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Summary Knowledge of meaning is knowledge of entailment patterns: To grasp the meaning of S is to grasp what it entails (and, of course, act accordingly)

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Presentation on theme: "Summary Knowledge of meaning is knowledge of entailment patterns: To grasp the meaning of S is to grasp what it entails (and, of course, act accordingly)"— Presentation transcript:

1 Summary Knowledge of meaning is knowledge of entailment patterns: To grasp the meaning of S is to grasp what it entails (and, of course, act accordingly) We have applied this idea to sentential connectives Result: A recursive, compositional characterization of how truth-conditions are channeled (‘project’) through syntactic structure using connectives We now know how an infinite set of entailments can be in principle captured through a finite, in fact, small machinery

2 A note on the bad and the ugly. In developing our theory of connectives, we have implicitly adopted the good (= classical logic/set theory). What about the bad and the ugly? The bad: i. Not (||IP|| t ) = 1- ||IP|| t ii. And (||IP 1 || t, ||IP 2 || t ) = MIN (||IP 1 || t, ||IP 1 || t ) iii. Or(||IP 1 || t, ||IP 2 || t ) = MAX (||IP 1 || t, ||IP 1 || t ) Multivalued logics

3 The ugly i.Not(||IP|| t ) = 1, if ||IP|| t = 0; Not(||IP|| t ) = 0, if ||IP|| t = 0; Not(||IP|| t ) is undefined otherwise ii.And (||IP 1 || t, ||IP 2 || t ) = 1 if both ||IP 1 || t, ||IP 2 || t are defined and true; And (||IP 1 || t, ||IP 2 || t ) = 0 if one of ||IP 1 || t, ||IP 2 || t is defined and false; And (||IP 1 || t, ||IP 2 || t ) is undefined otherwise iii. Or (||IP 1 || t, ||IP 2 || t ) = 1 if one of ||IP 1 || t, ||IP 2 || t is defined and true; And (||IP 1 || t, ||IP 2 || t ) = 0 if both of ||IP 1 || t, ||IP 2 || t are defined and false; And (||IP 1 || t, ||IP 2 || t ) is undefined otherwise. Strong Kleene logic (information based) Which of these modification of the good predicts better how truth conditions project (= how complex concepts behave)?

4 Are we really getting it right? Suppose (a) is true at some time t. Our approach predicts that sentences (b)-(c) would be true at any such time: (a) John doesn’t smoke (b) If John smokes, he is happy (c) John drinks or doesn’t smoke Yet, we certainly would not use (b) or (c) to describe what is going on in (a). Why?

5 Close encounters of the 4th kind: after truth conditions, entailments, (and presuppositions), Implicatures Words/sentences have an indefinite number of meanings a. The ham sandwich wants his bill b. Who stole the steak?The dog looks happy c. The man with the Martini is a bore [Context: you are at a party; a man is sipping a drink, which you know to be Vodka; nobody else is drinking: What did the speaker mean? Would you understand what was meant?] How is all this possible?

6 Meaning is composite. But it has a precise structure Some information comes directly from ‘core’ meaning. Other information rides on reasoning about the speakers intentions and what is known in the context. Both aspects need computation and logic … including jokes, lies, poems, etc.

7 In the beginning there is syntax… Sounds are arranged into patterns by syntax Words/morphemes are anchored to data points/information structures. Sentences have truth conditions that depend ultimately on what words refer to and how they are syntactically arranged [Compositionality]

8 Then, there is core meaning (which may not be what you think) ||John|| t = j ||snores|| t = s, where for any u, s(u) = 1 iff u does whatever it is that amunts to snoring || doesn’t || t = Neg’ || [John doesn’t snore] || t = [Neg’(||snore|| t )] (j) = Neg(s(j))

9 Then, comes use (Thanks to Paul Grice) It is ultimately least effortful to cooperate, when speaking. Be relevant Be truthful Do not under- or over-inform Be orderly Both core meaning and conditions of use are not ‘prescriptions’. We spontaneously conform to them

10 Layers of meaning Who stole the steak? The dog looks happy entails recursive semantics Some pet looks happy implicates maxims The dog stole the steak

11 Example 1: Quantity

12 General effect of quantity If IP contains IP’ but doesn’t entail it, the speaker isn’t sure that IP’ is true If John is home, he is watchning TV The speaker isn’t sure that John is home or that is watching TV Mary or Sue will bring the pizza Etc.

13 Example 2: inclusive vs. exclusive or. (a) If we get the money, we’ll hire either Mary or Sue (b) If we hire either Mary or Sue, we’ll be fine Against the ambiguity thesis No language has different words for exclusive vs. inclusive or. Cf: bank ambiguous in English but not in Italian (banca vs. banco) Piantina ambiguous in I but not in E (small plant vs map)

14 Exclusiveness as an implicature: Quantity again You’ll understand either Jesse or Gennaro Entailment: You’ll understand at least one of J and G Implicature: You won’t understand both Gricean reasoning: The speaker, who is well informed and opinionated, might have said You will understand both J and G He hasn’t; hence, he doesn’t believe it to be true

15 Relevance Implicatues A: Shall we smoke? B: There is a no smoking sign right there. Implicature: we should not smoke here. But specific properties of the context matter in a crucial way.

16 Relevance Implicature A:Shall we smoke? B: There is a no smoking sign right there. [New context: A and B are members of a civil disobedience group for the liberalization of Cannabis. They are cancer patients who want to get themselves arrested in order to call public attention to the issue.] Implicature: we should smoke here.

17 The mechanics of relevance impicatures A:Shall we smoke? B: There is a no smoking sign right there. B is apparently not addressing A’s question. B’s answer seems irrelevant (in violation to what normally happens) But that can’t really be… What can B be meaning, given what we know about the current situation?

18 The secret of temporal and: Manner Implicatures John met the love of his life and got married Implicature: He first met the love of his life and then got married Be orderly! Cancellability: …But not in that order. Compare: John met the love of his life and then got married. But not in that order.

19 Metaphors John is a riotJohn is a lot of fun Time fliesTime goes by quickly Metaphors as implicatures: they involve a blatant violation of quality. We ‘restore’ it by transfering some properties of literal field (e.g. the speed of flying) to the target field (e.g. the flow of time). Metaphors of this kind are ‘frozen’ (perhaps stored in your mental lexicon with their non literal meaning, or something).

20 Irony A: Its pouring outside What a nice day! B: How was Gennaro’s reading of Leopardi’s poem? He got the spelling perfectly. A is a blatant violation of quality B is a blatant violation of quantity

21 The big picture The multiple meanings of any single expression are computed from Reference of non logical words The reference of function words (‘logical component’) Truth conditions Use conditions (= Gricean maxims)

22 Implicatures are computed through: Literal (‘core’) meaning Conditions on use (maxims) Contextually salient information Background knowledge The typical trigger: some apparent violation of the maxims, which we refuse to take as such.

23 The special power of small words Reasoning uses truth functions: and, or, if, not… Reasoning can be machine executed The meaning of logical words in English does match pretty well their logical meaning… once conditions on use are factored in Logical words play multiple roles: –They glue together what we say = Project truth-conditions of atomic sentences to molecular ones –They are necessary to draw implicatures –They are probably necessary also in the understanding of non logical words [Test: try describing what snoring is without or’s, and’s, not’s, and if ‘s]

24 Where we stand Knowledge of meaning is knowledge of entailment/implicatures patterns We have applied this idea to sentential connectives (a subset of the functional lexicon) Result: A recursive, compositional characterization of how truth is channeled through syntactic structure A theory of how meanings goes beyond its core through conditions on use (/maxims)

25 More of the functional lexicon: Quantifiers in natural language. a. Every man smokes b. No man is an island c. The boy is tired Det: some, no, the, many, most, no more than three,... The category Det expresses quantifiers Quantifiers enable us to formulate general statements They enable us to talk about quantities They enable us to say things about individuals without having to name them

26 The components of a quantified sentence IP NPI’ Det N’ every man smokes Functions vs. sets: i.|| man || t (u) = 1 iff u is a man is t ii.|| smoke || t (u) = 1 iff u smokes in t iii. || man || t (u) = 1 iff u  {x: x is a man in t} iv.|| smoke || t (u) = 1 iff u  {x: x smokes in t} v. || man || t ≈ {x: x is a man in t} iv.|| smoke || t ≈ {x: x smokes in t}

27 every ||every man smokes|| t = 1 iff i. all the members of the class of men are also members of the class of smokers (the class of men is a subset of the class of smokers) ii. {x: x is a man in t }  {x: x smokes in t} iii. || man || t  || smokes || t iv. || every || t = 

28 How it all clicks together IP NPI’ Det N’ every man smokes Functions vs. sets: i.|| every|| t (X)(Y) = 1 iff {u: X(u) = 1}  {u: Y(u) = 1} ii.|| every N’ I’|| t = || every|| t (||N’|| t )(||I’|| t ) = iii. || every man smokes|| t = || every|| t (m)(s) iv.|| every|| t (m)(s) = 1 iff v. {u: m(u) = 1}  {u: s(u) = 1} iv.{u: u is a man at t}  {u: u smokes at t}

29 some || some man smokes|| t = T iff some member of the class of man is also a member of the class of smokers the intersection of the class of men with the class of smokers is non empty {x: x is a man in t }  {x: x smokes in t} ≠ 

30 no || no man smokes|| t = T iff no member of the class of man is also a member of the class of smokers the intersection of the class of men with the class of smokers is empty {x: x is a man in t }  {x: x smokes in t} = 

31 two || two men smoke|| t = T iff two members of the class of man are also members of the class of smokers the intersection of the class of men with the class of smokers contains two members ‘Exactly ’ interpretation: i. |{x: x is a man in t }  {x: x smokes in t}| = 2 ‘At least ’ interpretation ii. |{x: x is a man in t }  {x: x smokes in t}| ≥ 2 |A| = n, where n is the number of members of A

32 most || most men smoke|| t = T iff the majority of the members of the class of man are also members of the class of smokers the intersection of the class of men with the class of smokers contains more than one half of the class of men |{x: x is a man in t }  {x: x smokes in t}| > 1/2 |{x: x is a man in t }|

33 many || many men smoke|| t = T iff the members of the class of man that are also members of the class of smokers are ‘many ’ (more than n) |{x: x is a man in t }  {x: x smokes in t}| > n Q1. Does every man smoke entail many men smoke? Q2. Does many men smoke entail some men smoke?

34 few || few men smoke|| t = T iff the members of the class of men that are also members of the class of smokers are ‘few ’ (less than n) |{x: x is a man in t }  {x: x smokes in t}| < n For any N and any I’, there is no t such that || many N I’|| t = || few N I’|| t = T

35 at most two || at most two men smoke|| t = T the members of the class of man that are also members of the class of smokers are equal to or less than 2 |{x: x is a man in t}  {x: x smokes in t}| ≤ 2

36 the || the man smokes|| t has a value only if {x: x is a man in t } is a singleton; when that happens, then || the man smokes|| t = T iff {x: x is a man in t }  {x: x smokes in t}

37 The and presuppositions ||the|| t (||man|| t )(||snores|| t ) has a value iff ||man|| t (u) = 1 for just one u Whenever the(X)(Y) is defined, the(X)(Y) = every(X)(Y) i.e. true, if X  Y; and false if X  Y IP 1 presupposes IP 2 = ||IP 1 || t = 1 or 0 iff ||IP 2 || t = 1

38 Further presuppositions of quantifiers (a) Every cat is purring but there are no cats (b) Not every cat is purring but there are no cats (c) Many cats are purringbut there are no cats (d) Not many cats are meowing because there are no cats Every (A)(B) presupposes There are As Many(A)(B) does not.

39 A host of new predictions (a) Every cat purrs (b) Some cat purrs (c) Not every cat purrs (d) Not every cat doesn’t purr (e) No cat purrs (f) No cat doesn’t purr (a)entails (b) but not viceversa; (a) and (c) entail each other; (b) and (d) entail each other; (e) entails (d), if there are cats, etc.

40 Wonders of quantification Numbers and quantifiers are part of the functional vocabulary They emancipates us from having to name things They add more ‘glue’ to our thoughts They expose more of the logical structure of language

41 DP's in object position a. Lee loves every cat b. every cat Lee loves c. every cat i [Lee loves e i ] d. every’(cat’, {x i : Lee loves x i }) e. cat’  {x i : Lee loves x i }

42 The invisible movement hypothesis IPevery’(cat’, {x i : love (l, x i )}) NP i IP love (l, x i ) Det N NP VP N V NP Every,  cat, cat’ Lee, l loves, love’ e i, x i


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