Parry Syllogisms

Parry discusses an extension of Aristotle’s syllogistic that uses four nontraditional quantifiers. We show that his conjectured decision procedure for validity for the extended syllogistic is correct even if syllogisms have more than two premises. And we axiomatize this extension of the syllogistic. 1 Background and motivation Parry [2] discusses an extension of the syllogistic in which sentences are formed by using the quantifiers α, ι, η, ω in addition to the traditional quantifiers A, E, I, O. If a and b are terms and Q is a quantifier then Qab is a sentence. A sentence Qab is an affirmative sentence if Q is A, I, α, or ι; otherwise, it is a negative sentence. Our discussion of sentences Qab will be restricted to those in which a = b. As is customary Aab, Eab, Iab, and Oab are read as ‘All a are b’, ‘No a are b’, ‘Some a are b’, and ‘Some a are not b’, respectively. αab, ιab, ηab, and ωab may be read as ‘There is exactly one a and all a are b, and there is exactly one b and all b are a’, ‘There is exactly one b and all b are a’, ‘It is not true that ιab’, and ‘It is not true that αab’, respectively. (So ηab and ωab may be read as disjunctive sentences.)1 Parry conjectures a decision procedure for validity for this extension of the traditional syllogistic, given that syllogisms have no more than two premises. We show that his decision procedure is correct even if syllogisms have more than two premises. And we axiomatize the Parry syllogistic and thus Aristotle’s syllogistic as well.2 2 Preliminaries Definition 2.1 cd(Aab) (the contradictory of Aab) = Oab; cd(Iab) = Eab, cd(αab) = ωab; cd(ιab) = ηab; and cd(cd(x)) = x. Definition 2.2 A pair 〈W, v〉 is a model if and only if W is a nonempty set and v a function that maps terms into nonempty subsets of W and maps sentences into {t, f } where: Received January 7, 2000; revised April 19, 2000 PARRY SYLLOGISMS 415 (i) v(Aab) = t iff v(a) ⊆ v(b); (ii) v(Iab) = t iff v(a) ∩ v(b) = ∅; (iii) v(ιab) = t iff v(b) has exactly one member and v(Aba) = t; (iv) v(αab) = t iff v(ιab) = t and v(ιba) = t; and (v) v(x) = t iff v(cd(x)) = f . A set X of sentences is consistent if and only if there is a model 〈W, v〉 such that v assigns t to every member of X; otherwise X is inconsistent. X |= x if and only if X ∪ {cd(x)} (or X, cd(x) for short) is inconsistent. Definition 2.3 Sentence x is a superordinate of sentence y if and only if 〈x, y〉 has one of the following forms: 〈x, x〉; 〈αab, αba(ι[ab], A[ab], I[ab])〉 (where Q[ab] is Qab or Qba); 〈ιab, Aba(I[ab])〉; 〈Aab, I[ab]〉; or 〈Iab, Iba〉; or cd(y) is a superordinate of cd(x) in virtue of one of the above forms. x is a subordinate of y if and only if y is a superordinate of x. The following proofs use this fact: if x is a superordinate of y (or y is a subordinate of x) then x |= y (which is short for {x} |= y). 3 Decision procedure for validity Distribution is defined by the following table: a b Eab, αab d d Aab, ηab d Oab, ιab d Iab, ωab So, for example, a is distributed in Eab and b is undistributed in ωab. The following proofs use this fact: if x is a superordinate of y and term a is distributed in y then a is distributed in x. Definition 3.1 A set C of sentences is a chain if and only if it has form Q1[a1a2], . . . , Qn−1[an−1an], Qn[ana1], where each term ai occurs exactly twice and no term occurs twice in a sentence. Theorem 3.2 A chain C is inconsistent if and only if: (i) exactly one negative sentence occurs in C; (ii) every term is distributed at least once in C; and (iii) if η occurs in C so does α or ι.3 Proof: (Only if) Case 1: (i) is not satisfied. Subcase 1: No negative sentence occurs in C. Construct chain C′ by replacing every affirmative quantifier in C with α. C′ is consistent given model 〈{1}, v〉, where v assigns {1} to every term. So C is consistent since αab |= Qab if Q is affirmative. Subcase 2: More than one negative sentence occurs in C.