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Ling 130 Notes: Predicate Logic and English Syntax

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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. Montague’s thesis: Natural languages can be treated as interpreted formal languages. The language of Predicate Logic has lexicon, syntax, and semantics. • Lexicon for (a fragment of) English (note – I omit the lexical semantics): Grammatical Category Lexical Units Proper Names Joan, Pat, Waltham Pronouns he, she, it, they, him Common Nouns dog, chair, table, water Intransitive Verbs sleep, snore, swim, jump Transitive Verbs see, find, kiss, hug Ditransitive Verbs give, put, send Propositional/Clausal Verbs know, claim, believe Auxiliary Verbs will, would, could, must, might Determiners the, a, some, every, each, most, my, their Prepositions with, without, in, on, to, after, before Predicative Adjectives sad, happy, tall, green, vegetarian, former Manner Adverbs quickly, carefully, curiously, willingly Complementizers when, if, whether, although, that Conjunctions and, or, but 2 • Syntax. Syntax gives rules governing how to build sentences1. E.g., it contains rules similar to (8) and (9), abbreviated in (8’) and (9’) respectively. (8) If φ is a proper noun and ψ is an intransitive verb, then the sequence φψ (disregarding inflection) is a sentence. (8’) S → Npr Vintr (9) If ω is a proper noun, φ is an auxiliary verb and ψ is an intransitive verb, then the sequence ωφψ (disregarding inflection) is a sentence. (9’) S → Npr AUX Vintr (10) a. Bill walks. b. Bill will walk. c. * Bill walk will. (11) Grammatical sentences of English? a. The cat the nice is saw. b. Who do you wonder whether Mary arrived? c. Who do you think that saw Joanne? d. Who do you think that Joanne saw? e. Do you wanna come for lunch? f. Pat didn't see nobody. g. The child played with Sam and I yesterday. h. Colorless green ideas sleep furiously. i. The building is nice. j. The building the guy built is nice. k. The building the guy the woman hired built is nice. l. The building the guy the woman John met hired built is nice. • Semantics: Semantics specifies a semantic operation for each syntactic rule. The notation [[.]] which we have been using stands for “the meaning of” or “the denotation of”. That is, semantics is built in tandem with syntax (we will run into problems later). (12) If φ is a proper noun and ψ is an intransitive verb, then [[φψ ]] is true if and only if (iff) [[φ]]… [[ψ]]… 1 Descriptive vs. prescriptive grammar. Note that we are using the notion of English grammar (in particular, English syntax) in a descriptive and not in a prescriptive way. The job of a linguist is to construct a grammar that generates all and only the utterances that a given group of speakers consider well formed in their dialect. This grammar may coincide or not with prescriptive grammaticality. Also, grammaticality has to be distinguished from mere semantic anomaly (the form of the sentence is fine, though the meaning is strange) and processing difficulty. 3 3 Syntax Sentences are not just bags of words – it matters how they are put together. Here's a structurally ambiguous example from Groucho Marx in Animal Crackers: (13) a. One morning I shot an elephant in my pajamas. How he got into my pajamas I dunno. Syntactico-semantic units = constituents. We can express these meanings by grouping the words together in a particular way: (13) b. One morning I (shot (an elephant in my pajamas)). => the elephant was in my pajamas c. One morning I (shot (an elephant) in my pajamas). => I shot while wearing my pajamas This grouping reflects how the pieces of the sentence were put together by syntactic rules. • The following is the set of syntactic rules we start with. During the semester, some modifications will be made as we cover more types of sentences. (14) S →NP VP S →S Conj S NP →Npr NP →Det N' N' →Adj N' N' → N' PP N' →N' Conj N' N' →N VP →Adv VP VP →VP PP VP →Vtrans NP VP →Vintrans VP →V' to NP VP → Vclaus that S V' → Vditrans NP PP → P NP Npr → John, Bill, Brandeis University, he, she, it… N → cat, dog, boy, pig, table, binoculars… Det → a, the, some, my, all … Adj → tall, nice, green, … P → on, from, with, … Vtrans → see, like, help, kiss … Vclaus → know, claim, believe … Vditrans → give, assign, introduce … Vintr → run, sleep … Adv → quickly, soon … Conj → and, or, but … 4 • Running an example sentence: (15) The girl kissed the boy. S / \ NP VP / \ / \ Det N’ V NP / | | / \ The N kissed Det N’ | | | girl the N | boy • Trees and syntactico-semantic units: ◦ Every lexical item or word is a syntactico-semantic unit (minimal units). ◦ Every complete sentence is a syntactico-semantic unit (maximal unit). ◦ The tree structure of a sentence specifies all its intermediate units. (16) Every mother node α in a tree is a syntactico-semantic unit. In other words, for any node α, all the lexical material under its daughter branches forms a syntactico-semantic unit that excludes material under other non-daughter branches. The tree structure in (15) makes the following claims wrt the sentence The girl kissed the boy: the girl kissed the boy is a unit (in particular, a S). the girl is a unit (a NP). the boy is a unit (a NP). kiss(ed) the boy is a unit (a VP). the girl kiss(ed) is NOT a unit in this sentence. girl kiss(ed) the is NOT a unit in this sentence. etc. QUESTION 1: Draw the syntactic tree for the following sentence according to our grammar. Discuss what strings form a syntactico-semantic unit and what strings do not. (17) The tall man from Waltham helped the small boy. QUESTION 2: For each of the following sentences, our grammar allows us to generate more than one tree. For each sentence, draw all the possible trees and explain in your own words the meaning that each tree attributes to the sentence. (13) I shot an elephant in my pajamas. (18) John called a woman from my favourite country in Europe. (19) Some curious men and women from Waltham attended the meeting. 5 For the sentence (20), draw a tree that corresponds to each picture, below the picture: (20) I saw an owl with a telescope • Other ambiguities: (21) Only John likes his teacher. (22) Mary defended herself, and Sue did ▲ too. (23) Mary defended herself better than Sue did ▲. 4 Lambdas and types 4.1 Motivation: Set theory and thus, the semantics of Predicate Logic has problems: Problem 1: Russel’s paradox • There are several ways to define a set: ◦ list all its members {2,4,6,8} ◦ define via a property {x | x is an even positive number less than 10} • This last version is very convenient (powerful): ◦ {x | x is in this room now} ◦ {x | x is red in w100} ◦ {x | <x,m> ∈ [[kiss]] w17} But it can lead to problems, e.g. A={x | x ∉ x } QUESTION 3: is A ∈ A ? Problem 2: Verbs, prepositions, etc. not corresponding to the same sort of semantic thing What is the nature of syntactic categories, like verb, noun? • We cannot say “V always combines with one NP to make S”. Instead, we must define further categories: ◦ Vintransitive combines with one NP to make S, ◦ Vtransitive combines with two NPs to make S ◦ Vditransitive combines with three NPs to make S • Similarly, count and mass nouns behave differently: 6 ◦ Nmass is an NP ◦ Ncount combines with Det to make NP • Yet, there are syntactic reasons to call all verbs, “verbs” and all nouns, “nouns” ◦ All phrases built around a verb are VPs, all phrases built around nouns are NPs etc.
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