LIN3021 Formal Semantics Lecture 3

1. LIN3021 Formal SemanticsLecture 3 Albert Gatt

2. Aims • This lecture is divided into two parts: • We make our first attempts at formalising the notion of compositionality. • First, we’ll do it by providing a compositional interpretation of a couple of formal languages. • We’ll then look at Natural Language, and ask how we can model meaning in the same sort of way.

3. Syntax and semantics of formal systems • In defining a compositional semantics for formal languages like Propositional Logic, we: • Specify the syntax using a set of recursive rules of the following kind: • If α is a wff then γ = C(α) is a wff. • where C is some unary syntactic operation, such as logical negation • If α is a wff and β is a wff, then γ = F(α, β) is a wff. • where F is some binary syntactic operation, such as logical conjunction • Specify the semantics using recursive rules that correspond to the syntax: • If α is interpreted as [[α]], then then γ = C(α) is interpreted as M([[α]]) • where M is a semantic operation corresponding to C • If α is interpreted as [[α]] and β is interpreted as [[β]], then γ = F(α, β) is interpreted as [[γ]] = G([[α]], [[β]]) • where G is a semantic operation corresponding to F. • NB: this is not too different from the way we proceeded in describing the semantics of simple predicative expressions last week.

4. Part 1 Compositional semantics for propositional logic

5. Basic vocabulary • We assume infinitely many propositional variables for atomic propositions: • p, q, r, s, t,…, p1, q1, r1, s1,… • We also have the logical connectives: • ¬, , , ->,  • We also need the two parentheses ( and ) • No other symbol belongs to propositional logic.

6. If α is a wff then γ = C(α) is a wff. • If α is a wff and β is a wff, then γ = F(α, β) is a wff. Syntactic rules • Every propositional variable is a wff • thus, p is a formula of propositional logic • so is q, r etc • If α is a wff and β is a wff, then the following are also wffs: • ¬α • (α β) • (α β) • (α β) • (α β) • Only expressions constructed by these rules are wff’s.

7. The semantics of Propositional Logic • In Propositional Logic, we’re only dealing with atomic statements and combinations thereof • (we’re not looking at the structure of those statements) • All we can do with statements is assign them a true or false interpretation. • So our semantic rules must: • Specify the truth value of the atomic statements • Specify the way that complex statements can be evaluated as a function of their structure. • Let f be a function which assigns each atomic statement a truth value 1 (true) or 0 (false).

8. If α is interpreted as [[α]], then then γ = C(α) is interpreted as M([[α]]) • If α is interpreted as [[α]] and β is interpreted as [[β]], then γ = F(α, β) is interpreted as [[γ]] = G([[α]], [[β]]) Semantic rules Syntax Semantics Every prop. variable is a wff. If α is a wff and β is a wff, then the following are also wffs: ¬α (α β) (α β) (α β) (α β) Only expressions constructed by these rules are wff’s. If α is a wff, then [[α]] = f(α) If α is a wff and β is a wff, then: [[¬α]] = 0 if [[α]=1; 1 otherwise. [[(α β)]] = 1 iff [[α]] = 1 and [[β]] = 1 [[(α β)]] =1 iff [[α]] = 1 or [[β]] = 1 [[(α β)]] = 1 iff [[α]] = 0 or [[β]] = 1 [[α β]] = 1 iff [[α]] = [[β]] Note: these semantic rules basically describe the structure of the truth tables for the connectives.

9. Using the rules for compositional interpretation • In logic, every formula is unambiguous, i.e. can be assigned a unique derivation tree (compare this to NL). • This means the semantic rules can operate unambiguously as well. • Example: ¬(p  q) ¬(p q) (syn. 2a) 0 (sem. 2a) (p q) (syn. 2c) 1 (sem. 2c) q (syn. 1) 1 (sem.1) p(syn. 1) 0 (sem. 1)

10. Part 2 Compositional semantics for propositional logic

11. Basic vocabulary • individual constants: a, b, c,… • individual variables: x, y, z… • predicate variables: P, Q, R… • quantifiers: ,  Also, from propositional logic: • propositional variables: p, q, r… • logical connectives: ¬, , , ->,  • parentheses: ( )

12. Some conventions • We will use t1, t2, etc to stand for any individual term: • a variable; or • an individual constant. • We will use Greek letters (α, β…) to stand for well-formed formulas of predicate logic.

13. If α is a wff then γ = C(α) is a wff. • If α is a wff and β is a wff, then γ = F(α, β) is a wff. Syntactic rules • If t1, t2, …, tnare individual terms, and P is an n-place predicate, then P(t1, t2, …, tn) is a wff. • If α is a wff and β is a wff, then the following are also wffs: • ¬α • (α β) • (α β) • (α β) • (α β) • If α is a wff and v is a variable, then (v)αis a wff • If α is a wff and v is a variable, then (v)αis a wff • Only the formulas constructed with these rules are wff’s of predicate logic.

14. Semantics of Predicate Logic • For PL, we need to deal with variables and other individual terms. So it’s crucial to specify a model: • M = <U,I> where • U is our domain of individuals (what our variables range over) • I is our interpretation function • We’ll use last week’s model as our example: • U = {Isabel Osmond, Emma Bovary, Alexander Portnoy, Beowulf} • We also need to introduce a variable assignment function (denoted g). • Roughly, this is just a function that, given a domain U, assigns each variable in a formula some value in U.

15. Variable assignments • Consider: • x.clever(x) • “Some x is clever.” • x “stands for” some individual in the domain U. • To determine whether this is true, we need to determine some value for x – there must be at least one which makes the formula come out true. • Suppose our model stipulates that: • [[clever]]M= {A. Portnoy, E. Bovary, I. Osmond} • If we start with x = Beowulf, then the formula is false. • We need to test alternative values of x while we’re interpreting the formula.

16. If α is interpreted as [[α]], then then γ = C(α) is interpreted as M([[α]]) • If α is interpreted as [[α]] and β is interpreted as [[β]], then γ = F(α, β) is interpreted as [[γ]] = G([[α]], [[β]]) Semantic rules • If α is a constant, then [[α]]M,g = I(α) • Constants are fixed by our interpretation function • If α is a variable, then [[α]]M,g = g(α) • Variables are assigned by our variable assignment function • If P is an n-ary predicate and t1,...,tnare terms, then [[P(t1,...,tn)]] = 1 iff <[[t1]]M,g,...,[[tn]]M,g>  [[P]]M,g • Predicates are interpreted as ordered n-tuples • If α is a wff and β is a wff, then: • [[¬α]]M,g = 1 iff [[α]]M,g = 0 • [[α β]]M,g = 1iff [[α]]M,g = 1 and [[β]]M,g =1 • [[α β]]M,g =1 iff [[α]]M,g = 1 or [[β]]M,g =1 • [[α β]]M,g= 1 iff [[α]]M,g = 0 or [[β]]M,g =1 • [[α β]]M,g = 1 iff [[α]]M,g = [[β]]M,g • If α is a wff and v is a variable, then [[(v)α]]M,g = 1 iff for all x  U, [[α]]M,g[x/v]= 1 • If α is a wff and v is a variable, then [[( v)α]]M,g = 1 iff there is at least one x  U, [[α]]M,g[x/v]= 1 Essentially the same as for propositional logic, but we need to make reference to the model M and the assignment function g

17. A closer look... • If α is a wff and v is a variable, then [[(v)α]]M,g = 1 iff for all x  U, [[α]]M,g[x/v]= 1 • If α is a wff and v is a variable, then [[( v)α]]M,g = 1 iff there is at least one x  U, [[α]]M,g[x/v]= 1 • The notation [[α]]M,g[x/v] = 1 can be interpreted as a kind of instruction: • Find an assignment function which is identical to the original, except that the value for v is substituted for x.

18. A closer look Our model A formula U = {isabel, emma, Semantics} Constants: [[i]]M = Isabel [[e]]M = Emma [[s]]M = Semantics Predicates: [[person]]M = {isabel, emma} [[book]]M= {Semantics} [[read]]M= {<isabel,Semantics>} Initial variable assignment: g(x) = emma x[book(x)  read(i,x)] Isabel reads a book Only if x = Semantics is this formula true x[book(x)  read(i,x)] x book(x)  read(i,x) book(x) read(i,x) i read x book x

19. Interpreting the formula We proceed bottom-up: • [[book]]M,g = Semantics • [[x]M,g = emma • Therefore [[book(x)]]M,g= 0 x[book(x)  read(i,x)] x book(x)  read(i,x) book(x) read(i,x) read x book x

20. Interpreting the formula We proceed bottom-up: • [[book]]M,g = Semantics • [[x]M,g = emma • Therefore [[book(x)]]M,g= 0 • [[read]]M,g = {<isabel,Semantics>} • [[i]M,g = isabel • [[x]]M,g = emma • Therefore [[read(i,x)]]M,g = 0 x[book(x)  read(i,x)] x book(x)  read(i,x) book(x) read(i,x) read x book x

21. Interpreting the formula We proceed bottom-up: • [[book]]M,g = Semantics • [[x]M,g = emma • Therefore [[book(x)]]M,g= 0 • [[read]]M,g = {<isabel,Semantics>} • [[i]M,g = isabel • [[x]]M,g = emma • Therefore [[read(i,x)]]M,g = 0 • [[book(x)  read(i,x)]]M,g = 0 x[book(x)  read(i,x)] x book(x)  read(i,x) book(x) read(i,x) read x book x

22. The situation so far • We have a formula where we’ve set the value for the free variable x using our initial variable assignment g. • We know that its value is 0. • At this point, we find that the variable is bound by the existential quantifier. And our interpretation rule says: find some assignment g’ identical to the original, except that the value for the free variable x is substituted for some other value so that the formula comes out true on this assignment x[book(x)  read(i,x)] x book(x)  read(i,x) book(x) read(i,x) read x book x

23. The situation so far • Clearly, our assignment g doesn’t make the formula come out true. • Can we find another assignment g’ identical to g, except that x in g’ is a different individual, which is a book? • g’(x) = Semantics • On this assignment, the formula comes out true. x[book(x)  read(i,x)] x book(x)  read(i,x) book(x) read(i,x) read x book x (The point is that the initial assignment was needed to get us up the tree, but ultimately had no effect on the final interpretation.)

24. In general • To compositionally interpret Predicate Logic formulas, we need models and explicit consideration of how variables are assigned. • We’ve seen an example with existential quantification. • Things proceed in largely the same way with universal quantification, except that we have to check whether the formula is true for every value of the variable, on a given assignment.