Ling 130 Notes: Predicate Logic and English Syntax
Ling 130 Notes: Predicate Logic and English Syntax Sophia A. Malamud February 15, 2011 1 Characteristic functions The characteristic function of a set: (1) Let A be a set. The, charA, the characteristic function of A, is the function F such that, for any x∈ A, F(x)=1, and for any x ∉ A, F(x)=0. Schönfinkelization: (2) U = {a,b,c} (3) The relation "fond of": Rfond-of = {<a,b>, <b,c>, <c,c>} (4) The characteristic function of Rfond-of (takes all combinations of two elements, gives 0/1): <a,a> →0 <a,b> →1 <a,c> →0 <b,a> →0 CharRfond-of = <b,b> →0 <b,c> →1 <c,a> →0 <c,b> →0 <c,c> →1 (5) Turning n-ary functions into multiple embedded 1-ary functions: Schönfinkelization. a → 0 left-to-right schönfinkelization a →b → 1 c → 0 a → 0 b →b → 0 c → 1 a → 0 c →b → 0 c → 1 EXERCISE 1: Given the following Universe, spell out the characteristic function of the set in (6) and schönfinkelize it left-to-right. 1 (6) U = {d, e} (7) R = { <d,d,d>, <d,e,d>, <e,d,d>, <e,e,e>, <e,e,d> } 2 Syntactic composition A formal language is a set of strings - finite sequences of minimal units (words/morphemes, for natural languages) - with meaning. The "machine" that generates those strings and their corresponding meanings is its grammar. A grammar must specify the following three components: • A lexicon which contains every minimal unit with meaning (= every word, for this course) and its grammatical category; • a syntax, that is, a set of rules that tells you how the minimal units combine to form longer units, how this longer units combine to form yet longer units, and so forth until we form full complex sentences; and • a semantics, which determines what semantic operation or function corresponds to each syntactic rule and combines the “atomic” word meanings to build the meaning of the complete sentence.
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