1. Semantic competence: summary For any sentence S of our language, we know: 1.What the world would have to be like for it to be true (truth conditions) 2.What it entails 3.What it presupposes 4.What it implicates These are the semantic counterparts of intuitions about well formedness; they constitute our primary semantic data. We want a computationally tractable theory of (1)-(4) WE WILL MOSTLY FOCUS ON (1), (2) and eventually (4)

2. Getting started: Truth conditions Gennaro swims Gennaro is a linguist This is red Interpretation: -A mapping of expressions into meanings -Meanings: ways of getting from situations of use/times/contexts to informational values/data points

3. Gennaro swims Gennaro is a linguist This is red

4. Composition rules: i.|| [ TP NP T’]|| t = ||T’||(||NP||) ii.||Gennaro runs|| t = ||runs|| t (||Gennaro|| t ) = r(GC) which is 1 iff GC runs at t Truth conditions: a specification of how a sentence leads us from times/contexts t to whether that sentence holds in t

5. Meaning is compositional TP NP T’ N T AP This is A red ||this is red|| t = ||is red|| t (||this|| t ) = ||is|| t (||red|| t )(||this|| t ) = what the speaker is pointing at at t is red ||is|| t (A t ) = A t ||was|| t (A t ) = A t’, where t’ is some t that precedes t

6. Variations on the semantics of predicates and theories of concepts The Classical strategy (‘the good’): ||red|| t (u) = 1 iff the necessary and sufficient conditions for being red are met by u. Under this view, the function ||red|| t corresponds to/determines a set { x: ||red|| t (x) = 1} The Fuzzy strategy (‘the bad’): ||red|| t (u) = n, where n  [1,0] depending how close u is to focal red The Supervaluation strategy (‘the ugly’) ||red|| t (u) = 1, if u is certainly red, ||red|| t (u) = 0, if u is certainly not red; if u is neither, then ||red|| t (u) lacks a value

7. Comparatives: this is more red than that The Fuzzy strategy/ (the bad): r(u) > r(u’) The Supervaluation strategy (the ugly): It is impossible to extend r to r + so as to make r + (u’) = 1, without also making r + (u) = 1 The Classical strategy (the good) This is red = r d (u) = u is red to degree d This is more red than that = the degree d such that r d (u) = 1 is higher than the degree d’ such that r d’ (u’) = 1

9. Where we stand The beginning of a theory of Truth Conditions, in which they depend on the denotation of predicates and nouns. Our current model of model of the semantics of predicates: Functions from times and individuals into truth values (in three variants – the good, the bad, and the ugly) Three questions that this approach raises: i.Which predicate is associated with which function? ii. How does one determine whether ||red|| t (u) = 1 (what makes something red/a cat, etc.?) iii. What does a normally competent speaker know about the red- function?

10. Our three questions Referential DPs: t  individuals Predicates: t  characteristic functions from individuals into truth values i. Which predicate is associated with which function? This is determined by some historically established link between a phonological form and a particular function, sustained and transmitted by a spontaneous social convention This first question is relatively uncontroversial

11. Two more controversial questions ii. How do you determine whether ||cat|| t (u) = 1? = What makes something a cat? To be addressed in terms of the best theory of what cats are (a certain genetic template?); but also a matter of social practices/ecology (when are people willing to call a cat- embryo a cat? When does a dead cat cease to be a cat?) iii. What does a normally competent speaker know about the cat- function? What concept does the competent speaker associate with being a cat? Some way of computing a cat-function causally linked to cats and reliable enough for successful communication/survival

12. Two positions Externalism: only extra-mental cat-functions matter to language Internalism: only mind-internal cat-functions matter to language An internalist needs some story of how concepts are linked to their extra-mental manifestations. Not trivial. A classic externalist argument: Let w be your water-function; applied to some quantity x of clear liquid it returns 1; we then discover that x is not H2O. Do we still want to say that x is water? Internalist prediction: yes (x fits with the procedure) Externalist prediction: no (x is not a sample of what water is causally linked to)

13. In conclusion, what do we know about the meaning of content words? Something about: - their ‘logical type’ - how to link them up with the appropriate data set in our environment and/or to the corresponding cognitive structures Perhaps this isn’t all that much. But wait until we get to function words… (Which is what linguists have been doing anyhow): - how content words contribute to entailment, presuppositions, etc. and viceversa Good news: we don’t have to decide between exernalism and internalism to keep going

14. The power of little words a. not (doesn’t, isn’t), if, and, or b. i. John doesn’t smoke ii. it isn’t true that Lee or Kim smoke c. If Kim smokes, Lee smokes d. If Kim smokes and Lee sings, Sue isn’t happy You will find these words in every language And their syntax is quite rich…

15. A highly simplified starting point con 1 TP TPcon 1  if a. TP  TP con 2 TP con 2  and, or b.TP  con 3 TPcon 3  not TP it is not true that TP Lee smokes

16. How this syntax will need to be changed: Make it more X’-theoretic TP C TP TP if Sue sings Mary is happy TP CP TP C TP Mary is happy if Sue sings

17. Similarly, for other binary connectives Mary likes John or Mary likes Bill TP TP ConTP M likes J or M likes B Probably or is a head and the structure is binary: XP TP X’ X TP M likes J or M likes B

18. Constituent Coordination Mary likes John or Bill An ellipsis analysis: Mary likes John or Mary likes Bill [Only sentences get coordinated] A ‘generalized coordination’ analysis: XP  [XP Con XP], for any X [Any constituent gets coordinated]

19. Constituent (VP-) negation It is not true that John smokes TP Con IP it is not true that John smokes John doesn’t smoke TP NPT’ T negP neg VP V J does not smoke

20. Structural properties of connectives a. Structural ambiguity i. it is not true that Lee smokes and Kim smokes ii.  it is not true that  Lee smokes and Kim smokes  iii.  it is not true that Lee smokes  and Kim smokes  iv. There were… [[smart women] and men] vs [smart [women and men]] b. Recursiveness TP TP con TP TP con TP........................ [Lee smokes and [Sue smokes and … ]

21. Truth conditions for negative sentences 1  0 a. Lexicon: ||not|| t =0  1 b. || not TP || t = || not || t (|| TP || t ) c.It is not true that Leo smokes = 1 iff Leo smokes = 0 Consequences a. It is not true that it is not true that John is Italian b. John is Italian c. ||[not [not J is Italian]]|| t = ||not|| t (||[not J is Italian]]|| t ) = ||not||(||not||( J is Italian))|| t = ||J is Italian|| t d. ||[not [not [not J is Italian]]]|| t = ||[not [J is Italian]]|| t

22. Double (and triple) negations in real life A1: Nobody will come B: I doubt it it = that nobody will come A2: I don’t [doubt it] B = I doubt that nobody will come = (I believe that) it is false that nobody will come = (I believe that) somebody will come A2: I don’t doubt that nobody will come = I do not believe that it is not the case that nobody will come = nobody will come

23. All of a sudden: entailment

24. Cosmetics: VP negation Neg’ maps functions from individuals into truths values into their negative counterparts (concepts  negative concepts) For any t, and any u, [Neg’(||VP|| t )](u) = Neg (||VP|| t (u)) (a) John doesn’t run (b) [||doesn’t|| t (||run|| t )](j) = [Neg’(||run|| t )](j) = Neg(||run|| t (j)) So, ||John doesn’t run|| t = ||it is not true that John runs|| t

25. Truth conditions for connectives: conjunction 1 1  1 1 0  0 ||and|| t = 0 1  0 0 0  0 a. || TP 1 and TP 2 || t = ||and|| t (||TP|| t, ||TP|| t ) b. ||TP 1 and TP 2 || t = 1 iff || TP 1 || t = || TP 2 || t = 1 c. i. Leo is Italian and Lee is American ii. Lee is American iii. Lee is American and Leo is Italian (i) entails (ii) and it also entails (iii)

26. A problem: temporal interpretations of conjunctions

27. Truth conditions for connectives: disjunction 1 1  1 ||or|| t =1 0  1INCLUSIVE 0 1  1 0 0  0 a.|| TP 1 or TP 2 || t = ||or|| t (||TP 1 ||, || TP 2 || t ) b.|| TP 1 or TP 2 || t = 1 iff one of the following conditions holds: || TP 1 || t = 1 and || TP 2 || t = 0 || TP 1 || t = 0 and || TP 2 || t = 1 [|| TP 1 || t = 1 and || TP 2 || t = 1] John or Bill could lend us the money. Not Paul

28. Is or ambiguous? 1 1  0 ||or|| t =1 0  1exclusive 0 1  1 0 0  0 a.|| TP 1 or TP 2 || t = ||or|| t (||TP 1 ||, || TP 2 || t ) b.|| TP 1 or TP 2 || t = 1 iff one of the following conditions holds: || TP 1 || t = 1 and || TP 2 || t = 0 || TP 1 || t = 0 and || TP 2 || t = 1 They hired (either) Mary or Sue [false, if they hired both]

29. A complex intuition (a)Mary doesn’t like (both) Sue and Bill (b)Mary doesn’t like Sue or she doesn’t like Bill (or possibly she doesn’t like either) (c)It is not true that Mary likes Sue and that she (also) likes Bill (c) Neg (And (||M likes S|| t, ||M likes B|| t )) Is (a) [ = (c)] predicted to entail (b)?

30. Yes! (a)Neg (And (||M likes S||, ||M likes B||)) = (b)Or (Neg(||M likes S||), Neg(||M likes B||)) Proof that (a) entails (b):[by reductio/contraposition] For any t, assume ||(b)|| t = 0; then Neg(||M likes S|| t ) = 0 and Neg(||M likes B|| t ) = 0; if so, ||M likes S|| t = 1 and ||M likes B|| t = 1. But then, ||(a)|| t = 1. So there can’t be any t such that ||(a)|| t = 1 and ||(b)|| t = 0. Proof that (a) entails (b):[direct] For any t, assume ||(a)|| t = 1; then And(||M likes S|| t, ||M likes B|| t )) = 0; if so ||M likes S|| t = 0, or ||M likes B|| t = 0, or both. But any of these conditions suffices for ||(b)|| t = 1 (for if, e.g., ||M likes S|| t = 0, then Neg(||M likes || t ) = 1, etc.)

31. Truth conditions for connectives: conditionals Intuition: if A, B is true iff you can rule out with certainty that A = 1 and B = 0 E.g.: If Lee is happy, Kim is happy The speaker excludes that Lee is happy and Kim isn’t (though she may not know whether Lee is in fact happy) 1 1  1 ||if|| t = 1 0  0 0 1  1 0 0  1 || if TP 1 TP 2 || t = ||if||(||TP 1 ||, ||TP 2 || t ) iff it is not the case that: || TP 1 || t = 1 and || TP 2 || t = 0

32. A calculus of entailment a.TP 1 if TP 2 TP 3 TP 4 TP 5 Leo comes or Liz comes we will have fun b. TP 6 if TP 7 TP 8 Leo comes we will have fun i.Assumption ad absurdum: for some t, ||TP 1 || t = 1 and ||TP 6 || t = 0 ii.||TP 7 || t = 1 and ||TP 8 || t = 0, from (i) and sem. If iii.||TP 4 || t = 1 and ||TP 3 || t = 0, from (ii) and syntactic identity iv.||TP 2 || t = 1, from (iii) and sem. Or v.||TP 1 || t = 0, from (iii),(ii) and sem. If. BUT: this contradicts (i).

33. Further predictions (a)John won’t hire Mary or Sue (b)John wont hire Mary and won’t hire Sue (c)If John has enough money he will hire Mary and Sue (d)If John has enough money he will hire Sue (e)If John is in a bad mood, he is taciturn (f)If John is not taciturn, he is not in a bad mood (a) and (b) entail each other; (c) entails (d) but not viceversa; (e) and (f) entail each other.

34. 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