Lectures 4

1. Lectures 4 Ling 442

2. Exercises • Why isn’t because truth-functional? • How do we justify the semantics of ? (E.g. p or q in English usually means p or q but not both, but this is not explained by the standard semantics of .) • What does x ~[Sad(x)] mean? • How do you translate “Bill and Mary are happy” into Predicate Logic? • Are the following two equivalent (in meaning)? ~ [Happy (j) & Sad (b)], ~Happy (j) v ~Sad (b)

3. Some semantic concepts from Ch.1 • Entailment: The truth of S1 guarantees the truth of S2 • Contradiction: always false • Tautology/analytic statement: always true • Contingency/synthetic statement: not always true; not always false; you have to “look at the world” in order to determine its truth value

4. Intersective vs. non-intersective Adj. • Almost all attributive (prenominal) adjectives are non-intersective. • Big, small, wonderful, poor • E.g. big ant, small elephant, wonderful football player, poor chess player • Your text has examples like fake. This is a different type because e.g. a big ant is an ant, but a fake gun is not (really) a gun.

5. negation • Syntax: S  neg S • ⟦~p⟧ = true iff ⟦p⟧ = false

6. Natural language quantifiers • Everything • God made Seattle. • God made everything. • Semantics: For every thing x, God made x. • God made something. • Semantics: For some thing x, God made x. • Normally, English quantifiers are restrictive: e.g. every rabbit, some elephant

7. Quantifiers • A single expression designed to talk about multiple cases. • Quantifier x [ … x …] • You declare that you will talk about a particular type of quantification (all, some, etc.) • Then you say what variable the quantifier will bind (“specialize in”, “take care of”).

8. Predicate Logic Quantifiers • Unrestricted quantifiers (referring to the “whole domain of individuals”) • x [ … x …] Semantics: for each entity x (in the entire universe), [ … x …] is true • x [… x …] Semantics: for some entity x (in the entire universe), [ … x …] is true • Syntax is straightforward: S  x S; S  x S

9. Simple Examples with Quantifiers • x[Person(x)] • y[Animal(y)] • ~x[Person(x)] • ~y[Animal(y)]

10. English Q  PL Q translation • Every dog barks translates as x[Dog(x)  Barks(x)] (and not x [Dog(x) & Barks(x)]) • Some cat laughs translates as x[Cat(x) & Laughs(x)] (and not x[Cat(x)  Laughs(x)]) • No pig flies translates as ~x[Pig(x) & Flies(x)] or x[Pig(x)  ~Flies(x)]

11. Scope Ambiguity • When two different types of quantifiers occur in the same sentence (in English) they could produce scope ambiguity. • Predicate logic can disambiguate the “original” English sentence by providing two distinct translations.

12. Scope Ambiguity Disambiguated • Every boy likes some rock star. • x[Boy(x)  y[Rock-star(y) &likes(x, y)]] • y[Rock-star(y) &x[Boy(x)  likes(x, y)]]

13. Some Set-theoretic Stuff • A set is a collection of things and is defined solely in terms of the things (if any) contained in the “generic bag”. E.g. {Olympia, Austin, Salem, Sacramento, Boise, …} or, equivalently, {x | x is a …} A set can consist of “complex objects” such as sets. <a, b> is defined as {{a}, {a, b}} {<Seattle, WA>, <Portland, OR>, <LA, CA>, …}