Linguistics 401, section 3HaleSemantics: determiner meaningsNovember 9, 2007QuantifiersThe meaning of a determiner is naturally thought of as a relation between the set denotedby a common noun and that denoted by a postnominal predicate. Let A and B stand forthe meaning of the Nand the Irespectively.(1) a. all(A)(B)trueiffA ⊂ Bb. every(A)(B)trueiffA ⊂ Bc. some(A)(B)trueiffA ∩ B = ∅d. the(A)(B)trueiffA ⊂ B and |A| =1(2) IPNPDet NI= Det(N)(I) where Det is a quantifier.Although we have only considered single-word determiners so far, there are also multi-worddeterminers. The truth condition of [ Fewer than six NI] is that | N∩ I| < 6.(3)multiword quantifier truth conditionat least five AB true iff |A ∩ B|≥5all but three AB true iff |A|−3=|A ∩ B|Directional EntailingnessWhat happens when, through alternative word choice or the use of intersective adjectives,A or B get smaller while the quantifier D stays constant?Consider manipulating B by adding the word “fast.” The set of fast-walkers ought tobe a subset of the walkers. Exactly this restriction has been applied to the Imeaning inboth 4a and 4b. Curiously, 4b is a valid inference whereas 4a is invalid.(4) a. If every man walks, then every man walks fast.b. If no man walks, then no man walks fast.The pattern in 4 exposes a semantic distinction between “every” and “no.” “No” is down-ward entailing in its second argument, whereas “every” is not. The pattern in 5 confirmsthat “every” is upward entailing in its second argument, whereas “no” is not.(5) a. If every man walks, then every man moves.b. If no man walks, then no man moves.Is “every” downward entailing in its first argument? Yes. The validity of the inference in 6attests this property.(6) If every person walks, then every man walksConsider negative polarity items such as ever, at all, any and give a damn. Theyseem to only be acceptable under the influence of some negative word (example 7).1(7) a. I don’t want to go to London at all.b. ∗ IwanttogotoLondonatallc. She never gave a damn about youd. ∗ She gave a damn about youe. Susan seldom makes any comments in classf. ∗ Susan always makes any comments in classNPIs are acceptable as the A element with “every” but not in the B element.(8) a. Every fan who could ever dream of being on TV can be in the audience forTRL.b. Every voter who can conceive at all of a better world will support the Kyotoprotocol.(9) a. ∗ Every fan can ever be in the audience for TRL.b. ∗ Every voter will support the Kyoto protocol at all.(10) a. No fan can ever be in the audience for TRL.b. Few fans can ever be in the audience for TRL.(11) a. No voters will support the Kyoto protocol at all.b. Few voters will support the Kyoto protocol at all.A correct empirical claim about NPIs appears to be that they may grammatically appear indownward-entailing semantic environments; even with quantifiers like few that don’t appearto be negative in the way that n’t and no are.Comparative determiners such as more...than also seem to license negative polarity itemsas in 12. The examples in 13 suggest that it is only the first argument that licenses theNPIs, though.(12) a. Phil reads more books than any other student.b. Phil reads more books than I ever thought at all possible.(13) a. ∗ Phil reads more books ever than other students.b. ∗ Phil reads more books at all than other students.Is a comparative like more...than downward entailing in its first argument? The validity ofthe inferences in 14 supports this characterization.(14) a. If more students danced than teachers sang, then more students danced thanteachers sang a ballad.b. If more students danced a tango than teachers, then more students danced atango than nutty teachers.downward entailing in the first argument D(A)(B)andA⊂ A → D(A)(B)downward entailing in the second argument D(A)(B)andB⊂ B → D(A)(B)upward entailing in the first argument D(A)(B)andA ⊂ A→ D(A)(B)upward entailing in the second argument D(A)(B)andB ⊂ B→ D(A)(B)Figure 1: Concise definitions of directional entailingness2ConservativityA candidate semantic universal: natural language determiners are conserva tive relations(15) A relation Q named by a determiner is conservative if and only if, for any propertiesA and B, relation Q holds between A and B if and only if relation Q holds betweenA and the things in (A ∩ B)A conservative relation Q is one where Q(A)(B) ↔ Q(A)(A ∩ B). With conservative rela-tions, we only have to look at the part of predicate B that overlaps with A to check if therelation holds. Such relations “live on” A. Nonmembers of A are irrelevant.If human language determiner meanings are conservative, then sentences whose truthconditions are Q(A)(B) should be synonymous with sentences whose truth conditions areQ(A)(A ∩ B).constituent example kind of set-theoretic objectnames ‘W.’, Paul Wolfowitz individualsVPs snores, occupies Iraq sets of individuals that do that activityNs girl, Beatle set of individuals that have that propertyAs sleepy, trigger-happy set of individuals that have that property[NN[CPthat VP ] ] girl that sleeps, Beatle that is happy set of individuals that have both propertiesA compositional semantic rule for a class of relative clauses:(16) NouterNinnerCPCthatCVP= Ninner∩ VPAre determiners really conservative? Doing the experimentThe sentence 17 is true in models where the set of Beatles, BEATLE is a subset of the setof smokers, call it SM OKE.(17) Every Beatle smokes.Said another way, the truth condition for example 17 is that every holds between thetwo sets BEATLE and SM OKE. A quick look back at 1b confirms that the meaningof the word “every” amounts to the subset relation on the two sets A = BEATLE andB = SM OK E.If the meaning of “every” is a conservative relation, then according to the definition (15)example 17 should be synonymous with a sentence where “every” relates A = BEATLEand B =(BEATLE ∩ SM OKE).How can we make such a sentence?(18) Every [NBeatle ] [Iis a [N[NBeatle ] that [VPsmokes ] ] ]Indeed, 18 has truth conditions BEATLE ⊂ (BEATLE ∩ SM OKE). If “every” is reallyconservative, then 18 should be synonymous with 17.