INTRODUCTION to LOGIC Lecture 5 the Semantics of Predicate Logic

Outline

1 Validity. INTRODUCTIONTOLOGIC 2 Semantics for simple English sentences. Lecture 5 3 L2-structures. 4 Semantics for L -formulae. The Semantics of Predicate Logic 2 Dr. James Studd

We could forget about philosophy. Settle down and maybe get into semantics. Woody Allen ‘Mr. Big’

Introduction Introduction What of argument 2? Validity

Argument 2 Valid Recall the definition of validity for L1. Let Γ be a set of sentences of L and φ a sentence of L (1) Zeno is a tortoise. 1 1 (2) All tortoises are toothless. Definition Therefore, (C) Zeno is toothless. The argument with all sentences in Γ as premisses and φ as conclusion is valid if and only if there is no L1-structure Formalisation under which: (1) T a (i) all sentences in Γ are true; and (2) ∀x(T x → Lx) (ii) φ is false. (C) La We use an exactly analogous definition for L2, replacing ‘L1’ Dictionary: a: Zeno. T :. . . is a tortoise. L: . . . is toothless everywhere above with ‘L2’ It remains to define: L2-structure, truth in an L2-structure What is it for this L2-argument to be valid? Introduction Semantics in English Structures Semantics in English Structures interpret non-logical expressions. Start with a semantics for simple English sentences.

L1-structures ‘Bertrand Russell is a philosopher’ Non-logical expressions in L1: P, Q, R, . . .. The sentence is true (i.e.: its semantic value is: T). An L -structure A assigns each sentence letter a 1 . . . because of the relationship between the semantic values semantic value (speciﬁcally, a truth-value: T or F) of its constituents. L is a richer language. This calls for richer structures. 2 expression semantic value L2-structures ‘Bertrand Russell’ Russell Non-logical expressions: P 1,Q1,R1,... ‘is a philosopher’ the property of being a philosopher P 2,Q2,R2,...... because Russell has the property of being a philosopher. . . . . because |‘Bertrand Russell’| has |‘is a philosopher’|. a, b, c, . . . Notation An L2-structure A assigns each predicate and constant a semantic value (speciﬁcally, what?) When e is an expression, we write |e| for its semantic value.

Semantics in English Semantics in English Similarly: Semantic values for English expressions ‘Alonzo Church reveres Bertrand Russell’ is true iff Church stands in the relation of revering to Russell expression semantic value designator object In other words: unary predicate property (alias: unary relation) |‘Alonzo Church reveres Bertrand Russell’| = T iff binary predicate binary relation |‘Alonzo Church’| stands in |‘reveres’| to |‘Bertrand Russell’| Examples |‘Bertrand Russell’| = Russell |‘is a philosopher’| = the property of being a philosopher |‘reveres’| = the relation of revering We’ll take this one step further, by saying more about properties and relations. Semantics in English Semantics in English Properties Relations In logic, we identify properties with sets. Recall that we identify binary relations with sets of pairs. Property (alias: unary relation) Binary relation A unary relation P is a set of zero or more objects. A binary relation R is a set of zero or more pairs of objects. Specifically, P is the set of objects that have the property. R is the set of pairs hd, ei such that d stands in R to e.

Informally: d ∈ P indicates that d has property P . Informally: hd, ei ∈ R indicates that d bears R to e.

Example Example The property of being a philosopher The relation of revering = {hd, ei : d reveres e} = the set of philosophers = {d : d is a philosopher} Similarly: = {Descartes, Kant, Russell, . . . } A ternary (3-ary) relation is a set of triples (3-tuples). A quaternary (4-ary) relation is a set of quadruples (4-tuples). etc.

Semantics in English Atomic Sentences

Putting this all together: Semantics for atomic L2-sentences

‘Bertrand Russell is a philosopher’ is true The semantics for atomic L2-sentences is similar. iff |‘Bertrand Russell’| has |‘is a philosopher’| iff Russell ∈ the set of philosophers An L2-structure specifies semantic values for L2-expressions: Similarly: L2-expression semantic value constant: a object: |a|A ‘Alonzo Church reveres Russell’ is true sentence letter: P truth-value: |P | (i.e. T or F) iff |‘Alonzo Church’| stands in |‘reveres’| to |‘Russell’| unary predicate: P 1 unary relation: |P 1| (i.e. a set) iff hChurch, Russelli ∈ {hd, ei : d reveres e} binary predicate: P 2 binary relation: |P 2| (a set of pairs)

|P b| = T iff |b| has |P | iff |b| ∈ |P | |Rab| = T iff |a| stands in |R| to |b| iff h|a|, |b|i ∈ |R|

Notation: |e|A is the semantic value of e in L2-structure A. Atomic Formulae Atomic Formulae

Semantics for atomic L2-formulae Variable assignments

We have the semantics for L2-sentences like P a. Variable assignment What about L -formulae like P x? 2 A variable assignment assigns an object to each variable. In English: One can think of a variable assignment as an inﬁnite list The designator ‘Russell’ has a constant semantic value. Example: the assignment α. Pronouns, such as ‘it’, do not. xyzx y z x 1 1 1 2 ··· ‘it’ refers to diﬀerent objects depending on the context. Mercury Venus Venus Neptune Mars Venus Mars

Something similar happens in an L2-structure A: Notation a, b, c, . . . are assigned a constant semantic value in A. We write |x|α for the object α assigns to x. Variables: x, y, z, . . . are not. We use lower case Greek letters: α, β, γ for assignments.

α α α What object each variable denotes is speciﬁed with a e.g. |x| = Mercury; |y| =Venus; |x2| = Mars. variable assignment.

Atomic Formulae Atomic Formulae

Once x has been assigned an object, the semantics for P x Worked example are much like the semantics for P a Let L2-structure A be such that: α We write |e|A for the semantic value of expression e in the |a|A = Alonzo Church structure A under the variable assignment α. |b|A = Bertrand Russell

α α α α |P |A = {Frege, Russell} |P x|A = T iff |x| has |P |A (NB: |x|A = |x| ) α |R|A = {hChurch, Russelli} iff |x| ∈ |P |A α α α Let assignments α and β be such that: |Rxy|A = T iff |x| stands in |R|A to |y| x y z α α iff h|x| , |y| i ∈ |R|A α: Frege Russell Wittgenstein β: Church Church Church Note: semantic values of constants and predicates are unaffected by the assignment (e.g. |P |α = |P | , |a|α = |a| ). A A A A Compute the following:

α α β α |Rab|A = T iff h|a|A, |b|Ai ∈ |R|A |x|A = |x|A = |a|A = α α α β α |Rxb|A = T iff h|x| , |b|Ai ∈ |R|A |P y|A = |P y|A = |P b|A = |Rxy|α = |Rxy|β = |Rxb|α = Similarly for other atomic formulae. A A A Quantifiers Quantifiers

Semantics for quantifiers An L2-structure A specifies a non-empty set DA as the domain. An assignment over A assigns a member of DA to each variable. In English, the truth-value of a quantified sentence depends on how widely the quantifiers range. Semantics for ∀/∃ (first approximation): |∀xP x|A = T Almost everyone attended the first lecture. iff every member of DA has |P |A α iff every assignment α of x to a member of DA is such that |x| ∈|P |A α The context supplies a ‘domain’ telling us who ‘everyone’ iff every assignment α over A is such that |P x|A = T ranges over Similarly: Domain: the set of first-year Oxford philosophy students |∃xP x| = T Almost every first-year Oxford philosophy student attended A iff some member of DA has |P |A the first lecture. α iff some assignment α of x to a member of DA is such that |x| ∈ |P |A iff some assignment α over A is such that |P x|α = T Domain: the set of everyone in the world A Almost everyone in the world attended the first lecture. This is correct but the general case is more complex.

Quantiﬁers Quantiﬁers

α The semantics of quantifiers is complicated by the need to How to determine |∃yRxy|A? deal with multiple quantifiers in sentences such as ∀x∃yRxy α |∃yRxy|A = T Suppose we try to evaluate this as before under A with α iff some d in DA is such that |x| stands in |R|A to d domain DA α β iff some assignment β over A is such that |x| stands in |R|A to |y| |∀x∃yRxy|A = T α So we don’t have to keep track of multiple assignments: iff every assignment α over A is such that |∃yRxy|A = T α β To progress any further we need to be able evaluate ∃yRxy Say that β differs from α in y at most if |v| = |v| for all under an assignment α of an object to x. variables v with the possible exception of y.

α |∃yRxy|A = T iff some assignment β over A which differs from α in y at most β β is such that |x| stands in |R|A to |y| iff some assignment β over A which differs from α in y at most is such β that |Rxy|A = T Quantifiers Quantifiers

L2-structures Summary of semantics of L2

Here’s the full speciﬁcation of an L2-structure. Let A be an L2-structure and α an assignment over A. Atomic formulae An L2-structure A supplies two things n (1) a domain: a non-empty set D Let Φ be a n-ary predicate letter (n > 0) and let t1, t2, . . . be A variables or constants. (2) a semantic value for each predicate and constant. n α n |Φ |A is the n-ary relation assigned to Φ by A. L2-expression semantic value in A α |t|A is the object t denotes in A if t is a constant. constant: a object: |a|A α |t|A is the object assigned to t by α if t is a variable. sentence letter: P truth-value: |P |A ( = T or F) unary predicate: P 1 unary relation: |P 1| (i.e. a set) A 1 α α 1 2 2 (i) |Φ t1|A = T if and only if |t1|A ∈ |Φ |A binary predicate: P binary relation: |P |A (a set of pairs) 3 3 2 α α α 2 ternary predicate: P ternary relation: |P |A (a set of triples) |Φ t1t2|A = T if and only if h|t1|A, |t2|Ai ∈ |Φ |A etc. etc. 3 α α α α 3 |Φ t1t2t3|A = T if and only if h|t1|A, |t2|A, |t3|Ai ∈ |Φ |A etc.

Quantiﬁers Quantiﬁers

The semantics for connectives are just like those for L1. These are the semantic clauses for ∀v and ∃v. Semantics for connectives Quantifiers α α α β (ii) |¬φ|A = T if and only if |φ|A = F. (vii) |∀v φ|A = T if and only if |φ|A = T for all variable α α α assignments β over A differing from α in v at most. (iii) |φ ∧ ψ|A = T if and only if |φ|A = T and |ψ|A = T. α α α (viii) |∃v φ|α = T if and only if |φ|β = T for at least one (iv) |φ ∨ ψ|A = T if and only if |φ|A = T or |ψ|A = T. A A α α α variable assignment β over A differing from α in v at (v) |φ → ψ|A = T if and only if |φ|A = F or |ψ|A = T. α α α most. (vi) |φ ↔ ψ|A = T if and only if |φ|A = |ψ|A. Quantifiers Truth

Just one detail remains. 50 We haven’t yet said what it is for a sentence to be true in an L2-structure A. We’ve said what it is for a formula to be true in an L2-structure A under an assignment over A http://logicmanual.philosophy.ox.ac.uk α (We’ve deﬁned |φ|A; we want now to deﬁne |φ|A.) Fact about sentences The truth-value of a sentence does not depend on the assignment. α β For α and β over A: |φ|A = |φ|A (when φ is a sentence).

A sentence φ is true in an L2-structure A (in symbols: α |φ|A = T) iﬀ |φ|A = T for all variable assignments α over A. α equivalently: |φ|A = T for some variable assignment α over A.