\href {\doibase 10.1103/PhysRevA.89.032109} {\bibfield {journal} {\bibinfo {journal} {Physical Review A}\ }\textbf {\bibinfo {volume} {89}},\ \bibinfo {pages} {032109} (\bibinfo {year} {2014})},\ \Eprint {http://arxiv.org/abs/arXiv:1309.7188} {arXiv:1309.7188} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Pitowsky}(1983)}]{pitowsky-83}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Itamar}\ \bibnamefont {Pitowsky}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Deterministic model of spin and statistics},}\ }\href {\doibase 10.1103/PhysRevD.27.2316} {\bibfield {journal} {\bibinfo {journal} {Physical Review D}\ }\textbf {\bibinfo {volume} {27}},\ \bibinfo {pages} {2316--2326} (\bibinfo {year} {1983})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Meyer}(1999)}]{meyer:99}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {David~A.}\ \bibnamefont {Meyer}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Finite precision measurement nullifies the {K}ochen-{S}pecker theorem},}\ }\href {\doibase 10.1103/PhysRevLett.83.3751} {\bibfield {journal} {\bibinfo {journal} {Physical Review Letters}\ }\textbf {\bibinfo {volume} {83}},\ \bibinfo {pages} {3751--3754} (\bibinfo {year} {1999})},\ \Eprint {http://arxiv.org/abs/arXiv:quant-ph/9905080} {arXiv:quant-ph/9905080} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Schr{\"{o}}dinger}(1935)}]{schrodinger}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Erwin}\ \bibnamefont {Schr{\"{o}}dinger}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Die gegenw{\"{a}}rtige {S}ituation in der {Q}uantenmechanik},}\ }\href {\doibase 10.1007/BF01491891, 10.1007/BF01491914, 10.1007/BF01491987} {\bibfield {journal} {\bibinfo {journal} {Naturwissenschaften}\ }\textbf {\bibinfo {volume} {23}},\ \bibinfo {pages} {807--812, 823--828, 844--849} (\bibinfo {year} {1935})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {London}\ and\ \citenamefont {Bauer}(1983)}]{london-Bauer-1983}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Fritz}\ \bibnamefont {London}}\ and\ \bibinfo {author} {\bibfnamefont {Edmond}\ \bibnamefont {Bauer}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The theory of observation in quantum mechanics},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Quantum Theory and Measurement}}}\ (\bibinfo {publisher} {Princeton University Press},\ \bibinfo {address} {Princeton, NJ},\ \bibinfo {year} {1983})\ pp.\ \bibinfo {pages} {217--259},\ \bibinfo {note} {consolidated translation of French original~\cite{london-Bauer-1939}}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Svozil}(2004)}]{svozil-2003-garda}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Karl}\ \bibnamefont {Svozil}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum information via state partitions and the context translation principle},}\ }\href {\doibase 10.1080/09500340410001664179} {\bibfield {journal} {\bibinfo {journal} {Journal of Modern Optics}\ }\textbf {\bibinfo {volume} {51}},\ \bibinfo {pages} {811--819} (\bibinfo {year} {2004})},\ \Eprint {http://arxiv.org/abs/arXiv:quant-ph/0308110} {arXiv:quant-ph/0308110} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Howard}(2004)}]{Howard2004-HOWWIT}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Don}\ \bibnamefont {Howard}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Who invented the ``{C}openhagen interpretation''? a study in mythology},}\ }\href {\doibase 10.1086/425941} {\bibfield {journal} {\bibinfo {journal} {Philosophy of Science}\ }\textbf {\bibinfo {volume} {71}},\ \bibinfo {pages} {669--682} (\bibinfo {year} {2004})},\ \bibinfo {note} {proceedings of the 2002 Biennial Meeting of The Philosophy of Science Association. 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Mitchell}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Bohr}(1928)}]{bohr:28}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Niels}\ \bibnamefont {Bohr}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The quantum postulate and the recent development of atomistic theory},}\ }\href {\doibase 10.1038/121580a0} {\bibfield {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {121}},\ \bibinfo {pages} {580--590} (\bibinfo {year} {1928})},\ \bibinfo {note} {reprinted in Ref.~\cite[pp.~52-91]{bohr:61} and in \cite[pp.~87-126]{wheeler-Zurek:83}}\BibitemShut {NoStop}% \bibitem [{\citenamefont {{von Neumann}}(1932, 1996)}]{v-neumann-49}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {John}\ \bibnamefont {{von Neumann}}},\ }\href {\doibase 10.1007/978-3-642-61409-5} {\emph {\bibinfo {title} {{M}athematische {G}rundlagen der {Q}uantenmechanik}}},\ \bibinfo {edition} {2nd}\ ed.\ (\bibinfo {publisher} {Springer},\ \bibinfo {address} {Berlin, Heidelberg},\ \bibinfo {year} {1932, 1996})\ \bibinfo {note} {{E}nglish translation in Ref.~\cite{v-neumann-55}}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Zeilinger}(1999)}]{zeil-99}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Anton}\ \bibnamefont {Zeilinger}},\ }\bibfield {title} {\enquote {\bibinfo {title} {A foundational principle for quantum mechanics},}\ }\href {\doibase 10.1023/A:1018820410908} {\bibfield {journal} {\bibinfo {journal} {Foundations of Physics}\ }\textbf {\bibinfo {volume} {29}},\ \bibinfo {pages} {631--643} (\bibinfo {year} {1999})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Myrvold}(2011)}]{Myrvold2011237}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Wayne~C.}\ \bibnamefont {Myrvold}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Statistical mechanics and thermodynamics: A {M}axwellian view},}\ }\href {\doibase 10.1016/j.shpsb.2011.07.001} {\bibfield {journal} {\bibinfo {journal} {Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics}\ }\textbf {\bibinfo {volume} {42}},\ \bibinfo {pages} {237--243} (\bibinfo {year} {2011})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Specker}(1960)}]{specker-60}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ernst}\ \bibnamefont {Specker}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{D}ie {L}ogik nicht gleichzeitig entscheidbarer {A}ussagen},}\ }\href {\doibase 10.1111/j.1746-8361.1960.tb00422.x} {\bibfield {journal} {\bibinfo {journal} {Dialectica}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo {pages} {239--246} (\bibinfo {year} {1960})},\ \Eprint {http://arxiv.org/abs/http://arxiv.org/abs/1103.4537} {http://arxiv.org/abs/1103.4537} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Kochen}\ and\ \citenamefont {Specker}(1965{\natexlab{b}})}]{kochen3}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Simon}\ \bibnamefont {Kochen}}\ and\ \bibinfo {author} {\bibfnamefont {Ernst~P.}\ \bibnamefont {Specker}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The calculus of partial propositional functions},}\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science, Jerusalem}}}\ (\bibinfo {publisher} {North Holland},\ \bibinfo {address} {Amsterdam},\ \bibinfo {year} {1965})\ pp.\ \bibinfo {pages} {45--57}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Valentini}(2018)}]{Valentini-2018}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Antony}\ \bibnamefont {Valentini}},\ }\href@noop {} {\enquote {\bibinfo {title} {The {D}e {B}roglie-{B}ohm pilot-wave theory},}\ } (\bibinfo {year} {2018}),\ \bibinfo {note} {talk given for the lecture series on foundations of physics: scientific realism, on March 22, 2018}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Plato}(2000)}]{plato-republic}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibnamefont {Plato}},\ }\href {http://www.cambridge.org/academic/subjects/politics-international-relations/texts-political-thought/plato-republic} {\emph {\bibinfo {title} {The Republic}}},\ Cambridge Texts in the History of Political Thought\ (\bibinfo {publisher} {Cambridge University Press},\ \bibinfo {address} {Cambridge, UK},\ \bibinfo {year} {2000})\ \bibinfo {note} {translated by Tom Griffith, and edited by G. R. F. Ferrari}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Tkadlec}(2009, 2017)}]{tkadlec-2states-2017}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Josef}\ \bibnamefont {Tkadlec}},\ }\href@noop {} {\enquote {\bibinfo {title} {{\tt 2states}. {A} {P}ascal program},}\ } (\bibinfo {year} {2009, 2017}),\ \bibinfo {note} {private communication, electronic message from Aug 23nd, 2017}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Specker}(1990)}]{specker-ges}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ernst}\ \bibnamefont {Specker}},\ }\href {\doibase 10.1007/978-3-0348-9259-9} {\emph {\bibinfo {title} {Selecta}}}\ (\bibinfo {publisher} {Birkh{\"{a}}user Verlag},\ \bibinfo {address} {Basel},\ \bibinfo {year} {1990})\BibitemShut {NoStop}% \bibitem [{\citenamefont {London}\ and\ \citenamefont {Bauer}(1939)}]{london-Bauer-1939}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Fritz}\ \bibnamefont {London}}\ and\ \bibinfo {author} {\bibfnamefont {Edmond}\ \bibnamefont {Bauer}},\ }\href@noop {} {\emph {\bibinfo {title} {La theorie de l'observation en m\'ecanique quantique; {N}o.~775 of Actualit\'es scientifiques et industrielles: Expos\'es de physique g\'en\'erale, publi\'es sous la direction de {P}aul {L}angevin}}}\ (\bibinfo {publisher} {Hermann},\ \bibinfo {address} {Paris},\ \bibinfo {year} {1939})\ \bibinfo {note} {english translation in~\cite{london-Bauer-1983}}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Bohr}(1961)}]{bohr:61}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Niels}\ \bibnamefont {Bohr}},\ }\href@noop {} {\emph {\bibinfo {title} {Atomic Theory and the Description of Nature}}}\ (\bibinfo {publisher} {Cambridge University Press},\ \bibinfo {address} {London},\ \bibinfo {year} {1961})\BibitemShut {NoStop}% \bibitem [{\citenamefont {Wheeler}\ and\ \citenamefont {Zurek}(1983)}]{wheeler-Zurek:83}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {John~Archibald}\ \bibnamefont {Wheeler}}\ and\ \bibinfo {author} {\bibfnamefont {Wojciech~Hubert}\ \bibnamefont {Zurek}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Theory and Measurement}}}\ (\bibinfo {publisher} {Princeton University Press},\ \bibinfo {address} {Princeton, NJ},\ \bibinfo {year} {1983})\BibitemShut {NoStop}% \bibitem [{\citenamefont {{von Neumann}}(1955)}]{v-neumann-55}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {John}\ \bibnamefont {{von Neumann}}},\ }\href {http://press.princeton.edu/titles/2113.html} {\emph {\bibinfo {title} {Mathematical Foundations of Quantum Mechanics}}}\ (\bibinfo {publisher} {Princeton University Press},\ \bibinfo {address} {Princeton, NJ},\ \bibinfo {year} {1955})\ \bibinfo {note} {{G}erman original in Ref.~\cite{v-neumann-49}}\BibitemShut {NoStop}% \end{thebibliography}% \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %****** TRIANGLE ************************************************************************************************************************** %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Triangle in dimension 4 9 atoms 3 blocks 0 proper subsets of blocks 4 1 2 3 4 4 4 5 6 7 4 7 8 9 1 14 2-valued evaluations of atoms: 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 set of 2-valued evaluations of atoms: nonempty: yes unital: yes separating atoms: yes separating: yes 1s on nonorthogonal atoms (=OD if no noncomplete block): yes order determining: no for sets of atoms (ordered elements?) 4/8+9 7/2+3 4+7/2+3+4 1/5+6 1+4/4+5+6 1+7/1+2+3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Triangle states 1 2 3 4 5 6 7 8 9 1 1 0 0 0 1 0 0 0 0 2 1 0 0 0 0 1 0 0 0 3 0 1 0 0 1 0 0 1 0 4 0 1 0 0 1 0 0 0 1 5 0 1 0 0 0 1 0 1 0 6 0 1 0 0 0 1 0 0 1 7 0 1 0 0 0 0 1 0 0 8 0 0 1 0 1 0 0 1 0 9 0 0 1 0 1 0 0 0 1 10 0 0 1 0 0 1 0 1 0 11 0 0 1 0 0 1 0 0 1 12 0 0 1 0 0 0 1 0 0 13 0 0 0 1 0 0 0 1 0 14 0 0 0 1 0 0 0 0 1 * triangle logic dim=4 * V-representation begin 10 14 integer 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 end * cddlib: a double description library:Version 0.94g (March 23, 2012) * compiled for C double arithmetic. * Copyright (C) 1996, Komei Fukuda, fukuda@ifor.math.ethz.ch * roworder: lexmin ine_file: Inequalities H-representation linearity 3 11 12 13 begin 13 10 real 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 -2 -1 -1 0 1 1 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 -1 -1 0 0 0 1 -1 -1 -1 0 0 0 0 0 0 -1 1 1 1 1 0 0 0 0 0 0 -1 -1 -1 0 1 1 1 0 0 -1 2 1 1 0 -1 -1 0 1 1 end * Computation started at Sat Apr 21 13:23:42 2018 * ended at Sat Apr 21 13:23:42 2018 * Total processor time = 0 seconds * = 0 h 0 m 0 s %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %********************* Square ******************************* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Square in dimension 4 12 atoms 4 blocks 0 proper subsets of blocks 4 1 2 3 4 4 4 5 6 7 4 7 8 9 10 4 10 11 12 1 34 2-valued evaluations of atoms: 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 set of 2-valued evaluations of atoms: nonempty: yes unital: yes separating atoms: yes separating: yes 1s on nonorthogonal atoms (=OD if no noncomplete block): yes order determining: yes 01 1 0 0 0 1 0 0 1 0 0 0 0 02 1 0 0 0 1 0 0 0 1 0 0 0 03 1 0 0 0 0 1 0 1 0 0 0 0 04 1 0 0 0 0 1 0 0 1 0 0 0 05 1 0 0 0 0 0 1 0 0 0 0 0 06 0 1 0 0 1 0 0 1 0 0 1 0 07 0 1 0 0 1 0 0 1 0 0 0 1 08 0 1 0 0 1 0 0 0 1 0 1 0 09 0 1 0 0 1 0 0 0 1 0 0 1 10 0 1 0 0 1 0 0 0 0 1 0 0 11 0 1 0 0 0 1 0 1 0 0 1 0 12 0 1 0 0 0 1 0 1 0 0 0 1 13 0 1 0 0 0 1 0 0 1 0 1 0 14 0 1 0 0 0 1 0 0 1 0 0 1 15 0 1 0 0 0 1 0 0 0 1 0 0 16 0 1 0 0 0 0 1 0 0 0 1 0 17 0 1 0 0 0 0 1 0 0 0 0 1 18 0 0 1 0 1 0 0 1 0 0 1 0 19 0 0 1 0 1 0 0 1 0 0 0 1 20 0 0 1 0 1 0 0 0 1 0 1 0 21 0 0 1 0 1 0 0 0 1 0 0 1 22 0 0 1 0 1 0 0 0 0 1 0 0 23 0 0 1 0 0 1 0 1 0 0 1 0 24 0 0 1 0 0 1 0 1 0 0 0 1 25 0 0 1 0 0 1 0 0 1 0 1 0 26 0 0 1 0 0 1 0 0 1 0 0 1 27 0 0 1 0 0 1 0 0 0 1 0 0 28 0 0 1 0 0 0 1 0 0 0 1 0 29 0 0 1 0 0 0 1 0 0 0 0 1 30 0 0 0 1 0 0 0 1 0 0 1 0 31 0 0 0 1 0 0 0 1 0 0 0 1 32 0 0 0 1 0 0 0 0 1 0 1 0 33 0 0 0 1 0 0 0 0 1 0 0 1 34 0 0 0 1 0 0 0 0 0 1 0 0 * triangle logic dim=4 * V-representation begin 34 13 integer 1 1 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 end * cddlib: a double description library:Version 0.94g (March 23, 2012) * compiled for C double arithmetic. * Copyright (C) 1996, Komei Fukuda, fukuda@ifor.math.ethz.ch * roworder: lexmin ine_file: Inequalities H-representation linearity 4 13 14 15 16 begin 16 13 real 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 -1 -1 0 1 1 0 -1 0 1 -1 -1 -1 0 1 1 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 0 0 0 0 -1 1 1 1 1 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 1 1 1 0 0 0 0 0 -1 1 1 1 0 -1 -1 0 1 1 1 0 0 0 0 -1 -1 0 1 1 0 -1 -1 0 1 1 end * Computation started at Sun Apr 22 10:26:56 2018 * ended at Sun Apr 22 10:26:56 2018 * Total processor time = 0 seconds * %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %**************** pentagon/pentagram ************************************************************** %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pasting of 5 blocks 10 atoms 5 blocks 0 proper subsets of blocks 3 1 2 3 3 3 4 5 3 5 6 7 3 7 8 9 3 9 10 1 11 2-valued evaluations of atoms: 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 set of 2-valued evaluations of atoms: nonempty: yes unital: yes separating atoms: yes separating: yes 1s on nonorthogonal atoms (=OD if no noncomplete block): yes order determining: yes Blocks[{ {1,0,0,1,0,1,0,1,0,0}, {1,0,0,1,0,0,1,0,0,0}, {1,0,0,0,1,0,0,1,0,0}, {0,1,0,1,0,1,0,1,0,1}, {0,1,0,1,0,1,0,0,1,0}, {0,1,0,1,0,0,1,0,0,1}, {0,1,0,0,1,0,0,1,0,1}, {0,1,0,0,1,0,0,0,1,0}, {0,0,1,0,0,1,0,1,0,1}, {0,0,1,0,0,1,0,0,1,0}, {0,0,1,0,0,0,1,0,0,1} }] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %**************** Specker bug combo ************************************************************** %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'set linend on' 'c/a§ / 25 / * *' 'c/b§ / 26 / * *' 'c/a / 12 / * *' 'c/b / 13 / * *' 'c/c / 27 / * *' 'c/2§ / 15 / * *' 'c/3§ / 16 / * *' 'c/4§ / 17 / * *' 'c/5§ / 18 / * *' 'c/6§ / 19 / * *' 'c/7§ / 20 / * *' 'c/8§ / 21 / * *' 'c/9§ / 22 / * *' 'c/10§ / 23 / * *' 'c/11§ / 24 / * *' 'c/1§ / 14 / * *' %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Specker bug combo in 3 dim in dimension 3 a=12 b= 13 1'=1+13 .... 11'=11+13 a'= 25 b'=26 c=27 27 atoms 16 blocks 0 proper subsets of blocks 3 12 1 2 3 2 3 4 3 4 5 13 3 13 6 7 3 7 8 9 3 9 10 12 3 3 8 11 3 12 27 26 3 13 27 25 3 25 14 15 3 15 16 17 3 17 18 26 3 26 19 20 3 20 21 22 3 22 23 25 3 16 21 24 82 2-valued evaluations of atoms: 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Specker bug extension in 3 dim in dimension 3 a=12 b=13 a'= 14 b'=15 c=16 with TITS 16 atoms 9 blocks 0 proper subsets of blocks 3 12 1 2 3 2 3 4 3 4 5 13 3 13 6 7 3 7 8 9 3 9 10 12 3 3 8 11 3 12 16 15 3 13 16 14 22 2-valued evaluations of atoms: 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 set of 2-valued evaluations of atoms: nonempty: yes unital: yes separating atoms: yes separating: yes 1s on nonorthogonal atoms (=OD if no noncomplete block): no for atoms 12/13 order determining: no for sets of atoms (ordered elements?) 13/1+2 13+16/1+2 12/4+5 12/14 13/15 12+16/4+5 MatrixForm[Transpose[{ {0,0,1,0,1,0,1,0,0,0,0,1,0,1,0,0}, {0,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0}, {0,0,0,1,0,0,1,0,0,0,1,1,0,1,0,0}, {1,0,1,0,1,1,0,0,1,0,0,0,0,0,0,1}, {1,0,1,0,1,1,0,0,1,0,0,0,0,1,1,0}, {1,0,1,0,1,0,1,0,0,1,0,0,0,0,0,1}, {1,0,1,0,1,0,1,0,0,1,0,0,0,1,1,0}, {1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0}, {1,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1}, {1,0,0,1,0,1,0,1,0,1,0,0,0,1,1,0}, {1,0,0,1,0,1,0,0,1,0,1,0,0,0,0,1}, {1,0,0,1,0,1,0,0,1,0,1,0,0,1,1,0}, {1,0,0,1,0,0,1,0,0,1,1,0,0,0,0,1}, {1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0}, {0,1,0,0,1,1,0,1,0,1,0,0,0,0,0,1}, {0,1,0,0,1,1,0,1,0,1,0,0,0,1,1,0}, {0,1,0,0,1,1,0,0,1,0,1,0,0,0,0,1}, {0,1,0,0,1,1,0,0,1,0,1,0,0,1,1,0}, {0,1,0,0,1,0,1,0,0,1,1,0,0,0,0,1}, {0,1,0,0,1,0,1,0,0,1,1,0,0,1,1,0}, {0,1,0,0,0,0,0,1,0,1,0,0,1,0,1,0}, {0,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0}}]] { {0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0}, {1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1}, {1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1}, {0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Specker bug combo in 3 dim in dimension 3 a=12 b=13 with TIFS 13 atoms 7 blocks 0 proper subsets of blocks 3 12 1 2 3 2 3 4 3 4 5 13 3 13 6 7 3 7 8 9 3 9 10 12 3 3 8 11 14 2-valued evaluations of atoms: 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 1 set of 2-valued evaluations of atoms: nonempty: yes unital: yes separating atoms: yes separating: yes 1s on nonorthogonal atoms (=OD if no noncomplete block): no for atoms 12/13 order determining: no for sets of atoms (ordered elements?) 13/1+2 12/4+5 MatrixForm[Transpose[{ {0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0}, {1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1}}]] { {0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1}, {1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0}, {0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0}, {0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1}, {1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %**************** ACS *************************************************************************** %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a 1 2 b 2 3 4 a 5 b 6 7 7 10 4 5 29 23 3 21 23 4 28 22 22 19 3 b 8 9 9 11 5 10 12 13 13 31 29 28 30 15 15 14 11 6 33 17 17 20 21 7 34 27 27 26 23 22 24 25 25 35 9 15 17 1 13 16 1 16 18 19 16 32 8 25 1 27 ACS in dimension 3 a=36, b=37 37 atoms 26 blocks 0 proper subsets of blocks 3 36 1 2 3 37 2 3 3 4 36 5 3 37 6 7 3 7 10 4 3 5 29 23 3 3 21 23 3 4 28 22 3 22 19 3 3 37 8 9 3 9 11 5 3 10 12 13 3 13 31 29 3 28 30 15 3 15 14 11 3 6 33 17 3 17 20 21 3 7 34 27 3 27 26 23 3 22 24 25 3 25 35 9 3 15 17 1 3 13 16 1 3 16 18 19 3 16 32 8 3 25 1 27 8 2-valued evaluations of atoms: 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 set of 2-valued evaluations of atoms: nonempty: yes unital: no for atoms 2 13 15 16 17 25 27 36 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ { {1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1}, {1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1}, {1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1}, {1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1}, {1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0}, {1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0}, {1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0} } { {1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 1, 1, 1}, {1, 1, 0, 0, 1, 1, 0, 0}, {0, 0, 1, 1, 0, 0, 1, 1}, {0, 0, 0, 0, 1, 1, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0, 1, 1}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 1, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 1, 0, 0, 0}, {1, 1, 0, 0, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 1, 1, 1, 1}, {1, 0, 1, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 1, 1, 0, 0}, {1, 1, 0, 1, 1, 1, 0, 0}, {0, 1, 1, 1, 0, 0, 1, 1}, {1, 1, 1, 1, 0, 1, 0, 0}, {1, 1, 1, 1, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 0}, {1, 1, 1, 1, 1, 0, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0}, {1, 1, 1, 1, 0, 0, 0, 0} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% reduced ACS-TIF in dimension 3 a=29, b=31 35 atoms 24 blocks 0 proper subsets of blocks 3 29 1 2 3 31 2 3 3 4 29 5 3 31 6 7 3 7 10 4 3 3 21 23 3 4 28 22 3 22 19 3 3 31 8 9 3 9 11 5 3 10 12 13 3 28 30 15 3 15 14 11 3 6 33 17 3 17 20 21 3 7 34 27 3 27 26 23 3 22 24 25 3 25 35 9 3 15 17 1 3 13 16 1 3 16 18 19 3 16 32 8 3 25 1 27 13 2-valued evaluations of atoms: 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 set of 2-valued evaluations of atoms: nonempty: yes unital: no for atoms 16 MatrixForm[Transpose[ { {0,0,1,0,0,0,1,1,0,0,1,0,1,0,0,0,1,1,0,0,0,0,0,0,1,1,0,1,1,0,0,0,0,0,0}, {1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,1,0,0,1,0,1,0,0,0,1,1,1,1,1,1}, {1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0,0,0,0,1,1,1,1,1,1}, {1,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,1,1,1,1}, {1,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,0,1,1,1,1,1,1}, {1,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,1,1,0,0,1,1,0,0,0,1,0,0,1,1,1,1,1}, {1,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,1,0,0,0,0,0,0,1,1,1,1,1,1}, {1,0,1,1,0,1,0,1,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,1,1}, {1,0,1,1,0,1,0,0,1,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0}, {1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0,0,0,0,0,1,1}, {1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0,0,0,0,1,0,1}, {0,1,0,1,0,1,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0,0,0}, {0,1,0,0,1,0,1,1,0,0,0,0,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0,1,0,0,0,0,0,0,0} } ]] { {0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0}, {0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0}, {0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0}, {1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0}, {0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1}, {0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0}, {0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0}, {0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0}, {0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0}, {0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% reduced ACS with TITS in dimension 3 a=10 b=12 35 atoms 24 blocks 0 proper subsets of blocks 3 10 1 2 3 12 2 3 3 4 10 5 3 12 6 7 3 5 29 23 3 3 21 23 3 4 28 22 3 22 19 3 3 12 8 9 3 9 11 5 3 13 31 29 3 28 30 15 3 15 14 11 3 6 33 17 3 17 20 21 3 7 34 27 3 27 26 23 3 22 24 25 3 25 35 9 3 15 17 1 3 13 16 1 3 16 18 19 3 16 32 8 3 25 1 27 13 2-valued evaluations of atoms: 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 set of 2-valued evaluations of atoms: nonempty: yes unital: no for atoms 16 MatrixForm[Transpose[ { {0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,0,0,0,1,0,1,0}, {1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,1,0,0,1,0,1,0,0,1,1,0,1,1,1,1}, {1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0,0,0,0,1,1,1,1,1,1}, {1,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,1,1,1,1}, {1,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,0,1,1,1,1,1,1}, {1,0,1,1,0,1,0,1,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,1,0,0,0,1,1}, {1,0,1,1,0,1,0,0,1,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,1,0,1,0,1,0}, {1,0,1,1,0,0,1,1,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,1,0,0,1,0,1}, {1,0,1,1,0,0,1,0,1,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,1,0,1,1,0,0}, {1,0,1,0,1,1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0,0,1,0,0,1,1}, {1,0,1,0,1,0,1,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0,0,1,0,1,0,1}, {0,1,0,1,0,0,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,0,0,0,0,0}, {0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,0,1,1,0,0,0,0,1,0,0,0,0,0,0,0,1} } ]] { {0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0}, {0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0}, {0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1}, {0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0}, {0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0}, {1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1}, {1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0}, {0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0}, {0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0}, {1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0}, {0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0}, {1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0}, {0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ACCS in dimension 3 38 atoms 24 blocks 0 proper subsets of blocks 3 1 2 3 3 3 4 5 3 5 6 7 3 7 8 9 3 9 10 11 3 11 12 1 3 26 27 2 3 2 13 17 3 17 18 19 3 16 19 20 3 13 14 15 3 27 28 29 3 29 15 16 3 4 22 23 3 20 24 25 3 2 25 38 3 9 36 21 3 21 32 34 3 12 31 32 3 26 30 31 3 12 23 33 3 33 35 36 3 10 37 38 3 3 16 21 99 2-valued evaluations of atoms: 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 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1 0 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 set of 2-valued evaluations of atoms: nonempty: yes unital: yes separating atoms: yes separating: yes 1s on nonorthogonal atoms (=OD if no noncomplete block): no for atoms 1/7 1/15 1/19 1/20 1/21 1/29 3/33 10/33 13/21 15/25 17/21 19/27 21/25 21/26 21/27 21/38 order determining: no for sets of atoms (ordered elements?) 3/12+23 21/2 7/2+3 19/2+3 20/2+3 19+20/2+3 15/2+3 29/2+3 21/2+3 1/5+6 1/17+18 1/16 1/16+19 1/13+14 1/27+28 1/9+36 1+3/9+36 33/1+2 10/12+23 33/9+11 27/17+18 27/9+36 21/2+27 26/9+36 19/2+26 21/2+26 17/9+36 13/9+36 21/2+17 21/2+13 15/20+24 25/13+14 25/9+36 21/20+24 38/9+36 21/2+25 Pit-G3 logic in dimension 3 26 atoms 14 blocks 0 proper subsets of blocks 3 1 2 3 3 3 4 5 3 5 6 7 3 7 8 9 3 9 10 11 3 11 12 13 3 13 14 15 3 15 16 1 3 1 17 18 3 18 19 20 3 20 21 22 3 1 26 25 3 23 24 25 3 10 22 23 Tkadlec-unital logic in dimension 3 36 atoms 26 blocks 0 proper subsets of blocks 3 1 2 3 3 1 4 5 3 4 6 7 3 8 9 31 3 7 8 23 3 17 9 2 3 7 33 10 3 28 17 18 3 5 18 19 3 19 20 31 3 19 21 23 3 4 14 15 3 15 20 26 3 14 17 35 3 13 15 16 3 16 22 36 3 21 22 27 3 3 13 23 3 2 22 24 3 11 12 13 3 10 11 34 3 24 25 32 3 5 11 25 3 9 12 29 3 6 24 30 3 2 10 20 6 2-valued evaluations of atoms: 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 1 0 1 0 1 set of 2-valued evaluations of atoms: nonempty: yes unital: no for atoms 1 3 9 10 17 20 22 24 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'set linend on' 'c/,/ /* *' 'c/0/+0\lambda_{}/* *' 'c/1/+1\lambda_{}/* *' 'c/{}/1/ * 1 ' 'c/{}/2/ * 1 ' 'c/{}/3/ * 1 ' 'c/{}/4/ * 1 ' 'c/{}/5/ * 1 ' 'c/{}/6/* 1 ' 'c/{}/7/* 1 ' 'c/{}/8/* 1 ' 'c/{}/9/* 1 ' 'c/{}/{10}/* 1 ' 'c/{}/{11}/* 1 ' 'c/{}/{12}/* 1 ' 'c/{}/{13}/* 1 ' 'c/} //* *' 'c/{+/p_{ 1}= / 1 * ' 'c/{+/p_{ 2}= / 2 * ' 'c/{+/p_{ 3}= / 3 * ' 'c/{+/p_{ 4}= / 4 * ' 'c/{+/p_{ 5}= / 5 * ' 'c/{+/p_{ 6}= / 6 * ' 'c/{+/p_{ 7}= / 7 * ' 'c/{+/p_{ 8}= / 8 * ' 'c/{+/p_{ 9}= / 9 * ' 'c/{+/p_{10}= / 10 * ' 'c/{+/p_{11}= / 11 * ' 'c/{+/p_{12}= / 12 * ' 'c/{+/p_{13}= / 13 * ' 'c/{+/p_{14}= / 14 * ' 'c/{+/p_{15}= / 15 * ' 'c/{+/p_{16}= / 16 * ' 'c/{+/p_{17}= / 17 * ' 'c/{+/p_{18}= / 18 * ' 'c/{+/p_{19}= / 19 * ' 'c/{+/p_{20}= / 20 * ' 'c/{+/p_{21}= / 21 * ' 'c/{+/p_{22}= / 22 * ' 'c/{+/p_{23}= / 23 * ' 'c/{+/p_{24}= / 24 * ' 'c/{+/p_{25}= / 25 * ' 'c/{+/p_{26}= / 26 * ' 'c/{+/p_{27}= / 27 * ' 'c/{+/p_{28}= / 28 * ' 'c/{+/p_{29}= / 29 * ' 'c/{+/p_{30}= / 30 * ' 'c/{+/p_{31}= / 31 * ' 'c/{+/p_{32}= / 32 * ' 'c/{+/p_{33}= / 33 * ' 'c/{+/p_{34}= / 34 * ' 'c/{+/p_{35}= / 35 * ' 'c/{+/p_{36}= / 36 * ' 'c/{+/p_{37}= / 37 * ' 'c/0\lambda_1/ /* * ' 'c/+0\lambda_2/ /* * ' 'c/+0\lambda_3/ /* * ' 'c/+0\lambda_4/ /* * ' 'c/+0\lambda_5/ /* * ' 'c/+0\lambda_6/ /* * ' 'c/+0\lambda_7/ /* * ' 'c/+0\lambda_8/ /* * ' 'c/+0\lambda_9/ /* * ' 'c/+0\lambda_{10}/ /* * ' 'c/+0\lambda_{11}/ /* * ' 'c/+0\lambda_{12}/ /* * ' 'c/+0\lambda_{13}/ /* * ' 'c/1\lambda/\lambda/* * ' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/= /=/* *' 'c/=+/=/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/ +/+/* *' 'c/=+/=/* *' 'c/} /} ,\\/* *' %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ( **************************** Brute force enumeration of blocks/contexts, given 2-valued states *********************************) Blocks[x_] := ( (* number of 2-valued states *) n2s = Length[x]; (* number of atoms *) y = Transpose[x]; na = Length[y]; Print[n2s," two valued states"]; Print[na," atoms"]; (* reducing atoms by dropping unreachable ones: all 0 or all 1 *) yy = Union[y]; If[First[yy] == Table[0, {i, 1, n2s}], yyy = Drop[yy, 1]; yy = yyy, 0]; If[Last[yy] == Table[1, {i, 1, n2s}], yyy = Drop[yy, {Length[yy]}, 0]; yy = yyy, 0]; y=yy; na = Length[y]; Print[na," reachable atoms"]; pl={}; Do[ If[ y[[i1]] == Table[ 1 ,{i,1,n2s} ], (* Print[i1 ]; *) plaux=Append[pl,{i1}]; pl=plaux ,0]; Do[ If[ y[[i1]] + y[[i2]] == Table[ 1 ,{i,1,n2s} ], (* Print[i1,"-",i2]; *) plaux=Append[pl,{i1,i2}]; pl=plaux ,0]; Do[ If[ y[[i1]] + y[[i2]] + y[[i3]] == Table[ 1 ,{i,1,n2s} ], (* Print[i1,"-",i2,"-",i3 ]; *) plaux= Append[pl,{i1,i2,i3}]; pl=plaux,0]; Do[ If[ y[[i1]] + y[[i2]]+ y[[i3]] + y[[i4]] == Table[ 1 ,{i,1,n2s} ], (* Print[i1,"-",i2,"-",i3,"-",i4];*) plaux=Append[pl,{i1,i2,i3,i4}]; pl=plaux,0]; Do[ If[ y[[i1]] + y[[i2]]+ y[[i3]]+ y[[i4]]+ y[[i5]] == Table[ 1 ,{i,1,n2s} ], (* Print[i1,"-",i2,"-",i3,"-",i4,"-",i5]; *) plaux=Append[pl,{i1,i2,i3,i4,i5}]; pl=plaux,0]; ,{i5,i4+1,na}] ,{i4,i3+1,na}] ,{i3,i2+1,na}] ,{i2,i1+1,na}] ,{i1,1,na}]; Print[Length[Union[pl]]," secondary partitions"]; Print[Union[pl]]; ) %%%%%%%%%%%%%%%%%%%%% triangle D=4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Blocks[ { {1,0,0,0,1,0,0,0,0}, {1,0,0,0,0,1,0,0,0}, {0,1,0,0,1,0,0,1,0}, {0,1,0,0,1,0,0,0,1}, {0,1,0,0,0,1,0,1,0}, {0,1,0,0,0,1,0,0,1}, {0,1,0,0,0,0,1,0,0}, {0,0,1,0,1,0,0,1,0}, {0,0,1,0,1,0,0,0,1}, {0,0,1,0,0,1,0,1,0}, {0,0,1,0,0,1,0,0,1}, {0,0,1,0,0,0,1,0,0}, {0,0,0,1,0,0,0,1,0}, {0,0,0,1,0,0,0,0,1} } ] 9 reachable atoms 3 secondary partitions {{1,2,6,9},{1,3,7,8},{3,4,5,9}} %%%%%%%%%%%%%%%%%%%%% square D=4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Blocks[ { {1,0,0,0,1,0,0,1,0,0,0,0}, {1,0,0,0,1,0,0,0,1,0,0,0}, {1,0,0,0,0,1,0,1,0,0,0,0}, {1,0,0,0,0,1,0,0,1,0,0,0}, {1,0,0,0,0,0,1,0,0,0,0,0}, {0,1,0,0,1,0,0,1,0,0,1,0}, {0,1,0,0,1,0,0,1,0,0,0,1}, {0,1,0,0,1,0,0,0,1,0,1,0}, {0,1,0,0,1,0,0,0,1,0,0,1}, {0,1,0,0,1,0,0,0,0,1,0,0}, {0,1,0,0,0,1,0,1,0,0,1,0}, {0,1,0,0,0,1,0,1,0,0,0,1}, {0,1,0,0,0,1,0,0,1,0,1,0}, {0,1,0,0,0,1,0,0,1,0,0,1}, {0,1,0,0,0,1,0,0,0,1,0,0}, {0,1,0,0,0,0,1,0,0,0,1,0}, {0,1,0,0,0,0,1,0,0,0,0,1}, {0,0,1,0,1,0,0,1,0,0,1,0}, {0,0,1,0,1,0,0,1,0,0,0,1}, {0,0,1,0,1,0,0,0,1,0,1,0}, {0,0,1,0,1,0,0,0,1,0,0,1}, {0,0,1,0,1,0,0,0,0,1,0,0}, {0,0,1,0,0,1,0,1,0,0,1,0}, {0,0,1,0,0,1,0,1,0,0,0,1}, {0,0,1,0,0,1,0,0,1,0,1,0}, {0,0,1,0,0,1,0,0,1,0,0,1}, {0,0,1,0,0,1,0,0,0,1,0,0}, {0,0,1,0,0,0,1,0,0,0,1,0}, {0,0,1,0,0,0,1,0,0,0,0,1}, {0,0,0,1,0,0,0,1,0,0,1,0}, {0,0,0,1,0,0,0,1,0,0,0,1}, {0,0,0,1,0,0,0,0,1,0,1,0}, {0,0,0,1,0,0,0,0,1,0,0,1}, {0,0,0,1,0,0,0,0,0,1,0,0} } ] 34 two valued states 12 reachable atoms 4 secondary partitions {{1,2,6,12},{1,7,8,11},{3,4,5,12},{3,7,9,10}} %%%%%%%%%%%%%%%%%%%%% pentagon %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Blocks[{ {1,0,0,1,0,1,0,1,0,0}, {1,0,0,1,0,0,1,0,0,0}, {1,0,0,0,1,0,0,1,0,0}, {0,1,0,1,0,1,0,1,0,1}, {0,1,0,1,0,1,0,0,1,0}, {0,1,0,1,0,0,1,0,0,1}, {0,1,0,0,1,0,0,1,0,1}, {0,1,0,0,1,0,0,0,1,0}, {0,0,1,0,0,1,0,1,0,1}, {0,0,1,0,0,1,0,0,1,0}, {0,0,1,0,0,0,1,0,0,1} }] 5 secondary partitions {{1,4,10},{1,5,9},{2,3,10},{2,6,8},{5,6,7}} %%%%%%%%%%%%%%%%%%%%%%%% Specker Bug %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Blocks[ { {0,0,1,0,1,0,1,0,0,0,0,1,0}, {0,0,0,1,0,1,0,1,0,0,0,1,0}, {0,0,0,1,0,0,1,0,0,0,1,1,0}, {1,0,1,0,1,1,0,0,1,0,0,0,0}, {1,0,1,0,1,0,1,0,0,1,0,0,0}, {1,0,1,0,0,0,0,0,1,0,0,0,1}, {1,0,0,1,0,1,0,1,0,1,0,0,0}, {1,0,0,1,0,1,0,0,1,0,1,0,0}, {1,0,0,1,0,0,1,0,0,1,1,0,0}, {0,1,0,0,1,1,0,1,0,1,0,0,0}, {0,1,0,0,1,1,0,0,1,0,1,0,0}, {0,1,0,0,1,0,1,0,0,1,1,0,0}, {0,1,0,0,0,0,0,1,0,1,0,0,1}, {0,1,0,0,0,0,0,0,1,0,1,0,1} } ] 7 secondary partitions {{1,5,13},{1,9,11},{2,8,12},{2,9,10},{3,4,13},{4,7,12},{6,7,11}} %%%%%%%%%%%%%%%%%%%%%%%% Extended Specker Bug %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Blocks[ { {0,0,1,0,1,0,1,0,0,0,0,1,0,1,0,0}, {0,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0}, {0,0,0,1,0,0,1,0,0,0,1,1,0,1,0,0}, {1,0,1,0,1,1,0,0,1,0,0,0,0,0,0,1}, {1,0,1,0,1,1,0,0,1,0,0,0,0,1,1,0}, {1,0,1,0,1,0,1,0,0,1,0,0,0,0,0,1}, {1,0,1,0,1,0,1,0,0,1,0,0,0,1,1,0}, {1,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0}, {1,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1}, {1,0,0,1,0,1,0,1,0,1,0,0,0,1,1,0}, {1,0,0,1,0,1,0,0,1,0,1,0,0,0,0,1}, {1,0,0,1,0,1,0,0,1,0,1,0,0,1,1,0}, {1,0,0,1,0,0,1,0,0,1,1,0,0,0,0,1}, {1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0}, {0,1,0,0,1,1,0,1,0,1,0,0,0,0,0,1}, {0,1,0,0,1,1,0,1,0,1,0,0,0,1,1,0}, {0,1,0,0,1,1,0,0,1,0,1,0,0,0,0,1}, {0,1,0,0,1,1,0,0,1,0,1,0,0,1,1,0}, {0,1,0,0,1,0,1,0,0,1,1,0,0,0,0,1}, {0,1,0,0,1,0,1,0,0,1,1,0,0,1,1,0}, {0,1,0,0,0,0,0,1,0,1,0,0,1,0,1,0}, {0,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0} } ] 9 secondary partitions {{1,7,15},{1,11,13},{2,5,16},{2,10,14},{2,11,12},{3,6,15},{4,5,15},{6,9,14},{8,9,13}} %%%%%%%%%%%%%%%%%%%%%%%%% Bug combo %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Blocks[ { {0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0}, {1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0}, {1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1}, {0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0}, {0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0}, {0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0} } ] 82 two valued states 27 atoms 25 reachable atoms 15 secondary partitions {{1,10,25},{1,14,23},{2,9,25},{2,13,24},{2,14,22}, {2,17,21},{2,18,20},{3,8,25},{4,7,25},{4,18,19}, {5,6,25},{6,16,21},{8,12,24},{11,12,23},{15,16,19}} %%%%%%%%%%%%%%%%%%%%%%%%%% ACS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Blocks[ { {1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1}, {1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1}, {1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1}, {1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1}, {1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0}, {1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0}, {1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0}, {1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0} } ] 8 two valued states 37 atoms 28 nontrivial atoms 30 secondary partitions { {1,28},{2,27},{3,26},{4,25},{5,24},{6,23},{7,22},{8,21},{9,20},{10,19},{11,18},{12,17},{13,16},{14,15}, {1,4,24},{1,9,19},{2,3,24},{2,10,18},{5,6,22},{5,12,16},{6,8,19},{10,12,15}, {1,2,9,18},{1,4,6,22},{1,4,12,16},{1,9,12,15},{2,3,6,22},{2,3,12,16},{2,6,8,18},{6,8,12,15} } %%%%%%%%%%%%%%%%%%%%%%%% Tkadlec non-unital logic based on Schuette rays %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Blocks[{ {0,1,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,1,1,1,0,0,1,1,1,1}, {0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0,0,0,1,0,0,1,1,0,1,1,1,1,1,0,1,0}, {0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,1,0,1,1,1,0,0,1,1,0,1,1,1,0}, {0,1,0,0,1,1,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,1,0,0,1,1,1,0,0,1,1,1,1,0,0}, {0,1,0,0,1,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,1,1,1,1,1,1}, {0,1,0,0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,1,1,1,1,0,1,0,1} } ] 6 two valued states 24 reachable atoms 24 reachable atoms 26 secondary partitions {{1,24},{2,23},{3,22},{4,21},{5,20},{6,19},{7,18},{8,17},{9,16},{10,15},{11,14},{12,13}, {1,4,20},{1,11,13},{2,3,20},{2,10,14},{4,9,14},{5,7,17},{5,9,13},{6,7,14}, {1,2,10,13},{1,4,7,17},{1,4,9,13},{1,6,7,13},{2,3,7,17},{2,3,9,13}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% { {0,1,0,0,1}, {1,0,1,0,0}, {0,1,0,1,0}, {0,0,1,0,1}, {1,0,0,1,0} } A={ {0,0,0,0,0,0,0,1}, {1,0,0,0,0,0,0,0}, {0,1,0,0,0,0,1,0}, {0,0,1,0,0,0,0,0}, {0,0,0,1,0,0,0,0}, {0,0,0,0,1,0,0,0}, {0,0,0,0,0,1,0,0}, {0,0,0,0,0,0,1,0} } AdjacencyGraph[( A+Transpose[A] ), PlotTheme -> "Business"]