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Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams

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Modeling and Using Context (CONTEXT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9405))

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Abstract

This paper studies the logical context-sensitivity of Aristotelian diagrams. I propose a new account of measuring this type of context-sensitivity, and illustrate it by means of a small-scale example. Next, I turn toward a more large-scale case study, based on Aristotelian diagrams for the categorical statements with subject negation. On the practical side, I describe an interactive application that can help to explain and illustrate the phenomenon of context-sensitivity in this particular case study. On the theoretical side, I show that applying the proposed measure of context-sensitivity leads to a number of precise yet highly intuitive results.

Thanks to Hans Smessaert, Margaux Smets and three anonymous referees for their feedback on earlier versions of this paper. The author holds a Postdoctoral Scholarship from the Research Foundation–Flanders (FWO).

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Notes

  1. 1.

    Strictly speaking, the term ‘context-sensitive’ does not apply to the Aristotelian diagram itself, but to the fragment of formulas occurring in that diagram. Throughout this paper, however, I will be using this term both in a strict sense (as applying to fragments of formulas) and in a looser sense (as applying to Aristotelian diagrams).

  2. 2.

    So for all distinct \(\varphi ,\psi \in \mathcal {F}\), it holds that \(\mathsf {S} \not \models \varphi \), \(\mathsf {S} \not \models \lnot \varphi \), \(\mathsf {S}\not \models \varphi \leftrightarrow \psi \), and there exists a \(\varphi '\in \mathcal {F}\) such that \(\mathsf {S}\models \varphi '\leftrightarrow \lnot \varphi \).

  3. 3.

    The set \(\varPi _\mathsf {S}(\mathcal {F})\) is called a ‘partition’ because its elements are (i) jointly exhaustive (\(\mathsf {S}\models \bigvee \varPi _\mathsf {S}(\mathcal {F})\)), and (ii) mutually exclusive (\(\mathsf {S}\models \lnot (\alpha \wedge \beta )\) for distinct \(\alpha ,\beta \in \varPi _\mathsf {S}(\mathcal {F})\)).

  4. 4.

    The bitstring representation of \(\varphi \) is meant to keep track which formulas of \(\varPi _\mathsf {S}(\mathcal {F})\) enter into this disjunction. For example, if \(\varPi _\mathsf {S}(\mathcal {F}) = \{\alpha _1,\alpha _2,\alpha _3,\alpha _4\}\), then \(\varphi \) is represented by the bitstring 1011 iff \(\varphi \equiv _\mathsf {S} \alpha _1 \vee \alpha _3 \vee \alpha _4\).

  5. 5.

    Note the subtly different quantification patterns corresponding to these two limiting cases: minimal context-sensitivity corresponds to \(\exists \ell \in R_{|\mathcal {F}|}:\forall \mathsf {S}\in \mathcal {S}: |\varPi _\mathsf {S}(\mathcal {F})| = \ell \), while maximal context-sensitivity corresponds to \(\forall \ell \in R_{|\mathcal {F}|}:\exists \mathsf {S}\in \mathcal {S}: |\varPi _\mathsf {S}(\mathcal {F})| = \ell \).

  6. 6.

    As is well-known, in the language of first-order logic, these formulas can be formalized as \(\forall x(Ax\rightarrow Bx)\), \(\exists x (Ax\wedge Bx)\), \(\forall x(Ax\rightarrow \lnot Bx)\) and \(\exists x(Ax\wedge \lnot Bx)\), respectively.

  7. 7.

    Later in the paper, I will have more to say about when exactly a logical system can be considered ‘reasonable’ for a given fragment.

  8. 8.

    For example, in this system, the sentence ‘there are at least two As’ would not be formalized as \(\exists x \exists y (Ax \wedge Ay \wedge x\ne y)\), but simply as \(\exists x\exists y (Ax\wedge Ay)\): the syntactic difference between the variables x and y suffices to indicate that there is also a semantic difference between them, i.e. that they have distinct values.

  9. 9.

    We already encountered a similar situation in Sect. 2, where it was shown that each system in \(\mathcal {F}^\dag \) gives rise to a different Aristotelian square for \(\mathcal {F}^\dag \); see Fig. 2.

  10. 10.

    An Aristotelian octagon can be seen as consisting of 4 pairs of contradictory formulas (PCDs), and a square as 2 PCDs. The number of squares inside an octagon thus equals the number of ways in which one can select 2 PCDs out of 4 (without replacement), which is \(\left( {\begin{array}{c}4\\ 2\end{array}}\right) = \frac{4!}{2!2!} = 6\).

  11. 11.

    For reasons of space, a logical system such as \(\mathsf {FOL}(\{A1,A2,A3\})\) is abbreviated as ‘123’, and the bitstring length \(|\varPi _\mathsf {S}(\mathcal {F}^\ddag )|\) as \(\ell _\mathsf {S}\).

  12. 12.

    There certainly do exist systems \(\mathsf {S}\) such that \(|\varPi _{\mathsf {S}}(\mathcal {F}^\ddag )| = 4\). This is the case, for example, for the system \(\mathsf {S}^*\) that is obtained by adding to \(\mathsf {FOL}(\mathcal {AX})\) the additional axiom \(all(A,B)\vee all(A,\lnot B) \vee all(\lnot A,B) \vee all(\lnot A,\lnot B)\). Note, however, that \(\mathsf {S}^* \notin \mathcal {S}^\ddag \), and, more importantly, \(\mathsf {S}^*\) is far less reasonable than any of the systems in \(\mathcal {S}^\ddag \).

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  1. Center for Logic and Analytic Philosophy, KU Leuven, Leuven, Belgium

    Lorenz Demey

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  1. Department of Computer Science, Roskilde University, Roskilde, Denmark

    Henning Christiansen

  2. ENS-Pavillon Jardin, CNRS-Institut Jean-Nicod, Paris, France

    Isidora Stojanovic

  3. Department of Computer Science, University of Cyprus, Aglantzia, Cyprus

    George A. Papadopoulos

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Demey, L. (2015). Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams. In: Christiansen, H., Stojanovic, I., Papadopoulos, G. (eds) Modeling and Using Context. CONTEXT 2015. Lecture Notes in Computer Science(), vol 9405. Springer, Cham. https://doi.org/10.1007/978-3-319-25591-0_24

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