MSU LIN 401 - Semantics: review of compositional rules

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Linguistics 401, section 3HaleSemantics: review of compositional rulesNovember 13, 2007ReviewThe goal of semantics is to supply an account of what sentences (and their sub-pieces)actually mean. This account should be helpful in explicating relationships between sentencessuch as contradiction, synonymy (aka paraphrase) and entailment. This goal can beachieved using the concept ‘truth’.Truth-conditional semantics treats the meaning of a sentence S in terms its truth-conditions: the way the world would have to be in order for S to be true. For example, ifsomeone asks you what the sentence “Moldova is landlocked”means, you might well reply:(1)object sentence  Moldova is landlocked if and only if the nation named by “Moldova” has no border with any ocean.  truth conditionExamples like 1 are called ‘T-sentences’ [Davidson, 1967]. It’s important to keep the twoparts of the T-sentence straight; the distinction between the natural language sentencewhose meaning is to be explicated and the truth conditions that constitute the explicationof that meaning. The former are in the object language, the latter in the metalan-guage. Our metalanguage includes set-theoretical requirements that models will have tomeet in order for the sentence to be true in those models. It is often helpful for the meta-language to use an obviously different notation, so that it is clear that the translation fromobject language to metalanguage is not circular.notation idea example∈ membership 48 ∈{2, 4, 6, 8, 10, 12,...}|| cardinality |BEATLES| =4∅ empty set |∅| =0⊂ subset SEN IORS ⊂ MSUUGRAD∪ union MSUUGRAD= FRESH ∪ SOP H ∪ JUNIOR ∪ SEN IOR∩ intersection SU CCESSFU L = TALENTED ∩ HARDW ORKINGFigure 1: Set theory quick reference cardWe will legislate a solution to the problem of reference by defining a model to include1. the individuals that exist2. what words refer toFor instance, let NATION be the set of all countries in the world. This completes thedefinition of #1. Then, let us say that the object-language word “Moldova” refers to theset theoretical element m which is a member of NATION.Wemightsayalsothat“Italy”denotes a different element i of NATION, “France” denotes f ∈ NATION etc. The word“landlocked” shall denote the subset of NATION that in fact has no oceanfront property.All of this information can be summarized in table such as 2 (overleaf) where the denotationsof proper names of countries are specified to be model-theoretic individuals, while thedenotation of the adjective “landlocked” is a set.1(2)Moldova = mItaly = iFrance = fSlovakia = sMacedonia = alandlocked = {m, s, a}Writing up such a table indeed says what words refer to, accomplishing definition #2 andcompleting the specification of a model. Using the notation xfor “the interpretation of xon an assumed model” one can state truth conditions precisely.sentence (object language) truth condition (metalanguage)Moldova is landlocked Moldova∈ landlockedThe model in 2 satisfies this condition, because the denotation of “Moldova” is the el-ement m, the denotation of “landlocked” is the set {m, s, a} and indeed m ∈{m, s, a}.Correspondingly, this model fails to meet the truth condition for “France is landlocked”correctly deriving that sentence’s falsehood in this situation.A pleasant consequence of all of this is that the pretheoretic meaning-relationshipscontradition, synonymy and entailment now receive a systematic explanation. Twosentences S1andS2 are contradictory if and only if their truth conditions TC(S1) andTC(S2) cannot both be satisfied in the same model. For instance, 3a and 3b are con-tradictory in exactly this sense: given conventional meanings of the individual words,no model can satisfy both of their truth conditions.(3) a. Moldova is landlockedb. Moldova borders the Indian Ocean.Similarly, any model satisfying the truth conditions of an example like 4a will necessarilyalso satisfy the truth conditions for 4b. This provides an account of what entailment ac-tually is.(4) a. Mary was laughing and dancingb. Mary was laughing.The pre-theoretical sentence-relation synonymy can be analyzed as mutually-entailingtruth conditions. Any model satisfying the truth conditions of 5b will also be a modelthat satisfies the truth conditions of 5a and vice versa.(5) a. John sold a car to Maryb. Mary bought a car from John2Semantic compositionThe meaning of a sentence depends on its syntactic structure. To work out the truth con-ditions of 3a rules like the ones in 6 can be used. These rules state how the meaning of aparent node is defined in terms of the meaning of its children; the rules are “compositional.”The rules in 6b all basically say that the meaning of the parent is the same as the meaningof the child.(6) a. IPNP I= true iff NP is a proper name and NP∈ I.b. IIVP= VP VPV= V NPNNName= Name VV= VTransitive verbsThe simplest analysis is to view transitive verbs as denoting 2-place relations. These can berepresented by sets of pairs. For instance, it could be that Charles loves Camilla and Dianabut not Fergie. In model 7, Camilla loves him back reciprocally whereas Diana is fed up.(7)Charles = cFergie = fCamilla = aDiana = dloves = {(c, a), (c, d), (a, c)}The rule of semantic composition for transitives (8) roughly says “the denotation of a Vthat has an NP complement is the set of individuals occupying the first component of anypairs in the denotation of the V whose second component matches the meaning of the NP.”(8) VVNP= { X | V(X)(NP) } where NP is a proper name.3QuantifiersThe meaning of a determiner is naturally thought of as a relation between the meaningsborne by a common noun and those contributed by predicate verb phrases. This is thestandard semantics for quantified NPs. Let A and B stand for the meaning of the Nandthe Irespectively.(9) a. all(A)(B)trueiffA ⊂ Bb. some(A)(B)trueiffA ∩ B = ∅c. the(A)(B)trueiffA ⊂ B and |A| =1(10) IPNPDet NI= Det(N)(I) where Det is a quantifier.Intersective AdjectivesSome kinds of adjectives “narrow down” the denotation of a common noun.(11)


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