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1 Predicate Logic In what we ve discussed thus far, we haven t addressed other kinds of valid inferences: those involving quantification and predication. For example: All philosophers are wise Socrates is a philosopher Socrates is wise All psychiatrists are doctors All doctors are college graduates All psychiatrists are college graduates Brandon C. Look: Symbolic Logic II, Lecture 8 1
2 All Epicureans are charming bon vivants Some philosophers are Epicureans Some philosophers are charming bon vivants Brandon C. Look: Symbolic Logic II, Lecture 8 2
3 These arguments, of course, can be symbolized in the following way: x(px Wx) Ps Ws x(px Dx) x(dx Cx) x(px Cx) Brandon C. Look: Symbolic Logic II, Lecture 8 3
4 x(ex Cx) x(px Ex) x(ex Cx) Brandon C. Look: Symbolic Logic II, Lecture 8 4
5 Predicate Logic Grammar of Predicate Logic Primitive Vocabulary Connectives:,, and Variables x, y..., with or without subscripts For each n > 0, n-place predicates F, G..., with or without subscripts Individual constants (names) a, b..., with or without parentheses Just as all the truth functional connectives could be expressed by and, so can be expressed by what we have in our vocabulary, for xfx is equivalent to x Fx. Brandon C. Look: Symbolic Logic II, Lecture 8 5
6 Definition of wff (i) If Π is an n-placed predicate and α 1... α n are terms, then Πα 1... α n is a wff (ii) If φ, ψ are wffs, and α is a variable, then α, (φ ψ), and αφ are wffs (iii) Only strings that can be shown to be wffs using (i) and (ii) are wffs Quine: To be assumed as an entity is, purely and simply, to be reckoned as the value of a variable. ( On What There Is ) Brandon C. Look: Symbolic Logic II, Lecture 8 6
7 Variables are either free or bound. A variable is free iff it does not belong to a quantifier; a variable is bound iff it belongs to a quantifier. That is, a variable is bound if it is within the scope of a quantifier. For example, consider the following wffs: 1. Fx 2. xfx The occurrence of x in (1) is free; the second occurrence of x in (2) is bound. When a formula has no free occurrences of variables, it is closed ; otherwise it is an open formula. Brandon C. Look: Symbolic Logic II, Lecture 8 7
8 Predicate Logic Semantics of Predicate Logic Definition of a Model A PC-model is an order pair D, I such that: D is a non-empty set ( the domain ) I is a function ( the interpretation function ) obeying the following constraints: if α is a constant, then I (α) D if Π is an n-place predicate, then I (Π) = some set of n-tuples of members of D Brandon C. Look: Symbolic Logic II, Lecture 8 8
9 Models represent configurations of the world. N-place predicates are assigned to sets of n-tuples drawn from D. This set is the extension of the predicate of the model. Consider (again) Tarski s World. Brandon C. Look: Symbolic Logic II, Lecture 8 9
10 Our notion of truth will now be truth in a model or truth relative to a model D, I. And we will want a valuation function that assigns truth values to our propositions. But it is a little more complicated than that in predicate logic. Think first of the statement Fa, i.e., a is F. What we are saying is that there is some object of our domain that is among the set of F things; that is, that a is some object, call it u, and there is a subset, call it S, of the domain to which we assign the predicate F. So, if Fa is true, u S. Or, in saying Fa is true, we mean that I (a) I (F ). Brandon C. Look: Symbolic Logic II, Lecture 8 10
11 Definition of a Variable Assignment: g is a variable assignment for model D, I iff g is a function that assigns to each variable some object in D. And this in turn means that g is a function that assigns variables to objects in some domain. Per Sider, we will define g α u as the variable assignment that assigns u to α. Hence, g α u (α) = u. Brandon C. Look: Symbolic Logic II, Lecture 8 11
12 The denotation of a variable α relative to a model M (= D, I ) is defined as follows: { I (α) if α is a constant [α] M,g = g(α) if α is a variable Brandon C. Look: Symbolic Logic II, Lecture 8 12
13 Definition of Valuation: The PC valuation function, V M,g, for PC-model M (= D, I ) and variable assignment g, is defined as the function that assigns to each wff either 0 to 1 subject to the following constraints: 1. for any n-place predicate Π and any terms α 1... α n, V M,g (Πα 1... α n ) = 1 iff [α 1 ] M,g... [α n ] M,g I (Π) 2. for any wffs φ, ψ, and any variable α: 2.1 V M,g ( φ) = 1 iff V M,g (φ) = V M,g (φ ψ) = 1 iff V M,g (φ) = 0 or V M,g (ψ) = V M,g ( αφ) = 1 iff for every u D, V M,g α u (φ) = 1 Brandon C. Look: Symbolic Logic II, Lecture 8 13
14 Definition of Truth in a Model: φ is true in PC model M iff V M,g (φ) = 1, for each variable assignment g for M. Definition of Validity: φ is PC-valid ( PC φ ) iff φ is true in all PC-models. Brandon C. Look: Symbolic Logic II, Lecture 8 14
15 Deinition of Semantic Consequence: φ is a PC-semantic consequence of a set of wffs Γ ( Γ PC φ ) iff for every PC-model M and every variable assignment g for M, if V M,g (γ) = 1 for each γ Γ, then V M,g (φ) = 1. Brandon C. Look: Symbolic Logic II, Lecture 8 15
16 Predicate Logic Establishing Validity and Invalidity Let s do one of Sider s exercises: We are asked to show that x(fx (Fx Gx)). 1. So, suppose that V g ( x(fx (Fx Gx))) = 0) (for some assignment in some model). 2. Given (1), for some v D, V g x v (Fx (Fx Gx)) = Call one such v u. So, V g x u (Fx (Fx Gx)) = Given (3), V g x u (Fx (Fx Gx)) = 0 iff V g x u (Fx) = 1 and V g x u (Fx Gx)) = V g x u (Fx) = 1 iff [x] g x u I (F ). 6. But V g x u (Fx Gx)) = 0 iff V g x u (Fx) = 0 and V g x u (Gx) = And V g x u (Fx) = 0 iff [x] g x u / I (F ), which contradicts (5). Brandon C. Look: Symbolic Logic II, Lecture 8 16
17 Predicate Logic Axiomatic Proofs in PC Axiomatic System for PC: Rules: MP plus Universal Generalization (UG): φ αφ UG Axioms: PL1-PL3, plus: PC1 αφ φ(β/α) PC2 α(φ ψ) (φ αψ) Conditions: φ, ψ, and χ are any wffs of predicate logic, α is any variable, and β is any term φ(β/α) results from φ by correct substitution of β for α in PC2, no occurrences of variable α may be free in φ. Brandon C. Look: Symbolic Logic II, Lecture 8 17
18 So, let s do some problems. Prove x(fx Gx), x(gx Hx) PC x(fx Hx) 1. x(fx Gx) Premise 2. x(gx Hx) Premise 3. x(fx Gx) (Fa Ga) PC1 4. x(gx Hx) (Ga Ha) PC1 5. Fa Ga 1,3 MP 6. Ga Ha 2,4 MP 7. Fa Ha 5,6 HS 8. x(fx Hx) 7, UG The move from 7-8 via UG is justified because a does not occur in the premises, nor does it occur in x(fx Hx). Brandon C. Look: Symbolic Logic II, Lecture 8 18
19 Or the relatively easy proof of PC Fa xfx 1. x Fx Fa PC1 2. Fa x Fx PL 3. Fa xfx Def. of Brandon C. Look: Symbolic Logic II, Lecture 8 19
20 Here is another justification for what Sider calls Distribution: PC x(φ ψ) ( xφ xψ). 1. x(φ ψ) Premise 2. xφ Premise 3. x(φ ψ) (φ ψ) PC1 4. φ ψ 1,3 MP 5. xφ φ PC1 6. φ 2,5 MP 7. ψ 4,6 MP 8. xψ 7, UG 9. x(φ ψ), xφ xψ x(φ ψ) xφ xψ 9, DT 11. x(φ ψ) ( xφ xψ) 10, DT Brandon C. Look: Symbolic Logic II, Lecture 8 20
21 Predicate Logic Metalogic of PC Soundness and Completeness: As in the system of Propositional Logic, it can be shown that Γ PC φ iff Γ PC φ. Compactness Theorem: Let Γ be a set of wffs of language L. If for each finite subset of Γ there is a first-order structure making this subset of Γ true, then there is a first-order structure M that makes all the sentences true. (As Sider mentions, this is intuitively surprising, because it means that there is a true model even when Γ is infinite.) Brandon C. Look: Symbolic Logic II, Lecture 8 21
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