Formal Semantics Fall 2007 Predicate Logic and English Syntax

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1 Formal Semantics Fall 2007 Predicate Logic and English Syntax Sophia Malamud 1. Functions. Functions: (1) A relation R from A to B is a function from A to B (F:A B) iff: a. The domain of R is A. (except for partial "functions") Every member of A appears at least once as first member of a pair. b. Each element in the domain is paired just with one element in the range Every member of A appears at most once as first member of a pair. (2) Argument and value: F(a) = b (3) a. Functions from A to B are called functions into B. b. Functions from A to B such that the range of the function equals B are called onto B. (4) a. Functions from A to B where every member of B is assigned at most once to a member of A are called one-to-one. b. Otherwise, we'll call them many-to-one. (5) Functions that are at the same time onto and one-to-one are called one-to-one correspondences. The characteristic function of a set: (6) a. Let A be a set. The, char A, 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. b. Let F be a function with range {0,1}. Then, char F, the set characterized by F, is {x D F(x)=1} Schönfinkelization: (7) U = {a,b,c} (8) The relation "fond of": R fond-of = {<a,b>, <b,c>, <c,c>} (9) The characteristic function of R fond-of : <a,a> 0 <a,b> 1 <a,c> 0 <b,a> 0 Char Rfond-of = <b,b> 0 <b,c> 1 <c,a> 0 <c,b> 0 <c,c> 1 1

2 (10) Turning n-ary functions into multiple embedded 1-ary functions: Schönfinkelization. a. Left-to-right: b. Right-to-left:(inverse + left-to-r. shonf.) a 0 a 0 a b 1 a b 0 c 0 c 0 a 0 a 1 b b 0 b b 0 c 1 c 0 a 0 a 0 c b 0 c b 1 c 1 c 1 EXERCISE 1: Given the following Universe, spell out the characteristic function of the set in (11) and schönfinkelize it right-to-left. (12) U = {d, e} (13) 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, for the purposes of this course)-- 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. Remember Montague s thesis: Natural languages can be treated as interpreted formal languages. Bach s PC language: predicate calculus has lexicon, syntax, and semantics. J Lexicon for (a fragment of) English: Grammatical Category Proper Nouns Pronouns Nouns Intransitive Verbs Transitive Verbs Ditransitive Verbs Lexical Units Joan, Pat, Philadelphia He, she, it, they, him dog, chair, table, water sleep, snore, swim, jump see, find, kiss, hug give, put, send 2

3 Propositional Verbs Auxiliary Verbs Determiners Prepositions Adjectives Adverbs Complementizers Conjunctions know, claim, believe will, would, could, must, might the, a, some, every, each, most, my, their with, without, in, on, to, after, before sad, happy, tall, green, vegetarian, former quickly, carefully, very when, if, whether, although, that and, or, but J Syntax. Syntax gives rules governing how to build sentences. E.g., it contains rules similar to (14) and (15), abbreviated in (14 ) and (15 ) respectively. J 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. (14) (14 ) (15) (15 ) If φ is a proper noun and ψ is an intransitive verb, then the sequence φψ (disregarding inflection) is a sentence. S Npr Vintr 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. S Npr AUX Vintr (16) a. Bill walks. b. Bill will walk. c. * Bill walk will. (17) 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. J 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). 3

4 (18) If φ is a proper noun and ψ is an intransitive verb, then [[φψ]] is true if and only if (iff) [[φ]] [[ψ]] (19) If ω is a proper noun, φ is an auxiliary verb and ψ is an intransitive verb, then [[ωφψ]] is true if and only if (iff) [[ω]] [[φ]] [[ψ]] 3. Syntax. Sentences are not just bags of words it matters how they are put together. (20) Here's an example from Groucho Marx in Animal Crackers, where the sentence is structurally ambiguous.. One morning I shot an elephant in my pajamas. How he got into my pajamas I dunno. The ambiguity here centers on the prepositional phrase in my pajamas: does it modify the noun elephant, or the entire verb phrase? We can express these different meanings by grouping the words together in a particular way: One morning I (shot (an elephant in my pajamas)). I shot the elephant that was in my pajamas One morning I (shot (an elephant) in my pajamas). I shot the animal while wearing my pajamas This grouping reflects how the pieces of the sentence were put together by phrase structure rules. J 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. Diacritics: X* means that you can have as many X as you want (including none whatsoever). (21) S NP VP S S Conj S NP Npr NP Det N' N' Adj* N' PP* N' N' Conj N' N' N VP Adv* V' PP* V' Vtrans NP V' Vintrans PP P NP Npr John, Bill, Brandeis University, N cat, dog, boy, pig, table, binoculars Det a, the, some, my, 4

5 Adj tall, nice, green, P on, from, with, Vtrans see, like, help, kiss Vintr run, sleep Adv quickly, soon Conj and, or, but J Running an example sentence: (22) The girl kissed the boy. S NP VP \ Det N V / The N V NP / / \ girl kissed Det N the N boy J 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 syntactico-semantic units. (23) 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 (22) 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 is this sentence. Etc. J QUESTION: 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. (24) The tall man from Philadelphia helped the small boy. J QUESTION: For each of the following sentences, our grammar allows us to generate more than one tree. Each of those trees makes different claims about what the syntactico-semantic units of the sentence may be, and, in fact, each tree corresponds to a different meaning of the sentence. For each sentence, draw all the possible trees and explain in your own words the meaning that each tree 5

6 attributes to the sentence. (25) I shot an elephant in my pajamas. (26) John called a woman from my favorite country in Europe. (27) Some curious men and women from Philadelphia attended the meeting. For the sentence (28), draw a tree that corresponds to each picture, below the picture: (28) I saw an owl with a telescope J Other ambiguities: (29) Only John likes his teacher. (30) Mary defended herself, and Sue did too. (31) Mary defended herself better than Sue did. 4. The semantic rules: how the meanings of words are combined. J Combining a subject NP with the VP: (32) For any arbitrary NP, VP and world w S w = 1 iff [[NP]] w [[VP w NP VP 6

7 QUESTION 2: Take the denotation of kiss in w to be the set of pairs in (33). What is the denotation of VP in (34)? And what is the semantic contribution of the rule merging the V and the NPob into the VP? (33) [[kiss w100 = {<Ann Ann>, <Ann Betty>, <Betty Connor>} (34) S NPsu VP / Betty V NPob kisses Connor (35) For any arbitrary Vtrans, NP and world w VP w = Vtrans NP EXERCISE 2: With the 3-place predicate assign, we have three levels of NP embedding: least embedded (=NP subject), middle embedded (= NP indirect object), and most embedded (= NP direct object). For the least embedded, we can use the rule in (32). For the middle embedded, we can use the rule we gave in (35). What is the rule combining the most embedded NP (=the NP direct object) with the ditransitive verb assign? And what is the resulting denotation of V? (36) S NPsu VP / Ann V NPIO \ V NPDO (to) Connor assigned Ann (37) For any arbitrary Vditrans, NP and world w VP w = Vditrans NP 7