LING 501, Fall 2004: Quantification The universal quantifier Ax is conjunctive and the existential quantifier Ex is disjunctive Suppose the domain of quantification (DQ) is {a, b}. Then: (1) Ax Px Pa & Pb (2) Ex Px Pa Pb Consequently Ax Px = Ex Px, but not conversely. The logical relations among various propositions involving the unary predicate P in this DQ is diagrammed as follows. T Ex Px Ex ~ Px Pa Pb ~ Pa ~ Pb Ax Px Ax ~ Px If DQ has exactly one individual a, then Ax Px and Ex Px are each equivalent to Pa. T Ax Px = Ex Px = Pa Ax ~Px = Ex ~ Px = ~ Pa Finally, if DQ is empty, then Ax Px = Ax ~ Px = T and Ex Px = Ex ~ Px =. That is, if DQ is empty, Ax Px and Ex Px change places implicationally. This is because Ax Px is always one step above, and Ex Px one step below T. Ax Px = Ax ~ Px = T Ex Px = Ex ~ Px = In general, we can replace Ax by a conjunction of propositions involving constants and Ex by a disjunction of propositions involving constants if we can list the individual constants in DQ. This is only possible if the number of constants is finite, and we can distinguish them. For example, if DQ is the set of balls in an urn, but we can't tell one ball from another, then the proposition Ax Px can't be expressed by a conjunction of propositions involving constants, even though the number of balls in the urn is finite.
2 Interaction of Ax, Ex and negation Recall De Morgan's laws relating conjunction, disjunction and negation. (3) ~ [P & Q] [~ P ~ Q] (4) ~ [P Q] [~ P & ~ Q] Replacing P with Pa and Q with Pb, letting DQ = {a, b}, and then replacing [Pa & Pb] with Ax Px and [Pa Pb] with Ex Px, we obtain: (5) ~ Ax Px Ex ~ Px (6) ~ Ex Px Ax ~ Px Then negating both sides of both equivalences and canceling double negation, we obtain: (7) Ax Px ~ Ex ~ Px (8) Ex Px ~ Ax ~ Px That is Ax and Ex are interdefinable using negation. Interaction of Ax, Ex, & and The universal quantifier distributes over conjunction and the existential quantifier distributes over disjunction. That is: (9) Ax [Px & Qx] Ax Px & Ax Qx (10) Ex [Px Qx] Ex Px Ex Qx However the universal quantifier fails to distribute over disjunction and the existential quantifier fails to distribute over conjunction. That is: (11) Ax Px Ax Qx Ax [Px Qx], but not conversely. (12) Ex [Px & Qx] Ex Px & Ex Qx, but not conversely. These relations extend the following relation in propositional logic: (13) [[P & Q] [R & S]] [[P R] & [Q S]], but not conversely. Interaction of Ax and Ey Other extensions of the preceding relation are given by: (14) Ex Ay Pxy Ay Ex Pxy, but not conversely. (15) Ey Ax Pxy Ax Ey Pxy, but not conversely. However, the order of two universal or two existential quantifiers over different variables does not matter. (16) Ax Ay Pxy Ay Ax Pxy (17) Ex Ey Pxy Ey Ex Pxy So, Ax Ey Pxy and Ex Ay Pxy are distinct formulas logically, but neither Ax Ay Pxy and Ay Ax Pxy nor Ex Ey Pxy and Ey Ex Pxy.
3 Logical laws involving universal and existential quantifiers Let L be a first order language which contains at least one constant. Then: (18) For every open sentence Φx and constant i in L, Ax Φx = Φi (law of instantiation) (19) For every open sentence Φx and constant i in L, Φi = Ex Φx (law of generalization) From (18) and (19) and the transitivity of logical consequence, it follows that in any FOL with at least one constant, Ax Φx = Ex Φx. If L contains exactly one constant a, then Ax Φx Φa Ex Φx. Finally, if L contains no constants, then Ex Φx and Ax Φx T; hence in that case Ex Φ x = Ax Φ x. These relations were pointed out above. Next, suppose L contains just the two constants a and b. Then the laws of instantiation and generalization come to resemble the familiar propositional logic laws of simplification and addition involving conjunction and disjunction: (20) For every open sentence Φx in L, Ax Φx = Φa and Ax Φx = Φb (21) For every open sentence Φx in L, Φa = Ex Φx and Φb = Ex Φx If L has finitely many constants i 1, i n, then universal quantification is equivalent to conjunction over i 1, i n, and existential quantification to disjunction over i 1, i n (cf. (1) and (2)): (22) For every open sentence Φx and constants i 1, i n in L, Ax Φx Φi 1 & & Φi n (23) For every open sentence Φx and constants i 1, i n in L, Ex Φx Φi 1 Φi n Definition of the universal and existential quantifiers in terms of logical consequence The universal and existential quantifiers can be defined using their behavior under logical consequence as follows. (24) Definition of universal quantifier a. Ax Φx = Φi for every open sentence Φx and every constant i in L. b. If any sentence Tx Φx satisfies (24)a, then Tx Φx = Ax Φx. (25) Definition of existential quantifier a. If Φi = Ψ for some predicate Φ and every constant i in L, then Ex Φx = Ψ. b. If any sentence Tx Φx satisfies (25)a, then Tx Φx = Ex Φx. Definition of numerical quantifiers using logical consequence Numerical quantifiers can also be defined using their behavior under logical consequence. The definitions are modeled on the definition of the existential quantifier in (25). Note that these definitions place the nonidentity condition in the metalanguage of FOL rather than as a separate FOL proposition, as in standard treatments, e.g. W. Hodges, Logic, section 38. In a language with at least two distinct constants, 2x Φx = Ex Φ x, and in a language with infinitely many constants, Ax Φx = nx Φx, and nx Φx = (n-1)x Φx for every n > 2.
(26) Definition of 2x there are at least two xs such that : a. If {Φi, Φj} = Ψ for some predicate Φ and every pair of distinct constants {i, j} in L, then 2x Φx = Ψ. b. If any sentence Tx Φx satisfies (26)a, then Tx Φx = 2x Φx. (27) Schematic definition of nx there are at least n xs such that : a. If {Φi 1,, Φi n } = Ψ for some one place predicate Φ and every n-tuple of distinct constants {i 1,, i n } in L, then nx Φx = Ψ. b. If any sentence Tx Φx satisfies (27)a, then Tx Φx = nx Φx. The negation of nx, ~nx, may be glossed there are fewer than n xs such that (or there are at most (n 1) xs such that ), and written <nx. The quantifier n!x there are exactly n xs such that is equivalent to the conjunctive quantifier [nx & <(n+1)x]. For example E!x is equivalent to (Ex & <2x), and 2!x is equivalent to (2x & <3x). The quantifiers for all but one,... for all but n can be defined in terms of the exact existential or numerical quantifiers. (28) (A 1)x Φx E!x ~Φx (29) (A n)x Φx n!x ~Φx Scope of quantifiers As noted above, when two quantifiers occur in the same proposition, one may have scope over the other. For example in (30), the universal quantifier Ax has scope over the existential quantifier Ey, whereas in (31), the existential quantifier Ey has scope over the universal quantifier Ax. (30) AxEy P(x, y) (31) EyAx P(x, y) It is also noted that (30) is a logical consequence of (31), i.e. that EyAx P(x, y) = AxEy P(x, y), but not conversely (see (15)). For example, if there is some problem that every student solves (an instance of (31), where the variable x ranges over students, and y over problems, and R is the two-place relation solves ) then for every student, there is some problem that he or she solves (an instance of (30)). However, the converse is not true. If for every student, there is some problem that he or she solves, it doesn t follow that there is some problem that every student solves. Perhaps no one student solves every problem, but different students solve different problems such that every problem gets solved by one or another of the students. In considering the logical properties of formulas with two quantifiers, care must be taken as to which variable each quantifier binds. Although EyAx P(x, y) entails AxEy P(x, y), EyAx P(x, y) does not entail AyEx P(x, y). For example, if there is some problem that every student solves, it doesn t follow that for every problem there is some student who solves it. If an exam has a really easy problem in it that every student solves, but also a really hard one that no student solves, then the first sentence is true but the second is false. Some of the logical relations among doubly quantified formulas with universal and existential quantifiers may be diagrammed 4
5 as in (32). Since the relative scope of identical universal or existential quantifiers has no semantic effect (see (16) and (17)), we may write Axy for AxAy, and Exy for ExEy. (32) Hasse diagram for doubly universally or existentially quantified formulas ExyP(x, y) AxEyP(x, y) AyExP(x, y) EyAxP(x, y) ExAyP(x, y) AxyP(x, y) However, the scope of numerical quantifiers always matters, even if they are the same. For example 2x2y P(x, y) is logically independent of (neither implying nor implied by) 2y2x P(x, y). For example, suppose Alice, Bob and Carla are students and Alice read Moby Dick and The Scarlet Letter, Bob read The Scarlet Letter and A Tale of Two Cities, Carla read The Scarlet Letter, and no other reading took place. Then 2x2y P(x, y) (where x ranges over students, y over books and P is read ) is true but 2y2x P(x, y) is false. On the other hand, suppose Alice read Moby Dick, Bob read Moby Dick and The Scarlet Letter, Carla read The Scarlet Letter and no other reading took place. Then 2y2x P(x, y) is true but 2x2y P(x, y) is false. Note also that both 2x2y P(x, y) and 2y2x P(x, y) are false if Alice read Moby Dick and Bob read The Scarlet Letter and no other reading took place. However 2xEy P(x, y) and 2yEx P(x, y) are both true under that condition.