1. Semantic Typology for Human i-Languages Paul M. Pietroski, University of Maryland Dept. of Linguistics, Dept. of Philosophy

2. Human Languages acquirable by normal human children given ordinary courses of experience pair unboundedly many meanings with unboundedly many pronunciations how many types of meanings? one answer, via Frege on ideal languages: entity-denoters truth-evaluable sentences possible languages Human Languages if and are types, then so is

3. Human Languages acquirable by normal human children given ordinary courses of experience pair unboundedly many meanings with unboundedly many pronunciations how many types of meanings? one answer, via Frege on ideal languages: entity-denoters truth-evaluable sentences possible languages Human Languages S ADIE IS - A - HORSE (_)

4. Human Languages acquirable by normal human children given ordinary courses of experience pair unboundedly many meanings with unboundedly many pronunciations how many types of meanings? one answer, via Frege on ideal languages: entity-denoters truth-evaluable sentences possible languages Human Languages S ADIE SAW (_, _) > S OPHIE

5. Barbara Partee (2006), Do We Need Two Basic Types? …Carstairs-McCarthy argues that the apparently universal distinction in human languages between sentences and noun phrases cannot be assumed to be inevitable….His work suggests…that there is also no conceptual necessity for the distinction between basic types e and t…. If I am asked why we take e and t as the two basic semantic types, I am ready to acknowledge that it is in part because of tradition, and in part because doing so has worked well.… Some of Partees Suggested Ingredients for an Alternative eventish semantics for VPs: barked Bark(e, e) & Past(e) chase Chase(e, e, e) Heim/Kamp for indefinite NPs: a dog Dog(e) entity/event neutrality, and maybe predicate/sentence neutrality (cp. Tarski)

6. Human Languages acquirable by normal human children given ordinary courses of experience pair unboundedly many meanings with unboundedly many pronunciations how many types of meanings (basic or not)? one answer, via Frege on ideal languages: entity-denoters truth-evaluable sentences possible languages Human Languages if and are types, then so is

7. if and, then 0. (2) 1. (4) of 2.eight of eight of (32), including sixteen of and 3.64 of 64 of (1408), 128 of 128 of including > 1024 of and > 4.2816 of 2816 of 5632 of 5632 of (2,089,472), including 45,056 of 45,056 of > 1,982,464 of thats a lot of types

8. possible languages Fregean Languages with expression types:,, and if and are types, so is Fregean Languages with expression types:,, and if and are types, so is Human i-Languages Human i-Languages Level-n Fregean Languages with expression types:,, and the nonbasic types up to Level-n Level-n Fregean Languages with expression types:,, and the nonbasic types up to Level-n Pseudo-Fregean Languages with expression types:,, and a few of the nonbasic types Pseudo-Fregean Languages with expression types:,, and a few of the nonbasic types

9. Human Languages acquirable by normal human children given ordinary courses of experience pair unboundedly many meanings with unboundedly many pronunciations how many types of meanings (basic or not)? another answer: monadic predicates dyadic predicates possible languages Human Languages Horse(_) On(_, _) >

10. We can imagine/invent a language that has… (1) finitely many atomic monadic predicates: M 1 (_) … M k (_) (2) finitely many atomic dyadic predicates: D 1 (_, _) … D j (_, _) (3) boundlessly many complex monadic predicates Monad + Monad Monad Dyad + Monad Monad BROWN(_) + HORSE(_) BROWN(_)^HORSE(_) FAST(_) + BROWN(_)^HORSE(_) FAST(_)^BROWN(_)^HORSE(_)

11. We can imagine/invent a language that has… (1) finitely many atomic monadic predicates: M 1 (_) … M k (_) (2) finitely many atomic dyadic predicates: D 1 (_, _) … D j (_, _) (3) boundlessly many complex monadic predicates Monad + Monad Monad Dyad + Monad Monad Φ(_)^Ψ(_) applies to e iff Φ(_) applies to e and Ψ(_) applies to e ON(_, _) + HORSE(_) [ON(_, _)^HORSE(_)]

12. We can imagine/invent a language that has… (1) finitely many atomic monadic predicates: M 1 (_) … M k (_) (2) finitely many atomic dyadic predicates: D 1 (_, _) … D j (_, _) (3) boundlessly many complex monadic predicates Monad + Monad Monad Dyad + Monad Monad Φ(_)^Ψ(_) applies to e iff Δ(_, _)^Φ(_) applies to e iff Φ(_) applies to e and e bears Δ(_, _) to something Ψ(_) applies to e that Φ(_) applies to BROWN(_)^HORSE(_) [ON(_, _)^HORSE(_)]

13. Monad + Monad Monad Dyad + Monad Monad Φ(_)^Ψ(_) applies to e iff Δ(_, _)^Φ(_) applies to e iff Φ(_) applies to e and e bears Δ(_, _) to something Ψ(_) applies to ethat Φ(_) applies to FAST(_)^BROWN(_)^HORSE(_) [ON(_, _)^HORSE(_)] FAST(e) & BROWN(e) & HORSE(e) e[ON(e, e) & HORSE(e)] v[BETWEEN(x, y, z) & SOLD(z, w, v, x)] Triad & Tetrad Pentad v[Pentad(…v…)] Tetrad but & and v[…v…] permit a lot more than ^ and

14. Monad + Monad Monad Dyad + Monad Monad Φ(_)^Ψ(_) applies to e iff Δ(_, _)^Φ(_) applies to e iff Φ(_) applies to e and e bears Δ(_, _) to something Ψ(_) applies to ethat Φ(_) applies to FAST(_)^BROWN(_)^HORSE(_) [ON(_, _)^HORSE(_)] FAST(e) & BROWN(e) & HORSE(e) e[ON(e, e) & HORSE(e)] [AGENT(_, _)^HORSE(_)]^FAST(_)^RUN(_) e[AGENT(e, e) & HORSE(e)] & FAST(e) & RUN(e) [AGENT(_, _)^HORSE(_)^FAST(_)]^RUN(_) e[AGENT(e, e) & HORSE(e) & FAST(e)] & RUN(e)

15. Monad + Monad Monad Dyad + Monad Monad Φ(_)^Ψ(_) applies to e iff Δ(_, _)^Φ(_) applies to e iff Φ(_) applies to e and e bears Δ(_, _) to something Ψ(_) applies to ethat Φ(_) applies to PAST(_)^SEE(_)^ [THEME(_, _)^HORSE(_)] PAST(e) & SEE(e) & e[THEME(e, e) & HORSE(e)] PAST(_)^SEE(_)^ [THEME(_, _)^ RUN(_)^ [AGENT(_, _)^HORSE(_)]] PAST(e) & SEE(e) & e[THEME(e, e) & RUN(e) & e[AGENT(e, e)^HORSE(e)]] / \ saw / \ a horse / \ saw / \ / \ run a horse

16. How many types of meanings in human languages? How do the meanings combine? / \ / \ / \ / \ >, t> … … BROWN(_)^HORSE(_) [ON(_, _)^HORSE(_)] / \ brown horse possible languages Human Languages function application and (type-shifting or) a rule for adjunction

17. Human Languages acquirable by normal human children given ordinary courses of experience pair unboundedly many meanings with unboundedly many pronunciations in accord with language-specific constraints Bingley was ready _ to please _. Georgiana was eager _ to please _. Darcy was easy _ to please _. possible languages Human Languages ambiguous unambiguous (the other way)

18. Human Languages acquirable by normal human children given ordinary courses of experience pair unboundedly many meanings with unboundedly many pronunciations in accord with language-specific constraints hiker lost kept walking circles possible languages Human Languages

19. acquirable by normal human children given ordinary courses of experience pair unboundedly many meanings with unboundedly many pronunciations in accord with language-specific constraints The hiker who was lost kept walking in circles. The hiker who lost was kept walking in circles. Was the hiker who lost kept walking in circles? (meaning 2) possible languages Human Languages

20. acquirable by children unbounded but constrained and so presumably i-Languages in Chomskys (Church-inspired) sense function-in-intension vs. function-in-extension --a procedure that pairs inputs with outputs in a certain way --a set of ordered pairs (with no and where y z) possible languages Human Languages

21. acquirable by children unbounded but constrained and so presumably i-Languages in Chomskys (Church-inspired) sense function-in-intension vs. function-in-extension |x – 1| + (x 2 – 2x + 1) {…, (-2, 3), (-1, 2), (0, 1), (1, 0), (2, 1), …} λx. |x – 1| λx. + (x 2 – 2x + 1) λx. |x – 1| = λx. + (x 2 – 2x + 1) Extension[λx. |x – 1|] = Extension[λx. + (x 2 – 2x + 1)] possible languages Human Languages

22. acquirable by children unbounded but constrained biologically implemented procedures that pair meanings with sounds/gestures in a human way how many types of meanings? one hypothesis, via Frege on ideal languages: entity-denoters truth-evaluable sentences if and are types, then so is possible languages Human Languages

23. natural generative procedures how many types of meanings? one hypothesis, via Frege on ideal languages: if and, then THREE CONCERNS: available evidence suggests that… the proposed generative principle overgenerates (massively) Human Languages dont generate expressions of type possible languages Human Languages

24. if and, then one worry…overgeneration 0. (2) 1. (4) 2.eight of eight of sixteen of (32) 3.64 of 64 of 128 of 128 of 1024 of (1408) 4.2816 of 2816 of 5632 of 5632 of 45,056 of 45,056 of (2,089,471) 1,982,464 of possible languages Human Languages

25. natural generative procedures how many types of meanings? one hypothesis, via Frege on ideal languages: if and, then one worry…overgeneration >>>> λv. λw. λz. λy. λx. GRONK(x, y, z, w, v),, >> λZ. λY. λX. GLONK(X, Y, Z), >, e>??? ??? possible languages Human Languages x[X(x) v Y(x) v Z(x)] x[X(x)] & x[Y(x) & Z(x)]

26. Human Languages natural generative procedures how many types of meanings? one hypothesis, via Frege on ideal languages: if and, then one worry…overgeneration >>>> λw. λz. λy. λx. λe. SELL(e, x, y, z, w) >>> λz. λy. λx. λe. GIVE(e, x, y, z) >> λy. λx. λe. KICK(e, x, y) possible languages Human Languages claim: verbs dont provide evidence for supradyadic types

27. a linguist sold a car to a friend for a dollar (x) (y) (z) (w)

28. a linguist sold a friend a car for a dollar (x) (z) (y) (w)

29. a linguist sold a friend a car a dollar sold λz. λy. λw. λx. x sold y to z for w (x) (z) (y) (w) Why not just… λz. λy. λw. λx. λe. e was a selling by x of y to z for w a triple-object construction she her this that

30. (x) (y) (z) a thief jimmied a lock with a knife

31. (x) (y) (z) a thief jimmied a lock a knife Why not instead… jimmied λz. λy. λx. x jimmied y with z

32. (x) (y) (z) a rock betweens a lock a knife Why not… betweens λz. λy. λx. x is between y and z

33. (x) kicked (y) gave a linguist tossed a coin

34. (x) kicked (y) (z) gave a linguist tossed a coin to a friend

35. a linguist tossed a friend a coin tossed λz. λy. λx. x tossed y to z kicked λz. λy. λx. x kicked y to z (x) kicked (z) (y) Why not… λz. λy. λx. λe. e was a kicking by x of y to z

36. (y) kicked (x) a coin was tossed (by a linguist) tossed λy. λe. e was a tossing of y kicked λy. λe. e was a kicking of y if Kratzer and others are on the right track… representing an Agent is as optional as representing a Recipient

37. (x) kicked (y) a linguist tossed a coin if Kratzer and others are on the right track… kicked λy. λe. e was a kicking of y [AGENT(_, _)^LINGUIST(_)]^PAST(_)^ [KICK(_, _)^COIN(_)] or … ^PAST(_)^KICK(_)^ [PATIENT(_, _)^COIN(_)]

38. Chris hailed from Boston Why … and not… Chrisfrommed Boston λy. λx. λe. e was a hailing by x from y

39. Chris hailed from Boston Chris was taller than Sam Why … and not… Chrisfrommed Boston Chris talled Sam outheighted

40. Human Languages natural generative procedures how many types of meanings? one hypothesis, via Frege on ideal languages: if and, then THREE CONCERNS: it seems that… the proposed generative principle overgenerates (massively) Human Languages dont generate expressions of type possible languages Human Languages

41. Which expressions are of type ? NAMES ® © --|-- --|-- Robin = / \ Cruso = / \ Robin, t> = P. P( ® ) Cruso, t> = P. P(©) Robin = x. x = ® Cruso = x. x = © x. Robin(x) x. Cruso(x) [D1 Robin ] = x. Indexes(1, x) & Called(x, Robin)

42. Proper Nouns even English tells against the idea that lexical proper nouns are i-language expressions of type Every Tyler I saw was a philosopher Every philosopher I saw was a Tyler That Tyler stayed late, and so did this one There were three Tylers at the party, and Tylers are clever The Tylers are coming to dinner (That nice) Professor Tyler Burge was sitting next to John Jacob Jingleheimer Schmidt proper nouns seem to be of type, even if they are related to singular concepts of type

43. D(P) D(P) N(P) / \ / \ / \ D ND N N C(P) every tiger most tigers tiger(s) that I saw the Tyler some Tylers Tyler(s) this those ? / \ Tyler Burge / \ that / \ nice ? / \ Professor Burge / \ Prof. / \ Tyler Burge

44. Which expressions are of type ? V(P) / \ D(P) V(P) / \ / \ D N V D(P) that woman tossed / \ D N this coin if the nouns are of type then presumably, the indexed determiners are not of type ; if that and this are of type then presumably, the indices are not of type ; 1 2 [AGENT(_, _)^THAT(_)^1(_)^WOMAN(_)]^…

45. Which expressions are of type ? V(P) / \ / V(P) / / \ D(P) V D(P) she tossed \ it if the pronouns she and it are of type then presumably, the indices are not of type she = x. x is female 1 = x. Indexes(1, x) 1 2 [AGENT(_, _)^FEMALE(_)^1(_)]^…

46. Human Languages natural generative procedures how many types of meanings? one hypothesis, via Frege on ideal languages: if and, then THREE CONCERNS: it seems that… the proposed generative principle overgenerates (massively) Human Languages dont generate expressions of type possible languages Human Languages

47. Which expressions are of type ? SENTENCESS NP aux VP NP aux VP D + N D(P) V + D(P) V(P) every tree / \ saw / \ / \ D N D N V D(P) every tree every tree saw / \ D N every tree ? + ?? S / \ ? ?? S

48. Which expressions are of type ? SENTENCESS = TP T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary e. T e is (tenselessly) an event of John seeing Mary y. x. e. T e is (tenselessly) an event of x seeing y

49. Which expressions are of type ? SENTENCESS = TP T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary e. T e is (tenselessly) an event of John seeing Mary Why think TPs are of type instead of ? Is this expression of type, t> or type ?

50. Which expressions are of type ? SENTENCESS = TP T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary e. T e is (tenselessly) an event of John seeing Mary Why think TPs are of type instead of ?, t> E. T e[e is in the past & E(e) = T] e. T e is in the past

51. Which expressions are of type ? SENTENCESS = TP+ ? / \ T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary e. T e is (tenselessly) an event of John seeing Mary e. T e is in the past & … e. T e is in the past

52. Which expressions are of type ? ? / \ T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary e. T e is (tenselessly) an event of John seeing Mary e. T e is in the past & … e. T e is in the past ?? neg

53. Which expressions are of type ? SENTENCESS = TP T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary e. T e is (tenselessly) an event of John seeing Mary T e[e is in the past & …], t> E. T e[e is in the past & E(e) = T] vv is T a quantificational argument of V and a conjunctive adjunct?

54. Which expressions are of type ? SENTENCESS = TP (or TP+) V(P) / \ D(P) V(P) That / \ D(P) V(P) John/ \ V D(P) see Mary e. T e is (tenselessly) an event of John seeing Mary T That is (tenselessly) an event of J seeing M Tense may be needed (in matrix clauses). But does it do two semantic jobs: adding time information via the e-variable, like the adjunct yesterday; and closing the e variable, as if tense is the 3 rd argument of a verb that cant take a 3 rd argument? if T is (semantically) the verbs 3 rd argument, then why not…

55. Which expressions are of type ? SENTENCESS = TP T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary e. T e is (tenselessly) an event of John seeing Mary T e[PAST(e) & …], t> E. T e[PAST(e) & E(e) = T] God likes Fregean Semantics theory of tense (e < RefTime) & (RefTime = SpeechTime)

56. Which expressions are of type ? Maybe None: T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary PastSeeingOfMaryByJohn(_) a monadic predicate M can be polarized into a predicate +M that applies to e iff M applies to something or a predicate -M that applies to e iff M applies to nothing +Polarized | _ is such that [PastSeeingOfMaryByJohn(_)]

57. But what about Quantification? / \ / \ ran > every cow Not at all clear that the external argument of every cow is or even can bean expression of type

58. / \ Fido / \ chased / \ every cow / \ / \ today Bessie ran / \ today / \ / \ today / \ ran every cow

59. / \ / \ today / \ ran every cow / \ / \ today (i)t 1 ran every cow 1

60. / \ / \ / \ every cow which ran / \ / \ today (i)t 1 ran every cow 1 Why not… very syncategorematic

61. / \ today / \ Fido / \ chased / \ every cow

62. / \ today / \ Fido / \ chased / \ every cow, > We can discuss the difficulties for this kind of view in Q&A. But my point is not that an analysis of quantifiers cannot be preserved. My point is that there is no argument here for the standard typology, especially given doubts about., >

63. Human Quantification: Still Puzzling But maybe… every is a plural Dyad: EVERY(_, _) applies to iff the Xs include the Ys every cow is a plural Monad: [EVERY(_, _)^THE-COWS(_)] applies to the Xs iff the Xs include the cows every cow ran is a plural Monad: [EVERY(_, _)^THE-COWS(_)]^RAN(_) applies to the Xs iff the Xs include the cows & the Xs ran

64. Human Quantification: Still Puzzling But maybe… most is a plural Dyad: MOST(_, _) applies to iff the Ys that are Xs outnumber the Ys that are not Xs most cows is a plural Monad: [MOST (_, _)^THE-COWS(_)] applies to the Xs iff the cows that are Xs outnumber the cows that are not Xs most cows ran is a plural Monad: [MOST(_, _)^THE-COWS(_)]^RAN(_) applies to the Xs iff the cows that are Xs outnumber the other cows

65. Human Quantification: Still Puzzling But maybe… every is a plural Dyad: EVERY(_, _) applies to iff the Xs include the Ys every cow is a plural Monad: [EVERY(_, _)^THE-COWS(_)] applies to the Xs iff the Xs include the cows every cow ran is a plural Monad: [EVERY(_, _)^THE-COWS(_)]^RAN(_) applies to the Xs iff the Xs include the cows & the Xs ran

66. / \ t 1 ran every cow polarity for any assignment A of values to variables… applies to e iff there was an event of A1 running applies to e iff e was an event of A1 running

67. / \ t 1 ran every cow polarity for any assignment A of values to variables… applies to e iff A1 ran so if e is the value of 1 (but A is otherwise the same), then the polarized predicate applies to e iff e ran

68. / \ t 1 ran every cow polarity for any assignment A of values to variables… applies to e iff A1 ran and we can define a Tarski Relation such that TARSKI(e, Polarity[t 1 ran], 1) iff e ran

69. / \ t 1 ran every cow polarity for any assignment A of values to variables… applies to e iff A1 ran TARSKI(e, Polarity[t 1 ran], 1) A*:A* 1 A{Satisfies(A*, Polarity[t 1 ran]) & (e = A*[1])}

70. / \ t 1 ran every cow polarity for any assignment A of values to variables… applies to e iff e ran we can define a Tarski Relation such that TARSKI(e, Polarity[t 1 ran], 1) iff e ran syncategorematic, but honestly so

71. Human Quantification: Still Puzzling But maybe… every is a plural Dyad: EVERY(_, _) applies to iff the Xs include the Ys every cow is a plural Monad: [EVERY(_, _)^THE-COWS(_)] applies to the Xs iff the Xs include the cows every cow ran is a plural Monad: [EVERY(_, _)^THE-COWS(_)]^RAN(_) applies to the Xs iff the Xs include the cows & the Xs ran

72. Lots of Further Issues Quantification Homework (including conservativity) Mary saw John, and John didnt see Mary lexicalization of singular and relational concepts lexical inflexibilities *Chris sneezed the baby *Chris put the letter Gleitman-esque acquisition of verbs Your Objection Here

73. Human Languages acquirable by normal human children given ordinary courses of experience generatively pair meanings with gestures in accord with human constraints possible languages Fregean Languages with expression types:,, and if and are types, so is Fregean Languages with expression types:,, and if and are types, so is Human i-Languages Human i-Languages

74. possible languages Fregean Languages with expression types:,, and if and are types, so is Fregean Languages with expression types:,, and if and are types, so is Human i-Languages Human i-Languages Level-n Fregean Languages with expression types:,, and the nonbasic types up to Level-n Level-n Fregean Languages with expression types:,, and the nonbasic types up to Level-n Pseudo-Fregean Languages with expression types:,, and a few of the nonbasic types Pseudo-Fregean Languages with expression types:,, and a few of the nonbasic types

75. Semantic Typology for Human I-Languages THANKS!

76. Human Quantification: Still Puzzling But maybe... DET(_, _) applies to only if the Xs are among the Ys EVERY(_, _) applies to only if the Xs are the Ys MOST(_, _) applies to only if #(X) > #(Y) #(X)

77. Monad + Monad Monad Dyad + Monad Monad Φ(_)^Ψ(_) applies to e iff Δ(_, _)^Φ(_) applies to e iff Φ(_) applies to e and e bears Δ(_, _) to something Ψ(_) applies to ethat Φ(_) applies to FAST(_)^BROWN(_)^HORSE(_) ON(_, _)^HORSE(_) FAST(e) & BROWN(e) & HORSE(e) e[ON(e, e)^HORSE(e)] HORSE(e) s[EXEMPLIFIES(e, s) & HORSEY(s)] & i[AT(s, i) & NOW(i)]

78. Which expressions are of type ? Maybe None / \ Mary saw John / \ > and John saw Mary

79. / \ Mary saw John / \ before John saw Mary [Before(_, _)^JohnSeeMary(_)] Past(_)^MarySeeJohn(_) & …

80. / \ Mary saw John / \ before John saw Mary and f[Before(e, f) & f is an event of John seeing Mary] e was an event of Mary seeing John & …