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This discussion explores the possibility of distinguishing a tighter notion of contrariety evident in the Square of Opposition, especially in its modal incarnations, than as that binary relation holding statements that cannot both be true, with or without the added rider ‘though can both be false’. More than one theorist has voiced the intuition that the paradigmatic contraries of the traditional Square are related in some such tighter way—involving the specific role played by negation in contrasting them—that distinguishes them from other pairs of incompatible statements constructed from the same conceptual materials. Prominent among examples, these other nonstandard pairs are the ‘new contraries’ presented by Robert Blanché’s hexagon(s) of opposition. With special, though not exclusive, attention to these cases, we investigate whether contrariety in the distinguished sense can be captured by adding to the incompatibility condition the further demand that the pair of statements concerned can be represented as the results of applying some sentence operator to the content in its scope, for one of the pair, and, for the other, the application of that same operator to the negation of that content. For one of the two cases, a Blanché case, of nonstandard contrariety singled out for attention, the question of whether such a representation is available is settled at the end of Section 4, and then in a more satisfying way in Section 5, though for the other case, noticed by Peter Simons, the question remains open, after some tentative discussion in one subsection, 6.2, of an Appendix (Section 6).
This discussion explores the possibility of distinguishing a tighter notion of contrariety evident in the Square of Opposition, especially in its modal incarnations, than as that binary relation holding statements that cannot both be true, with or without the added rider ‘though can both be false’. More than one theorist has voiced the intuition that the paradigmatic contraries of the traditional Square are related in some such tighter way—involving the specific role played by negation in contrasting them—that distinguishes them from other pairs of incompatible statements constructed from the same conceptual materials. Prominent among examples, these other nonstandard pairs are the ‘new contraries’ presented by Robert Blanché’s hexagon(s) of opposition. With special, though not exclusive, attention to these cases, we investigate whether contrariety in the distinguished sense can be captured by adding to the incompatibility condition the further demand that the pair of statements concerned can be represented as the results of applying some sentence operator to the content in its scope, for one of the pair, and, for the other, the application of that same operator to the negation of that content. For one of the two cases, a Blanché case, of nonstandard contrariety singled out for attention, the question of whether such a representation is available is settled at the end of Section 4, and then in a more satisfying way in Section 5, though for the other case, noticed by Peter Simons, the question remains open, after some tentative discussion in one subsection, 6.2, of an Appendix (Section 6).
After some generalities about connections between functions and relations in Sections 1 and 2 recalls the possibility of taking the semantic values of n-ary Boolean connectives as n-ary relations among truth-values rather than as n-ary truth functions. Section 3, the bulk of the paper, looks at correlates of these truth-value relations as applied to formulas, and explores in a preliminary way how their properties are related to the properties of "logical relations" among formulas such as equivalence, implication (entailment) and contrariety (logical incompatibility), concentrating for illustrative purposes on binary logical relations such as those just listed. To avoid an excess of footnotes, some points have been deferred to an Appendix as "Longer Notes". K E Y W O R D S exhaustification, logical relations, truth functions | INTRODUCTIONOur interest here will be on relations among truth-values and relations holding among sentences (or formulas) in virtue of their truth-values. In particular, the concern will be with such relations as are systematically associated with truth functions. The setting will be bivalent, with the two truth-values taken as T and F . A valuation for a language is any mapping assigning one of these values to each formula of the language. 1 1 All object languages considered here are sentential, with their formulas constructed by suitable iterated application of primitive connectives (varying from language to language) to a denumerable set, Π, of sentence letters (or propositional variables) p 1 ,p 2 ,… (usually written as p,q,…). Lest the restriction to bivalent valuations be thought unduly restrictive, recall that every consequence relation ' on such a languageand not just those with a two-element (strongly) characteristic matrixis determined by a class V of such valuations, meaning by this that it is of the form ⊧ V , defined thus: for any set Γ [ fAg of formulas, Γ ⊧ V A iff for no v V, do we have vðCÞ ¼ T for each C Γ, while vðAÞ ¼ F . We use the customary notational abbreviations in connection with consequence relations, "A, B ⊧ V C" for "fA, Bg ⊧ V C", " ⊧ V A for ; ⊧ V A", and so forth. When ⊧ V is being thought of as a relation of logical consequencefor this or that logicone typically expects it to respect substitutions, a property secured by requiring that for any Π o ⊆ Π there is some v V such that vðp i Þ ¼ T iff p i Π 0 . The informal use, here, of "respecting substitutions" is made precise in the definition of substitutioninvariance for consequence relations in note 22 below. This condition is satisfied whenever no constraint is imposed on the treatment of sentence letters by the valuations in V, but only on compound formulas, as with the main choices of interest below, the most prominent of which is the class of all Boolean valuations.
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