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.
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
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∴
All Epicureans are charming bon vivants Some philosophers are Epicureans Some philosophers are charming bon vivants
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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)
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∴
∀x(Ex → Cx) ∃x(Px ∧ Ex) ∃x(Ex ∧ Cx)
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Predicate Logic Grammar of Predicate Logic
Primitive Vocabulary I
Connectives: →, ∼, and ∀
I
Variables x, y . . ., with or without subscripts
I
For each n > 0, n-place predicates F , G . . ., with or without subscripts
I
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.
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Definition of wff (i) If Π is an n-placed predicate and α1 . . . αn are terms, then Πα1 . . . αn 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”)
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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.
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Predicate Logic Semantics of Predicate Logic
Definition of a Model A PC-model is an order pair hD, I i such that: I I
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
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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.
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Our notion of truth will now be “truth in a model” — or truth relative to a model hD, I 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 ).
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Definition of a Variable Assignment: g is a variable assignment for model hD, I 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 “guα ” as the variable assignment that assigns u to α. Hence, guα (α) = u.
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The denotation of a variable α relative to a model M (= hD, I i) is defined as follows: ( I (α) if α is a constant [α]M ,g = g (α) if α is a variable
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Definition of Valuation: The PC valuation function, VM ,g , for PC-model M (= hD, I 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 α1 . . . αn , VM ,g (Πα1 . . . αn ) = 1 iff [α1 ]M ,g . . . [αn ]M ,g ∈ I (Π) 2. for any wffs φ, ψ, and any variable α: 2.1 VM ,g (∼ φ) = 1 iff VM ,g (φ) = 0 2.2 VM ,g (φ → ψ) = 1 iff VM ,g (φ) = 0 or VM ,g (ψ) = 1 2.3 VM ,g (∀αφ) = 1 iff for every u ∈ D, VM ,guα (φ) = 1
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Definition of Truth in a Model: φ is true in PC model M iff VM ,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.
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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 VM ,g (γ) = 1 for each γ ∈ Γ, then VM ,g (φ) = 1.
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Predicate Logic Establishing Validity and Invalidity
Let’s do one of Sider’s exercises: We are asked to show that p∀x(Fx → (Fx ∨ Gx))q. 1. So, suppose that Vg (∀x(Fx → (Fx ∨ Gx))) = 0) (for some assignment in some model). 2. Given (1), for some v ∈ D, Vgvx (Fx → (Fx ∨ Gx)) = 0. 3. Call one such v “u”. So, Vgux (Fx → (Fx ∨ Gx)) = 0. 4. Given (3), Vgux (Fx → (Fx ∨ Gx)) = 0 iff Vgux (Fx) = 1 and Vgux (Fx ∨ Gx)) = 0.
5. Vgux (Fx) = 1 iff [x]gux ∈ I (F ). 6. But Vgux (Fx ∨ Gx)) = 0 iff Vgux (Fx) = 0 and Vgux (Gx) = 0.
7. And Vgux (Fx) = 0 iff [x]gux ∈ / I (F ), which contradicts (5). Brandon C. Look: Symbolic Logic II, Lecture 8
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Predicate Logic Axiomatic Proofs in PC
Axiomatic System for PC: I
I
Rules: MP plus Universal Generalization (UG): φ UG ∀αφ Axioms: PL1-PL3, plus: PC1 ∀αφ → φ(β/α) PC2 ∀α(φ → ψ) → (φ → ∀αψ)
Conditions: I
φ, ψ, and χ are any wffs of predicate logic, α is any variable, and β is any term
I
φ(β/α) results from φ by “correct substitution” of β for α
I
in PC2, no occurrences of variable α may be free in φ.
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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).
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Or the relatively easy proof of `PC Fa → ∃xFx 1. 2. 3.
∀x ∼ Fx →∼ Fa Fa →∼ ∀x ∼ Fx Fa → ∃xFx
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PC1 PL Def. of ∃
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Here is another justification for what Sider calls Distribution: `PC p∀x(φ → ψ) → (∀xφ → ∀xψ)q. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
∀x(φ → ψ) ∀xφ ∀x(φ → ψ) → (φ → ψ) φ→ψ ∀xφ → φ φ ψ ∀xψ ∀x(φ → ψ), ∀xφ ` ∀xψ ∀x(φ → ψ) ` ∀xφ → ∀xψ ∀x(φ → ψ) → (∀xφ → ∀xψ)
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Premise Premise PC1 1,3 MP PC1 2,5 MP 4,6 MP 7, UG 1-8 9, DT 10, DT
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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.)
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