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Journal of Logic, Language and Information

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01.12.2014

Logical Geometries and Information in the Square of Oppositions

verfasst von: Hans Smessaert, Lorenz Demey

Erschienen in: Journal of Logic, Language and Information | Ausgabe 4/2014

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Abstract

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams (such as the square) and, on a more abstract level, the Aristotelian geometry (a set of logical relations). We then introduce two new logical geometries (and their corresponding diagrams), and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams.

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For the sake of readability, some technical remarks and results have been placed in an Appendix; they are not essential for our main line of argumentation.

For a more exhaustive historical overview, see Parsons (2006) and Seuren (2010, chapter 5).

In this paper, the term ‘Aristotelian’ is used in a strictly technical sense, to distinguish the Aristotelian geometry and its diagrams from other kinds of geometries and diagrams that will be introduced later. Hence, by calling a certain relation ‘Aristotelian’ we do not mean to imply that Aristotle himself acknowledged that relation; similarly, by calling a certain diagram ‘Aristotelian’ we do not mean to imply that Aristotle himsef drew such a diagram, or even defended its validity. (For a detailed account of the historical origins of the square, see Londey and Johanson (1984).) Finally, the discussion of the problems in the Aristotelian geometry in Sect. 3.1 should not be seen as a piece of historical Aristotle scholarship, but rather as the systematic development of a new perspective on some issues in contemporary logic.

It is well-known that in the presence of classical negation, each of \(\wedge \) and \(\rightarrow \) can be defined in terms of the other: \(\varphi \rightarrow \psi = \lnot (\varphi \wedge \lnot \psi )\), and \(\varphi \wedge \psi = \lnot (\varphi \rightarrow \lnot \psi )\). We will return to this remark in Sect. 3.3.

It is well-known that \(\lnot (\lnot \varphi \wedge \lnot \psi )\) is equivalent to \(\varphi \vee \psi \), but we choose to stick with the first notation, because it more clearly expresses the idea of \(\varphi \) and \(\psi \) being false together.

It should be clear that we do not view the Aristotelian relations in terms of properties of the formulas they relate such as quantity and quality, as is done in many historical studies on Aristotelian logic (Parry and Hacker 1991). For more about this difference, see Demey (2012c, pp. 328–329).

So \({\mathsf {S}} \not \models \varphi _i\), \({\mathsf {S}} \not \models \lnot \varphi _i\), and \({\mathsf {S}}\not \models \varphi _i\leftrightarrow \varphi _j\), for \(1 \le i \ne j \le n\).

It follows immediately from Definition 1 that the first three relations are symmetric, and are therefore represented in Fig. 1 by lines without arrows. We represent \(\varphi \) and \(\psi \) being in subalternation by means of an arrow going from \(\varphi \) to \(\psi \), classically referred to as the ‘superaltern’ and ‘subaltern’, respectively.

Tautologies are subaltern and subcontrary to any contingent formula. Conversely, contradictions are superaltern and contrary to any contingent formula.

For a more detailed discussion of the connection between Aristotelian diagrams and Hasse diagrams, see Smessaert (2009) and Demey and Smessaert (2014).

The operators \({\mathsf {O}}\) and \({\mathsf {P}}\) in the deontic square stand for ‘obligatory’ and ‘permitted’, respectively.

Note that we will not consider squares for the quantifiers, and thus sidestep the notoriously difficult issue of existential import (Chatti and Schang 2013; Parsons 2006; Read 2012a; Seuren 2012a), since the informativity account to be developed here is entirely independent of it.

Note that Fig. 3a contains the formula \(\Diamond p\wedge \Diamond \lnot p\), which is usually taken to express the (metaphysical) contingency of \(p\). This notion of contingency is distinct from the logical notion of contingency that is used in Definition 2 of Aristotelian diagram (see Footnote 7). Although the formula \(\Box p\) expresses that \(p\) is metaphysically necessary, it is itself logically contingent (i.e., \({\mathsf {S5}}\not \models \Box p\) and \({\mathsf {S5}}\not \models \lnot \Box p\)), and is thus perfectly allowed to occur in Aristotelian diagrams. Finally, note that the octagon in Fig. 3c can be seen as the ‘sum’ of the hexagons in Fig. 3a and b.

This does not mean that these extensions do not have any applications at all. For example, Horn (1990) uses various hexagons to study Gricean maxims and conversational implicatures, while Jaspers (2012) uses the Sesmat-Blanché hexagon to analyze the structure of the color categories from a logical, linguistic and cognitive perspective.

For example, this intuition seems to be implicit in the following remarks: “familiarity with the square is useful for logicians today as a kind of lingua franca, when adapted as a shorthand to express logical relations in specialized applied logics with specialized domains” (Jacquette 2012, p. 81), and “the square [...] is a compact way of representing various logical relations between formulas, and thus serves as an illustration of the underlying logic’s expressive and deductive powers” (Demey 2012c, p. 314).

Note that both examples involve a non-contingent formula. This is not a coincidence: if we focus on contingent formulas, the Aristotelian relations are mutually exclusive (Demey 2012c, Lemma 3.2).

Non-contradiction is clearly different from the Aristotelian relation of subalternation. First of all, there exist pairs of formulas—such as \((p,\lnot \lnot p)\) and \((p,q)\)—which are in non-contradiction, but not in subalternation. Furthermore, if two contingent formulas \(\varphi \) and \(\psi \) are in subalternation, they will also be in non-contradiction, but that characterization would miss the key point that the truth values of \(\varphi \) and \(\psi \) are not independent (if \(\varphi \) is true, then \(\psi \) has to be true as well).

‘Opposition geometry’ (Definition 3) is a technical term, on a par with ‘Aristotelian geometry’ (Definition 1) and ‘implication geometry’ (Definition 4), and should thus not be confused with the general framework of oppositional geometry developed by Moretti (2012b). Furthermore, note that Definition 3 is similar in spirit to Moretti (2009a, 2012a) and Schang (2012)’s ‘question–answer semantics’; however, they propose this as a semantics for the Aristotelian geometry, and thus fail to fully distinguish between the relations of subalternation and non-contradiction (recall Footnote 17). Recently, however, Schang (2012a, 2013) has shown that subalternation can be understood in terms of ‘contradictories of contraries’ (in the sense that \(SA(\varphi ,\psi )\) iff there is a formula \(\theta \) such that \(C(\varphi ,\theta )\) and \( CD (\theta ,\psi )\); see item 6b of our Lemma 3), and used this fact to argue that subalternation is an Aristotelian relation after all.

Obviously, the indices come from the canonical way of displaying a truth table for a binary connective.

The contrast within both geometries between (three kinds of) black lines on the one hand and a grey line on the other is motivated by informativity considerations that will be presented later in the paper.

The arrow’s head indicates the direction of truth propagation. In the case of \( LI \), this direction matches the direction of the arrow itself, but in the case of \( RI \), they differ. For example, \( LI (\Box p, p)\) is visualized as \(\Box p\) —\(\blacktriangleright p\), because both the \( LI \)-relation and truth propagation go from \(\Box p\) to \(p\); however, \( RI (\Diamond p, p)\) is visualized as \(\Diamond p\) —\(\blacktriangleleft p\), because the \( RI \)-relation goes from \(\Diamond p\) to \(p\), but truth is propagated from \(p\) to \(\Diamond p\).

Note that this set includes the original Aristotelian relations, i.e., \({\fancyscript{AG}} \subseteq {\fancyscript{G}}\).

For more background on group theory, see Rotman (1995), in particular p. 55ff. and p. 345ff.

These facts were already known by the thirteenth century logician Peter of Spain, who called them the ‘law of contraries’ (Horn 2010).

Lemma 3 consists of an a- and a b-series, which describe the effects of negating the first, resp. the second argument of a given relation. The symmetry breaking/creating required to connect the opposition and implication geometries is manifested in the fact that exactly one argument is negated. This is to be contrasted with Lemma 2, in which both arguments are negated, and the geometries are kept apart (opposition relations are connected with opposition relations, implication relations with implication relations).

It should be emphasized that the rich structure of the opposition and implication geometries does not primarily consist in the individual items of Lemmas 1–3, but rather in the fact that they interact with each other in interesting ways. These interactions can concisely be described using the language of group theory, as illustrated in Remarks 3–4 and Remarks 7–8.

Definition 6 might look cumbersome, because it involves quantifying over formulas. However, it follows immediately from Definitions 3–4 that if \(R(\varphi ,\psi )\), \(S(\varphi ,\psi )\), \(R(\varphi ',\psi ')\) and \(S(\varphi ',\psi ')\), then for \(1\le i\le 4:\models \lnot \varDelta _i(\varphi ,\psi ) \; \Leftrightarrow \;\; \models \lnot \varDelta _i(\varphi ',\psi ')\). This shows that the quantification over formulas in Definition 6 is ‘innocent’: if there exists at least one pair of formulas \((\varphi ,\psi )\) standing in the relations \(R\) and \(S\) for which it holds that \(\models \lnot \varDelta _i(\varphi ,\psi )\), then this holds for all such pairs of formulas.

Theorem 1 also has a partial converse, which is of less importance for the sake of our argument; more information about this converse can be found in Lemma 7 and Remark 9 in the Appendix.

Of course, the \(\models \)- and \(\not \models \)-parts together are sufficient.

Seuren (2010, p. 49) defines the Aristotelian relations in a similar way. Sanford (1968) compares the usual definition (cf. Definition 1) with that of Bocheński, and judges the former to be preferable.

The information ordering \(\le _i\) is not antisymmetric, because from \(\sigma \le _i \tau \) and \(\tau \le _i \sigma \) it follows that the statements \(\sigma \) and \(\tau \) are equally informative (i.e., \({\mathbb {I}}(\sigma ) = {\mathbb {I}}(\tau )\)), but not that they are identical (i.e., not \(\sigma = \tau \)).

Definition 8 involves a universal quantification over \(\varphi \) and \(\psi \), i.e., it makes use of a \(\forall \forall \)-pattern. One might wonder whether other quantification patterns could be used here. A promising candidate seems to be the \(\exists \exists \)-pattern: define \(R \le _i^\exists S\) iff \(\exists \varphi \in {\fancyscript{L}}_{\mathsf {S}}:\exists \psi \in {\fancyscript{L}}_{\mathsf {S}}:R(\varphi ,\psi ) \le _i S(\varphi ,\psi )\). However, one can show that this \(\le _i^\exists \)-ordering fails to make any distinctions within the two geometries, in the sense that for all \(R,S \in {\fancyscript{OG}}\) it holds that \(R \le _i^\exists S\) (and similarly for \({\fancyscript{IG}}\)). To see this, note that by applying Lemma 4 to the formulas \(p\) and \(\lnot p\), we find for all \(R,S \in {\fancyscript{OG}}\) that \({\mathbb {I}}(R(p,\lnot p)) = {\fancyscript{C}}_{\mathsf {S}} \supseteq {\fancyscript{C}}_{\mathsf {S}}= {\mathbb {I}}(S(p,\lnot p))\), and thus \(R(p,\lnot p) \le _i S(p,\lnot p)\), from which it follows that \(R \le _i^\exists S\). Finally, one might consider the \(\forall \exists \)- and \(\exists \forall \)-patterns, but these asymmetrical patterns have even less intuive appeal. Thanks to an anonymous referee for suggesting us to explore this in more depth.

The only cross-geometry informativity statements that hold are \( NCD \le _i^\forall R\) and \( NI \le _i^\forall R\), for all relations \(R \in {\fancyscript{OG}}\cup {\fancyscript{IG}}\). We will return to such cross-geometry statements in Sect. 5.1; in particular, see Definition 9.

This is reflected in the code in Fig. 5, which visualizes these two relations in grey instead of black (recall Footnote 20).

For the notion of ‘level’ in a Boolean algebra, or, more generally, in a poset, see Engel (1997, p. 7).

More precisely, a formula in level \(L_i\) has 1 contradictory, \(2^{n-i} - 1\) contraries, \(2^i - 1\) subcontraries, and \((2^{n-i} - 1)(2^i - 1)\) noncontradictories. Note that \(1 < \{2^{n-i} - 1, 2^i - 1\} < (2^{n-i}-1)(2^i - 1)\) iff \(1 < i < n - 1\), i.e., a formula yields the right comparative results iff it does not belong to \(L_0, L_1, L_{n-1}\) or \(L_n\). Finally, note that if \(i \approx \frac{n}{2}\), then \(2^{n-i} - 1 \approx 2^{i} - 1\), i.e., formulas sitting (approximately) in the middle level of \({\mathbb {B}}_n\) indeed have (approximately) the same number of contraries and subcontraries.

The opposition relations are thus typically defined in terms of \(\wedge _{\mathbb {B}}\) and \(\vee _{\mathbb {B}}\), while the implication relations are typically defined in terms of \(\le _{\mathbb {B}}\). This suggests that the distinction between the opposition and implication geometries is analogous to the distinction between the algebraic and order-theoretic perspectives on Boolean algebras (Davey and Priestley 2002, pp. 33–41). Furthermore, it is well-known that both perspectives are equivalent to each other (via \(x \le _{\mathbb {B}} y \Leftrightarrow x \wedge _{\mathbb {B}} y = x\)); this is analogous to the connection between the opposition and implication geometries described in Lemma 3.

Theorems 3 and 4 apply only ‘locally’ to \({\fancyscript{OG}}\) and \({\fancyscript{IG}}\), respectively; cf. Footnote 33.

Since this discussion applies to all Aristotelian diagrams, it rightly belongs in this subsection. The next subsection, in contrast, will distinguish between various particular Aristotelian diagrams, e.g. the concrete square, the concrete Sesmat-Blanché hexagon, etc.

This argument is made fully precise in Definitions 11 and 12 and Lemma 8 in the Appendix.

From a theoretical perspective, the case of \( LI / RI \) described above (which is based on the equivalence \( LI (\varphi ,\psi ) \Leftrightarrow RI (\psi ,\varphi )\); cf. Lemma 1) seems to be exactly similar to the case of \(C/ SC \) (which is based on the equivalence \(C(\varphi ,\psi ) \Leftrightarrow SC (\lnot \varphi ,\lnot \psi )\); cf. Lemma 2) and to that of \(C/ LI \) (which is based on the equivalence \(C(\varphi ,\psi ) \Leftrightarrow LI (\varphi ,\lnot \psi )\); cf. Lemma 3). This might suggest that \( SC \) and \( LI \) can unproblematically be left out of the Aristotelian diagrams as well. From a visual perspective, however, the latter two cases are entirely different from the first. The \( LI / RI \) case does not require considering formulas other than \(\varphi \) and \(\psi \); hence, in the diagrams, the \( RI \) relations occur in exactly the same place as the original \( LI \) relations (but in the reverse direction). On the other hand, the \(C/ SC \) and \(C/ LI \) cases require considering formulas other than \(\varphi \) and \(\psi \), viz., (\(\lnot \varphi \) and) \(\lnot \psi \); hence, in the diagrams, the \( SC \) and \( LI \) relations occur in other places than the original \(C\) relations.

Note that Fig. 8 shows the same three diagrams as Fig. 6, but in a different order: we will henceforth put the Aristotelian diagram in between the opposition and implication diagrams, to reflect \({\fancyscript{AG}}\) being hybrid between \({\fancyscript{OG}}\) and \({\fancyscript{IG}}\).

Although its combinatorial and informational properties have not been systematically explored so far, the notion of unconnectedness as such has surfaced at various places in the literature, usually under the label ‘logical independence’; for example, see Hughes (1987, p. 99), Sion (1996, p. 36), Béziau (2003, p. 226) Karger (2003, p. 435), Seuren (2010, p. 50), Campos-Benítez (2012, pp. 101ff.), Jacquette (2012, p. 86) and Read (2012b, p. 104).

For example, Campos-Benítez states that “independent sentences [...] are not contrary neither subcontrary nor contradictory or subaltern: they have no relationship at all” (2012, p. 103).

In other words, Theorem 8 characterizes the absence of any Aristotelian relation in a positive way, viz., as the joint presence of an opposition and an implication relation.

This might explain why some authors, while acknowledging the existence of this relation, deny its logical relevance. For example, according to Seuren, unconnectedness is “a legitimate relation between L-propositions producing truth under certain conditions, yet [...] plays no role [...] in any logic” (2010, p. 50). Similarly, Béziau thinks that by treating unconnectedness as a ‘real’ logical relation, “we are going too far and confusing here negation with distinction” (2003, p. 226).

These diagrams fit into an exhaustive typology that is currently being developed (Smessaert and Demey 2014). This typology includes several other types of Aristotelian diagrams, which have various proportions of unconnectedness among their relations.

Note that \(\fancyscript{F}'_4 = \fancyscript{F}_8 - \fancyscript{F}_{4}\), i.e., the squares in Fig. 13 can be seen as the result of ‘subtracting’ the classical squares in Fig. 9 from the corresponding Béziau octagons in Fig. 12.

Note that the second problem involves contingent formulas (such as \(p\) and \(\Diamond p\wedge \Diamond \lnot p\)), and thus occurs both in the Aristotelian geometry and its diagrams. In contrast, we showed above that non-contingency is a necessary condition for the first problem, which thus never occurs in the diagrams (since these contain only contingent formulas).

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Titel: Logical Geometries and Information in the Square of Oppositions
verfasst von: Hans Smessaert
Lorenz Demey
Publikationsdatum: 01.12.2014
Verlag: Springer Netherlands
Erschienen in: Journal of Logic, Language and Information / Ausgabe 4/2014
Print ISSN: 0925-8531
Elektronische ISSN: 1572-9583
DOI: https://doi.org/10.1007/s10849-014-9207-y

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