1. Quantified sentences TP NPT’ Det N’ every man smokes i.|| man || t (u) = 1 iff u  {x: x is a man in t} ii.|| smoke || t (u) = 1 iff u  {x: x smokes in t} iii. ||every man smokes|| t = ||every|| t (|| man|| t )(|| smokes|| t ) = 1 iff iv. {x: x is a man in t }  {x: x smokes in t}

2. Many new predictions

3. Visually, using Venn’s diagrams man smokes blond j

4. Visually, using Venn’s diagrams man smokes j

5. Many, many new predictions

6. With Venn diagrams mansmokes

7. With Venn diagrams man blond smokes

8. How properties of entailment affect things: Or again

9. More contexts 1. No one will see Mary or Sue 2. Everyone will see either Mary or Sue 3. Everyone who’ll see either M or S will report to me 4. If you are lucky you’ll see either Mary or Sue 5. If you’ll see either Mary or Sue, you are lucky 6. John is angry at either Mary or Sue 7. John is angrier than either Mary or Sue

10. More contexts 1. No one will see Mary or SueIn 2. Everyone will see either Mary or SueEx 3. Everyone who’ll see either M or S will report to meIn 4. If you are lucky you’ll see either Mary or SueEx 5. If you’ll see either Mary or Sue, you are luckyIn 6. John is angry at either Mary or SueEx 7. John is angrier than either Mary or SueIn

11. From a set to its supersets: Upward entailing contexts Bill smokes MurattiBill smokes Bill belongs to the setBill belongs to the set of of Muratti smokerssmokers 7 Muratti smokers smokers

12. From a set to its subsets: Downward entailing contexts Bill doesn’t smoke  Bill doesn’t smoke MurattiBill does not belong to the set of smokersto the set of Muratti smokers 7 7 Muratti smokers smokers

13. Structural factors in the interpretation of or 1.Everyone will see either Mary or SueEx 2.Everyone who’ll see either M or S will report to meIn 3. If you are lucky you’ll see either Mary or SueEx 4. If you’ll see either Mary or Sue, you are luckyIn Every A Bif A, B DE UE DE UE Every man smokes and drinks  every man smokes Every man smokes  Every blond man smokes If J is nervous he eats pizza  If J is nervous he eats If J eats, he doesn’t gain weigh  If J eats pizza, he doesn’t gain weigh

14. The interpretation of or is polarity sensitive

15. Why? I will see M or Sue 00  0 10  1 01  1 11  1 11  0 Ex Ex or more informative I won’t see M or S 0 0  1 1 0  0 0 1  0 1 1  0 1 1  1 Ex Ex or less informative

16. Spontaneous logicality of language We tend to interpret or exclusively ( = add the implicature of exclusiveness) when it leads to more information (strengthening) We tend to interpret or inclusively (= not add the implicature) when it leads to more information

17. Other Polarity Sensitive phenomena: the distribution of any and ever a. *There was any bird in the garden a'. There wasn’t any bird in the garden b. *Some student ever read my papers b'. No student ever read my papers c. *Many students ever read my papers c'. Few students ever read my papers Some NPIs also have ‘Free Choice’ uses a.i. You may read any book ii. * You may ever read my books b. i. Read any bookii. * Ever read my book FC is related to but distinct from Negative Polarity

18. Other cases: conditionals and avversatives a.If there were any cookies left for him to eat, by now he will have eaten them b. * If Leo yesterday stayed home, there were any cookies left for him to eat c. If Leo yesterday stayed home, there were cookies left for him to eat d. J believes that M will ever show up e. J doubts that M will ever show up

19. Distribution of NPIs with Dets a. No student ever read a paper of mine b. No student that ever read a paper of mine believed what I said Few, less than 10, at most 5 c. At most five students in this class read any paper of mine d. * Some student ever read a paper of mine The strange case of every a.* Every student who likes pasta will ever buy any tomato sauce b. Every student who likes any pasta dish will buy tomato sauce DetNPVP any/ ever

20. Det NP VP any/ever DET NPVP Some * * a * * at least three * * many * * no   at most two   no more than two   every  *

21. Relevance of DE vs UE a. No student smokesa' No student smokes the pipe b. No student smokesb’. No blond student smokes Few, no more than n, at most n.... c. At most three students will pass c’. At most three students will get an A d. At most three students will pass d’. At most three first year students will pass e. Some student smokes the pipee’. Some student smokes f. Some first year student smokesf’. Some student smokes

22. Classification of Dets in terms of UE/  vs DE/  DET NPVP Some   a   at least three   many neither  no   at most two   no more than two   every  

23. The perfect match:  with  DET NP VP Some */  */  a */  */  at least three */  */  many */neither */  no  /   /  at most two  /   /  no more than two  /   /  every  /  */ 

25. Why are NPI sensitive to DE? A detour through even How many papers will you have graded by tomorrow? a.If I am in luck, maybe even 20 b.If I am out of luck, not even one c.* If I am in luck/out of luck, may be even one d.If I am lucky, by tomorrow I will have graded twenty papers and even one honors thesis

26. What does even mean? Even p: p [and p is the least likely ALT] Even John showed up John showed up [and John showing up < (less likely) x showing up, for other salient x’s]

27. How many papers will you have graded by tomorrow?

28. Back to Negative Polarity Items What if any meant something like even one? a. There are any cookies left c. There is even one cookie left Assertion: there is one cookie left There being one cookies left < entails There being n cookies left  CONTRADICTION

29. Why NPIs like DE contexts a.There aren’t any cookies left b.Assertion: There isn’t even one cookie left c.even [not [there is one cookie left]] d. ALT: There aren’t n cookies left (c) entails every ALT. Therefore, it is consistent (and non trivial)

30. Summary If any = even one and we are good logicians its distribution follows: any yields contradictions (or trivialities) in UE contexts

31. When the secret even shows up

32. So logical properties of language (being DE vs. UE) determine..

33. So, things look pretty good…

34. Why are words like and, every, even ‘logical’? Possible answers: … because their meaning is independent of what we are talking about (the domain of things under consideration) …and it manifests itself exclusively through the inferences they give rise to

35. Languages ‘come with’ a spontaneous logic …Which might well be the key to our capacity to form and process meaning/information But there is a lot of work to do: - what is the range of variation in function words? - how are function words acquired? - is there (a natural) logic independently of language?

36. Acquisition of logical meanings /e//o//ogni//persino/  EVEN ORAND

37. Acquisition of logical meanings /e//o//ogni//persino/  EVEN ORAND Capacity for abstraction OR: only looks at truth values of sentences  : looks at concepts as sets

38. Importance of abstraction A: Mom, nobody will come to my party B: I doubt it [ it = that nobody will come] A: I don’t [doubt it] neg (neg (neg (people will come))) Without the capacity of looking at sentences in terms of their truth value, logic/recursion is powerless

39. So MEANING REFERENCE + LOGIC + ABSTRACTION Reference: the capacity to anchor expressions to data points/information structures Logic: the capacity to reason Abstraction: the capacity to ignore aspects of reference

40. What’s next John saw Bill John saw that if Bill moved, he would fall What is the meaning of that-clauses? John must leave John cannot stay How do ‘modals’ work? …and much more