symbolic logic ii - lecture 8look/phi520-lecture8.pdfbrandon c. look: symbolic logic ii, lecture 8...

Predicate Logic

In what we’ve discussed thus far, we haven’t addressed other kindsof valid inferences: those involving quantification and predication.For example:

All philosophers are wiseSocrates is a philosopher

∴ Socrates is wise

All psychiatrists are doctorsAll doctors are college graduates

∴ All psychiatrists are college graduates

Brandon C. Look: Symbolic Logic II, Lecture 8 1

All Epicureans are charming bon vivantsSome philosophers are Epicureans

∴ Some philosophers are charming bon vivants


These arguments, of course, can be symbolized in the followingway:

∀x(Px →Wx)Ps

∴ Ws

∀x(Px → Dx)∀x(Dx → Cx)

∴ ∀x(Px → Cx)


∀x(Ex → Cx)∃x(Px ∧ Ex)

∴ ∃x(Ex ∧ Cx)


Predicate LogicGrammar 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 withoutsubscripts

I Individual constants (names) a, b . . ., with or withoutparentheses

Just as all the truth functional connectives could be expressed by∼ and →, so ∃ can be expressed by what we have in ourvocabulary, for ∃xFx is equivalent to ∼ ∀x ∼ Fx .


Definition of wff

(i) If Π is an n-placed predicate and α1 . . . αn are terms, thenΠα1 . . . 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 bereckoned as the value of a variable.” (“On What There Is”)


Variables are either free or bound. A variable is free iff it does notbelong to a quantifier; a variable is bound iff it belongs to aquantifier. That is, a variable is bound if it is within the scope of aquantifier.

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.


Predicate LogicSemantics of Predicate Logic

Definition of a Model A PC-model is an order pair 〈D ,I 〉 suchthat:

I D is a non-empty set (“the domain”)I 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


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.


Our notion of truth will now be “truth in a model” — or truthrelative to a model 〈D ,I 〉. And we will want a valuation functionthat assigns truth values to our propositions. But it is a little morecomplicated than that in predicate logic.

Think first of the statement Fa, i.e., “a is F”. What we are sayingis that there is some object of our domain that is among the set ofF things; that is, that a is some object, call it “u”, and there is asubset, call it “S”, of the domain to which we assign the predicateF . So, if Fa is true, u ∈ S . Or, in saying Fa is true, we mean thatI (a) ∈ I (F ).


Definition of a Variable Assignment: g is a variable assignmentfor model 〈D ,I 〉 iff g is a function that assigns to each variablesome object in D .

And this in turn means that g is a function that assigns variablesto objects in some domain. Per Sider, we will define “gαu ” as thevariable assignment that assigns u to α. Hence, gαu (α) = u.


The denotation of a variable α relative to a model M (= 〈D ,I 〉)is defined as follows:

[α]M ,g =

{I (α) if α is a constant

g(α) if α is a variable


Definition of Valuation: The PC valuation function, VM ,g , forPC-model M (= 〈D ,I 〉) and variable assignment g , is defined asthe function that assigns to each wff either 0 to 1 subject to thefollowing 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) = 02.2 VM ,g (φ→ ψ) = 1 iff VM ,g (φ) = 0 or VM ,g (ψ) = 12.3 VM ,g (∀αφ) = 1 iff for every u ∈ D , VM ,gα

u(φ) = 1


Definition of Truth in a Model: φ is true in PC model M iffVM ,g (φ) = 1, for each variable assignment g for M .

Definition of Validity: φ is PC-valid (“�PC φ”) iff φ is true in allPC-models.


Deinition of Semantic Consequence: φ is a PC-semanticconsequence of a set of wffs Γ (“Γ �PC φ”) iff for every PC-modelM and every variable assignment g for M , if VM ,g (γ) = 1 foreach γ ∈ Γ, then VM ,g (φ) = 1.


Predicate LogicEstablishing 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 someassignment in some model).

2. Given (1), for some v ∈ D ,Vgxv

(Fx → (Fx ∨ Gx)) = 0.

3. Call one such v “u”. So, Vgxu

(Fx → (Fx ∨ Gx)) = 0.

4. Given (3), Vgxu

(Fx → (Fx ∨ Gx)) = 0 iff Vgxu

(Fx) = 1 andVgx

u(Fx ∨ Gx)) = 0.

5. Vgxu

(Fx) = 1 iff⟨[x ]gx

u

⟩∈ I (F ).

6. But Vgxu

(Fx ∨ Gx)) = 0 iff Vgxu

(Fx) = 0 and Vgxu

(Gx) = 0.

7. And Vgxu

(Fx) = 0 iff⟨[x ]gx

u

⟩/∈ I (F ), which contradicts (5).

�


Predicate LogicAxiomatic Proofs in PC

Axiomatic System for PC:

I Rules: MP plus Universal Generalization (UG):φ

∀αφ UG

I 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.


So, let’s do some problems. Prove∀x(Fx → Gx), ∀x(Gx → Hx) `PC ∀x(Fx → Hx)

1. ∀x(Fx → Gx) Premise2. ∀x(Gx → Hx) Premise

3. ∀x(Fx → Gx)→ (Fa→ Ga) PC14. ∀x(Gx → Hx)→ (Ga→ Ha) PC15. Fa→ Ga 1,3 MP6. Ga→ Ha 2,4 MP7. Fa→ Ha 5,6 HS8. ∀x(Fx → Hx) 7, UG

The move from 7-8 via UG is justified because a does not occur inthe premises, nor does it occur in ∀x(Fx → Hx).


Or the relatively easy proof of `PC Fa→ ∃xFx

1. ∀x ∼ Fx →∼ Fa PC12. Fa→∼ ∀x ∼ Fx PL3. Fa→ ∃xFx Def. of ∃


Here is another justification for what Sider calls Distribution:`PC p∀x(φ→ ψ)→ (∀xφ→ ∀xψ)q.

1. ∀x(φ→ ψ) Premise2. ∀xφ Premise

3. ∀x(φ→ ψ)→ (φ→ ψ) PC14. φ→ ψ 1,3 MP5. ∀xφ→ φ PC16. φ 2,5 MP7. ψ 4,6 MP8. ∀xψ 7, UG9. ∀x(φ→ ψ),∀xφ ` ∀xψ 1-8

10. ∀x(φ→ ψ) ` ∀xφ→ ∀xψ 9, DT11. ∀x(φ→ ψ)→ (∀xφ→ ∀xψ) 10, DT


Predicate LogicMetalogic of PC

Soundness and Completeness:As in the system of Propositional Logic, it can be shown thatΓ `PC φ iff Γ �PC φ.Compactness Theorem:Let Γ be a set of wffs of language L. If for each finite subset of Γthere is a first-order structure making this subset of Γ true, thenthere is a first-order structure M that makes all the sentences true.(As Sider mentions, this is intuitively surprising, because it meansthat there is a true model even when Γ is infinite.)


symbolic logic ii - lecture 8look/phi520-lecture8.pdfbrandon c. look: symbolic logic ii, lecture 8...

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