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) 8x(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 j‘Bertrand Russell’j has j‘is a philosopher’j. 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 jej 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) j‘Alonzo Church reveres Bertrand Russell’j = T iff binary predicate binary relation j‘Alonzo Church’j stands in j‘reveres’j to j‘Bertrand Russell’j Examples j‘Bertrand Russell’j = Russell j‘is a philosopher’j = the property of being a philosopher j‘reveres’j = 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 2 P indicates that d has property P . Informally: hd; ei 2 R indicates that d bears R to e. Example Example The property of being a philosopher The relation of revering = fhd; ei : d reveres eg = the set of philosophers = fd : d is a philosopherg Similarly: = fDescartes, Kant, Russell, . g 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 j‘Bertrand Russell’j has j‘is a philosopher’j iff Russell 2 the set of philosophers An L2-structure specifies semantic values for L2-expressions: Similarly: L2-expression semantic value constant: a object: jajA ‘Alonzo Church reveres Russell’ is true sentence letter: P truth-value: jP j (i.e. T or F) iff j‘Alonzo Church’j stands in j‘reveres’j to j‘Russell’j unary predicate: P 1 unary relation: jP 1j (i.e. a set) iff hChurch, Russelli 2 fhd; ei : d reveres eg binary predicate: P 2 binary relation: jP 2j (a set of pairs) jP bj = T iff jbj has jP j iff jbj 2 jP j jRabj = T iff jaj stands in jRj to jbj iff hjaj; jbji 2 jRj Notation: jejA 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 jxjα 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. jxj = Mercury; jyj =Venus; jx2j = 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 jejA for the semantic value of expression e in the jajA = Alonzo Church structure A under the variable assignment α. jbjA = Bertrand Russell α α α α jP jA = fFrege, Russellg jP xjA = T iff jxj has jP jA (NB: jxjA = jxj ) α jRjA = fhChurch, Russellig iff jxj 2 jP jA α α α Let assignments α and β be such that: jRxyjA = T iff jxj stands in jRjA to jyj x y z α α iff hjxj ; jyj i 2 jRjA α: Frege Russell Wittgenstein β: Church Church Church Note: semantic values of constants and predicates are unaffected by the assignment (e.g. jP jα = jP j , jajα = jaj ). A A A A Compute the following: α α β α jRabjA = T iff hjajA; jbjAi 2 jRjA jxjA = jxjA = jajA = α α α β α jRxbjA = T iff hjxj ; jbjAi 2 jRjA jP yjA = jP yjA = jP bjA = jRxyjα = jRxyjβ = jRxbjα = 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 8=9 (first approximation): j8xP xjA = T Almost everyone attended the first lecture. iff every member of DA has jP jA α iff every assignment α of x to a member of DA is such that jxj 2jP jA α The context supplies a ‘domain’ telling us who ‘everyone’ iff every assignment α over A is such that jP xjA = T ranges over Similarly: Domain: the set of first-year Oxford philosophy students j9xP xj = T Almost every first-year Oxford philosophy student attended A iff some member of DA has jP jA the first lecture. α iff some assignment α of x to a member of DA is such that jxj 2 jP jA iff some assignment α over A is such that jP xjα = 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 j9yRxyjA? deal with multiple quantifiers in sentences such as 8x9yRxy α j9yRxyjA = T Suppose we try to evaluate this as before under A with α iff some d in DA is such that jxj stands in jRjA to d domain DA α β iff some assignment β over A is such that jxj stands in jRjA to jyj j8x9yRxyjA = T α So we don’t have to keep track of multiple assignments: iff every assignment α over A is such that j9yRxyjA = T α β To progress any further we need to be able evaluate 9yRxy Say that β differs from α in y at most if jvj = jvj for all under an assignment α of an object to x. variables v with the possible exception of y. α j9yRxyjA = T iff some assignment β over A which differs from α in y at most β β is such that jxj stands in jRjA to jyj iff some assignment β over A which differs from α in y at most is such β that jRxyjA = 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, .