1. Semantic Typology and Composition Paul M. Pietroski, University of Maryland Dept. of Linguistics, Dept. of Philosophy

2. Human children naturally acquire languages that somehow generate boundlessly many expressions that connect “meanings” with “pronunciations” in accord with certain constraints. Bingley is eager to please. (a) Bingley is eager to please relevant parties. #(b) Bingley is eager to be pleased by relevant parties. Bingley is easy to please. #(a) Bingley can easily please relevant parties. (b) Bingley can easily be pleased by relevant parties. 2

3. Human children naturally acquire languages that somehow generate boundlessly many expressions that connect “meanings” with “pronunciations” in accord with certain constraints. The senator called the donor from Texas. (a) The senator called the donor, and the donor was from Texas. (b) The senator called the donor, and the call was from Texas. #(c) The senator called the donor, and the senator was from Texas. 3

4. Human Languages acquirable by normal children given ordinary courses of experience somehow generate unboundedly many expressions that connect “meanings” with “pronunciations” what are these meanings? 4

5. Human Languages acquirable by normal children given ordinary courses of experience somehow generate unboundedly many expressions that connect “meanings” with “pronunciations” what types of meaning do human linguistic expressions exhibit? (1) Fido(5) every cat (2) chase(6) chase every cat (3) every(7) Fido chase every cat (4) cat(8) Fido chased every cat. 5

6. Human Languages acquirable by normal children given ordinary courses of experience somehow generate unboundedly many expressions that connect “meanings” with “pronunciations” what are the Human Meaning Types? standard answer, via Frege’s conception of ideal languages (i) a basic type, for entity denoters (ii) a basic type, for truth-value denoters (iii) if and are types, then so is Fido Fido chased every cat. 6

7. F ELIX IS - A - CAT (_) F IDO CHASED (_, _) > F ELIX since and are types, meanings of type can be “abstracted” from the meanings of sentences (type ) that contain a name (type ) since and are types, meanings of type > can be “abstracted” from the meanings of sentences that contain two names 7

8. F ELIX IS - A - CAT (_) F IDO CHASED (_, _) > F ELIX RAN (_) EVERY (_, _) > CAT (_) …and so on “composition” (reverse abstraction) via Function Application / \ 8

9. Human Languages acquirable by normal children given ordinary courses of experience somehow generate unboundedly many expressions that connect “meanings” with “pronunciations” what are the Human Meaning Types? standard answer, via Frege’s conception of ideal languages (i) a basic type, for entity denoters (ii) a basic type, for truth-value denoters (iii) if and are types, then so is That’s a lot of types 9

10. 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 at Level 5, more than 5 x 10 12 10

11. Human Languages acquirable by normal children given ordinary courses of experience somehow generate unboundedly many expressions that connect “meanings” with “pronunciations” what are the Human Meaning Types? standard answer, via Frege’s conception of ideal languages (i) a basic type, for entity denoters (ii) a basic type, for truth-value denoters (iii) if and are types, then so is Languages Human Languages That’s a lot of types 11

12. Human Languages acquirable by normal children given ordinary courses of experience somehow generate unboundedly many expressions that connect “meanings” with “pronunciations” what are the Human Meaning Types? another answer (i) a type, for monadic predicates (ii) a type, for dyadic predicates (iii) all complex expressions are of type Languages Human Languages horse, brown, run on, from, cause 12

13. Outline for Rest of the Talk say a little more about the suggested alternative to a “Fregean” semantic typology then focus on why we need an alternative – if and are types, then so is (way too many) – a basic type, for entity denoters of (unattested) – a basic type, for truth-value denoters (unattested) if time permits, brief discussion of quantification – serious difficulties for the idea that ‘every cat’ is of type and ‘every’ is of type > – alternatives available if we don’t insist that and are basic 13

14. Outline for Rest of the Talk say a little more about the suggested alternative to a “Fregean” semantic typology then focus on why we need an alternative – if and are types, then so is (way too many) – a basic type, for entity denoters of (unattested) – a basic type, for truth-value denoters (unattested) if time permits, brief discussion of quantification (otherwise, conversations about quantification later) 14

15. We can invent languages that have boundlessly many expressions that exhibit… no semantic typology (see Tarski) finitely many semantic types (2, 3, …, a million, …) endlessly many semantic types (see Frege and Church) But where are human languages located along this dimension of potential variation? In addressing the empirical questions, it helps to have some “typologically spare” languages as possible models. 15

16. We can imagine a language whose expressions are limited to… (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 BROWN(_) + HORSE(_)  BROWN(_)^HORSE(_) FAST(_) + BROWN(_)^HORSE(_)  FAST(_)^BROWN(_)^HORSE(_) 16

17. We can imagine a language whose expressions are limited to… (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 for each entity: Φ(_)^Ψ(_) applies to it if and only if Φ(_) applies to it, and Ψ(_) applies to it 17

18. We can imagine a language whose expressions are limited to… (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 ON(_, _) + HORSE(_)   [ON(_, _)^HORSE(_)] for each entity: Φ(_)^Ψ(_) applies to it if and only if Φ(_) applies to it, and Ψ(_) applies to it |________| |_______ (thing that is) on a horse

19. We can imagine a language whose expressions are limited to… (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 ON(_, _) + HORSE(_)   [ON(_, _)^HORSE(_)] for each entity: Φ(_)^Ψ(_) applies to it if and only if Φ(_) applies to it, and Ψ(_) applies to it (thing that is) on a horse # thing that a horse is on 19

20. We can imagine a language whose expressions are limited to… (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 for each entity: Φ(_)^Ψ(_) applies to it if and only if Φ(_) applies to it, and Ψ(_) applies to it for each entity:  [ Δ(_, _)^Ψ(_)] applies to it if and only if it bears Δ to something that Ψ(_) applies to 20

21. FAST(_)^BROWN(_)^HORSE(_)  [ON(_, _)^HORSE(_)] is like is like FAST(e) & BROWN(e) & HORSE(e)  e[ON(e’, e) & HORSE(e)] But ‘&’ and ‘  e[…e…]’ allow for much more. FAST(e) & BROWN(e’) & HORSE(e’’)  e[ON(e, e’) & HORSE(e)]  e[BETWEEN(e’, e, e’’) & SOLD(e’’’, e’’’’, e’’’’’, e)] And human languages are not this permissive. 21

22.  [ AGENT(_, _)^HORSE(_) ] ^EAT(_)^FAST(_) is like  e [ AGENT(e’, e) & HORSE(e) ] & EAT(e’) & FAST(e’)  [ AGENT(_, _)^FAST(_)^HORSE(_) ] ^EAT(_) is like  e [ AGENT(e’, e) & FAST(e) & HORSE(e) ] & EAT(e’) ] We don’t need variables to capture the meanings of ‘horse eat fast’ and ‘fast horse eat’. 22

23. SEE(_)^  [ THEME(_, _)^HORSE(_) ] is like SEE(e’) &  e [ THEME(e’, e) & HORSE(e) ] SEE(_)^  [ THEME(_, _)^  [ AGENT(_, _)^HORSE(_) ] ^EAT(_) ] is like SEE(e’’) &  e’ [ THEME(e’’, e’) &  e [ AGENT(e’, e)^HORSE(e) ] & EAT(e’) ] We don’t need variables to capture the meanings of ‘see a horse’ and ‘see a horse eat’. 23

24. What are the Human Meaning Types? --two basic types, and --endlessly many derived types of the form --a monadic type --a dyadic type, for finitely many atomic expressions -- can combine with to form -- +  +  24

25. Outline for Rest of the Talk ✔ say a little more about the suggested alternative to a “Fregean” semantic typology then focus on why we need an alternative – if and are types, then so is (way too many) – a basic type, for entity denoters of (unattested) – a basic type, for truth-value denoters (unattested) if time permits, brief discussion of quantification (otherwise, conversations about quantification later) 25

26. 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 26

27. But even below Level Five… λy.λx.Predecessor(x, y) Level Two function of type λy.λx.Precedes(x, y) another function of the same type λR.TRANSITIVE[R] Level Three function of type, t> | | | 3 2 0 λy.λx.Precedes(x, y) = ANCESTRAL[λy.λx.Predecessor(x, y)] λR.ANCESTRAL[R] Level Three function of type, > 27

28. But even below Level Five… λy.λx.Predecessor(x, y) Level Two function of type λy.λx.Precedes(x, y) = ANCESTRAL[λy.λx.Predecessor(x, y)] λR.ANCESTRAL[R] Level Three function of type, > ANCESTRAL-OF[λy.λx.Precedes(x, y), λy.λx.Predecessor(x, y)] λR.λR’.ANCESTRAL-OF[R’, R] Level Four function of type,, t> | | | 4 2 3 28

29. Frege had to invent a language that supported abstraction on relations The plate outweighs the knife. The plate is something which outweighs the knife. The knife is something which the plate outweighs. *Outweighs is somerelat which the plate the knife. Three precedes four. Three is something that precedes four. λx.Precedes(x, 4) Four is something that three precedes. λx.Precedes(3, x) *Precedes is somerelat that three four.λR.R(3, 4) 29

30. if and, then 0. (2) 1. (4) 2. > …(32), including and 3. >> …(1408), including > and, > 4. >>> … (2,089,472), including >> and,, t> 30

31. a linguist sold a car to a friend for a dollar Can Human Lexical Items have “Level Four Meanings”? (sold a friend a car for a dollar) whatever the order of arguments, the concept SOLD, which differs from GAVE, is plausibly (at least) tetradic 31

32. a linguist sold a car a friend a dollar Can Human Lexical Items have “Level Four Meanings”? x y z w (she sold this him that) So why not…  λy. λz. λw. λx. x sold y to z for w 32

33. Can Human Lexical Items have “Level Four Meanings”? λZ. λY. λX. GLONK(X, Y, Z)  x[X(x) v Y(x) v Z(x)]  x[X(x) & Y(x)] &  x[Y(x) & Z(x)] Glonk cat friendly dog 33

34. if and, then 0. (2) 1. (4) 2.(32), including and 3.(1408), including > and, > 4. (2,089,472), including >> and,, t> and >> maybe some kind of “resource limitation” keeps us from going beyond Level Three 34

35. Can Human Lexical Items have Level Three Meanings? F IDO CHASED (_, _) > G ARFIELD R OMEO GAVE (_, _) > G ARFIELD J ULIET 35

36. Romeo gave it to Juliet Romeo kicked the rock to Juliet Romeo kicked Juliet the rock but double-object constructions do not show that verbs can have Level Three Meanings

37. a thief jimmied a lock with a knife

38. (x) (y) (z) he jimmied it that a thief jimmied a lock a knife Why not instead… ‘jimmied’  λz. λy. λx. x jimmied y with z The concept JIMMIED is plausibly (at least) triadic. So why isn’t the verb of type >>?

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

40. Still, one might think that many verbs do have Level Three Meanings… F IDO BARK (_, _) F IDO CHASE (_, _) > G ARFIELD - ED (_) 40

41. Can Human Lexical Items have Level Three Meanings? CHASE (_, _) > G ARFIELD Saying that expressions of type can be modified by expressions of type is like positing a covert Level 4 element. And why does the modifier skip over the thing chased, applying instead to the chase? INTO - A - BARN, > THE - SENATOR FROM - TEXAS 41

42. Garfield was chased > if the meaning of ‘chase’ is at Level Three, then a “passivizer” would also be at Level Four:, > Kratzer and others “sever” agent-variables from verb meanings: ‘chase’  λy. λe. e is a chase of y Garfield was chased >

43. CHASE (_, _) > G ARFIELD INTO - A - BARN > F IDO “active voice head” Level Three But if the posited verb meaning is below Level Three, do we really need the covert Level Three element? 43

44. CHASE (_, _) > G ARFIELD INTO - A - BARN AGENT F IDO 44

45. CHASE (_, _) G ARFIELD INTO - A - BARN AGENT F IDO Are names really expressions of type ? 45

46. CHASE (_, _) G ARFIELD INTO - A - BARN AGENT F IDO 46

47. Do Human i-Languages have expressions of type ? Initially tempting hypothesis: proper names are of type ® Robin  --|-- / \ But alternatives have been considered. Robin  x. x = ® x. Robinizes(x) Robin, t>  P.P( ® ) P.the(x):Robinizes(x).P(x) [D1 Robin ]  x. Indexes(1, x) & Called(x, ‘Robin’) 47

48. Do Human i-Languages have expressions of type ? Initially tempting hypothesis: proper names are of type ® Robin  --|-- / \ But alternatives have been considered for good reasons. Neptune is a planet, but Vulcan isn’t. Every Tyler at the party was tall, and every philosopher was a Tyler. That Tyler stayed late, and so did this one. There were three Tylers at the party, and Tylers are clever. We sat next to (that nice) Professor Tyler Burge. 48

49. Do Human i-Languages have expressions of type ? Another tempting hypothesis: deictic pronouns are of type ® that/she/him  --|-- / \ But we need alternatives. That planet is bright. This 1 trumps that 2. / \ That planet / \ This 1 / \ planet / \ that 2 /woman/… 49

50. Do Human i-Languages have expressions of type ? T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary  e. e is (tenselessly) a John-see-Mary event Why think tensed phrases denote truth values? Why think the tense morpheme is of type E.  e[Past(e) & E(e)] as opposed to e. Past(e) S  NP aux VP 50

51. Do Human i-Languages have expressions of type ? T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary  e. e is (tenselessly) a John-see-Mary event Why think the tense morpheme is of type E.  e[Past(e) & E(e)] a quantifier… | …that is also a conjunctive adjunct to V? 51

52. Do Human i-Languages have expressions of type ? T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary  e. e is (tenselessly) a John-see-Mary event e. Past(e)  e. Past(e) & e is a John-see-Mary event ?? / \ ? 52  Longer Story 1 1. Everybody needs some version of the following Tarskian idea: a predicate that is satisfied by some but not all things can be “polarized” into a predicate that is satisfied by everything or nothing.

53. Do Human i-Languages have expressions of type ? T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary  John-see-Mary(_)  Past(_)^John-see-Mary(_) Pol(P) / \ +  +[ Past(_)^John-see-Mary(_)] suppose that a monadic predicate M can be “polarized” into a monadic predicate + M that applies to e iff M applies to something....… applies to e if and only if there was an event of John seeing Mary 53 ‘such that John saw Mary’ is sentential; but it is a sentential predicate, not a truth-value denoter

54. Do Human i-Languages have expressions of type ? T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary  John-see-Mary(_)  Past(_)^John-see-Mary(_) Pol(P) / \ −  −[ Past(_)^John-see-Mary(_)] suppose that a monadic predicate M can be “polarized” into a monadic predicate + M that applies to e iff M applies to something or a monadic predicate - M that applies to e iff M applies to nothing ….… applies to e if and only if there was no event of John seeing Mary 54

55. Outline for Rest of the Talk ✔ say a little more about the suggested alternative to a “Fregean” semantic typology ✔ then focus on why we need an alternative – if and are types, then so is (way too many) – a basic type, for entity denoters of (unattested) – a basic type, for truth-value denoters (unattested) if time permits, brief discussion of quantification – serious difficulties for the idea that ‘every cat’ is of type and ‘every’ is of type – alternatives available if we don’t insist that and are basic 55

56. …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…. --Barbara Partee, “Do We Need Two Basic Types?” Next Question: Why take or as basic types? little or no independent support for the claim that Human Languages (as opposed to certain languages of thought) generate denoters treating names and deictic pronouns as predicates isn’t hard Tarski showed us how to treat sentences as predicates, and the grammatical status of sentences is unclear 56

57. 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 57

58. 1. 2. 58

59. Semantic Typology for Human I-Languages THANKS!

60. Extra Slides Follow 60

61. Suppose that a monadic predicate M can be “polarized” into a monadic predicate + M that applies to e if and only if M applies to something BROWN(_)^HORSE(_) applies to e if and only if e is both brown and a horse +[BROWN(_)^HORSE(_)] applies to e if and only if something is both brown and a horse 61

62. Suppose that a monadic predicate M can be “polarized” into a monadic predicate + M that applies to e if and only if M applies to something a monadic predicate - M that applies to e if and only if M applies to nothing BROWN(_)^HORSE(_) applies to e if and only if e is both brown and a horse −[BROWN(_)^HORSE(_)] applies to e if and only if nothing is both brown and a horse 62

63. Suppose that a monadic predicate M can be “polarized” into a monadic predicate + M that applies to e if and only if M applies to something a monadic predicate - M that applies to e if and only if M applies to nothing +[BROWN(_)^HORSE(_)] applies to e if and only if (e is such that) there is a brown horse −[BROWN(_)^HORSE(_)] applies to e if and only if (e is such that) there is no brown horse cp. ~  x[Brown(x) & Horse(x)] cp.  x[Brown(x) & Horse(x)] 63

64. / \ / \ ran > every cat / \ Fido / \ chased / \ every cat / \ Fido / \ / \, et> chased / \ every cat We can try assigning a higher type to ‘chased’. But then (I claim): adjuncts like ‘into the barn’ and passive constructions will lead us to posit Level Four meanings.

65. / \ which / \ Fido / \ chased __ But then how do we explain the unambiguity of strings like ‘every cat which Fido chased’ #Every cat is one which Fido chased. / \ / \ > every cat / \ 1 / \ Fido / \ chased __ Heim&Kratzer

66. / \ which / \ + / \ Fido / \ chased __ / \ / \ > every cat / \ 1 / \ Fido / \ chased __ / \ / \ every cat / \ + / \ Fido / \ chased __ 1 1 Maybe the external argument of a raised quantifier like ‘every cat’ is a polarized predicate, while a relative clause is a more ordinary (“unsentential”) predicate that can apply to some things without applying to all things Heim&Kratzer

67. But as a refinable first approximation, suppose that… ‘every’ is a plural Dyad: EVERY(_, _) applies to if and only if the Xs include the Ys ‘every cat’ is a plural Monad:  [EVERY(_, _)^THE-CATS(_)] applies to the Xs if and only if the Xs include the cats ‘Fido chased every cat’ is a plural Monad:  [EVERY(_, _)^THE-CATS(_)]^FIDO-CHASED-THEM(_) applies to the Xs if and only if the Xs include the cats, and Fido chased the Xs Technical Details Remain (see Conjoining Meanings) 67

68. Fido chase t 1 every cat + past Technical Details Remain (see Conjoining Meanings) applies to entity e  if and only if (e is such that) Fido chased whatever whatever A assigns to t 1 so relative to any assignment that assigns e to the variable, the polarized predicate applies to e if and only if Fido chased e but relative to each assignment A of values to variables…

69. Fido chase t 1 every cat + past Technical Details Remain (see Conjoining Meanings) and it’s not (too) hard to formulate Tarski-style composition principles according to which…  applies to the Xs if and only if (i) the Xs include the cats, and (ii) Fido chased the Xs (i.e., for each of the Xs, the polarized predicate applies to that entity when that entity is the value of the variable)

70. Fido chase t 1 every cat + past Technical Details Remain (see Conjoining Meanings) and it’s not (too) hard to formulate Tarski-style composition principles according to which…  applies to the Xs if and only if (i) the Xs include the cats, and (ii) Fido chased the Xs +  applies to e if and only if Fido chased the cats

71. i before e function in intension a procedure that pairs inputs with outputs in a certain way function in extension a set of ordered pairs (with no and where y ≠ z) |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)] 71

72. i before e i-Language a procedure that connects meanings with pronunciations in a certain way, thereby respecting certain constraints Bingley is eager to please. (a) Bingley is eager to please relevant parties. #(b) Bingley is eager to be pleased by relevant parties. Bingley is easy to please. #(a) Bingley can easily please relevant parties. (b) Bingley can easily be pleased by relevant parties. 72

73. i before e i-Language a procedure that connects meanings with pronunciations in a certain way, thereby respecting certain constraints ‘i’ connotes: intensional/procedural; implementable, and hence constrained; internal/individual, rather than external/public e-language a language in any other sense—e.g., a suitable set of meaning-pronunciation pairs 73

74. i before e i-Language a procedure that connects meanings with pronunciations in a certain way, thereby respecting certain constraints Human i-Languages Human i-Languages Finite i-Languages Finite i-Languages Gruesome i-Languages Gruesome i-Languages Less Permissive i-Languages Less Permissive i-Languages More Permissive i-Languages More Permissive i-Languages | unbounded “sane” i-Languages | 74

75. / \, > / \ ran every cat / \ today Level 3 / \, et> / \ every dog,, et> chase / \ every cat today Level 4 and without quantifier raising, severing external quantifiers still requires high types

76. / \, > / \ ran every cat / \ today / \, et> / \ every dog, > chase / \ every cat today / \ see,, et>> Level 4

77. V(P) / \ D(P) V(P) That / \ D(P) V(P) John/ \ V D(P) see Mary  e. e is (tenselessly) a John-see-Mary event  That is a John-see-Mary event 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 can’t take a 3 rd argument? if T is (semantically) the verb’s 3 rd argument, then why not… 77

78. T(P) / \ T V(P) past / \ D(P) V(P) John/ \ V D(P) see Mary   e[PAST(e) & John-see-Mary(e)], t> E.  e[PAST(e) & E(e)] “God likes Fregean Semantics” theory of tense | (e < RefTime) & (RefTime = SpeechTime)  e. John-see-Mary(e) 78

79. … … i-Languages Fregean i-Languages: expressions of the types:,, and if and are types, so is Fregean i-Languages: expressions of the types:,, and if and are types, so is Human i-Languages Human i-Languages Level-n Fregean i-Languages: expressions of the types:,, and the nonbasic types up to Level n Level-n Fregean i-Languages: expressions of the types:,, and the nonbasic types up to Level n Psuedo-Fregean Languages: expressions of the types:,, and some nonbasic types Psuedo-Fregean Languages: expressions of the types:,, and some nonbasic types 79

80. A Pseudo-Fregean Language Might Have a Dozen Types >, t>, >, > 80

81. / \ Mary saw John/ \ > and John saw Mary / \ Mary saw John / \ before John saw Mary   [Before(_, _)^PastJohnSeeMary(_)]   [ PastMarySeeJohn(_) & … And 81