1. Problems of syntax-semantics interface ESSLLI 02 Trento

2. summary The need for lambda calculus From Montague grammar to categorial grammar Lambek calculus Curry-Howard isomorphism Proof-nets Extensions (and restrictions) of L Extended proof-nets

3. Jackendoff Where (narrow) syntax has structural relations such as head-to-complement, head-to-specifier, and head-to-adjunct, conceptual structure has structural relations such as predicate-to- argument, category-to-modifier, and quantifier- to-bound variable. Thus, although conceptual structure undoubtedly constitutes a syntax in the generic sense, its units are not NPs, VPs, etc. […] In particular, unlike syntactic and phonological structures, conceptual structures are purely relational, in the sense that linear order plays no role.

4. recall: Montague grammars Truth-conditional approach: –sentence logical formula (true or false) –noun phrase term (constant, variable, complex term) But what for other linguistic expressions? –verb open atomic formula? –but how to combine? kiss(x,y) composed with p and m gives: kiss(p,m) or kiss(m,p)?

5. fortunately : lambda calculus constants, variables : -terms If M and N are -terms, then (M N) [or M(N)] is a -term,(application) If M is a -term and if x is a variable, then x.M is a -term (abstraction) + -reduction : ( x.M, N) M[N/x]

6. Goal : x (child(x) play(x)) Identical to : ( P.[ x (enfant(x) P(x))] u.play(u)) therefore : every child = P.[ x (child(x) P(x))] Identical to : ( Q. P.[ x (Q(x) P(x))] v.child(v)) therefore: every = Q. P.[ x (Q(x) P(x))] Example : how to extract the « meaning » of quantifiers?

7. other quantifiers a, an = Q. P.[ x (Q(x) P(x))] no = Q. P.[ x (Q(x) P(x))]

8. But we cannot apply anything to anything… x is a -term (x x) is a -term x.(x x) is a -term ( x.(x x) x.(x x)) is a -term But ( x.(x x) x.(x x)) (no end to the reduction : the normalisation process does not stop)

9. « Intransitive verbs » apply to nominal entities (and they give propositions) « Transitive verbs » apply to nominal entities (and they give intransitive verbs…) « Propositional verbs » apply to propositions (and they give propositions) « Adjectives » apply to nominal entities (and they give nominal entities)

10. Typed -calculus Constants and variables of type a are - terms of type a if M is a -term of type and N a - term of type a, then (M N) is a -term of type b If M is a -term of type b and if x is a variable of type a, then x. M is a -term of type

11. In other words:

12. Correspondance syntactic categories – semantic types sentences VP, IV NP, PN TT verbal adverbs VI/VI CN (common noun) sentential adverbs preposition propositional verb intentional verb article t e ou bien, t>,t>, >, >,t>,, >> >, >,, t>>

13. syntax For each syntactic category A, the set P A of all expressions of category A contains at least the set B A of the « dictionary words » of category A, If P A and if P B, then, in some cases to enumerate, F(, ) for some function F belongs to some set P C.

14. Example of rule S2 : if P T/CN and if P CN, then, F 2 (, ) P T, where F 2 (, ) = *, where * = except if is equal to a and if the first word of begins by a vowel, in which case * = an Remark : T is the category of terms, example : a man, an aristocrat

15. Example of rule S4 : if P T and if P VI, then F 4 (, ) P t, where F 4 (, ) = *, where * is obtained from by replacing the first verb by its 3rd person singular form Example : = John, = walk, F 4 (, ) = John walks

16. Montagovian analysis John seeks a unicorn S1 : a T/CN, unicorn CN S2 : F 2 (a, unicorn) = a unicorn T S1 : seek VI/T S5 : F 5 (seek, a unicorn) = seek a unicorn VI S1 : John T S4 : F 4 (John, seek a unicorn) = John seeks a unicorn t

17. John seeks a unicorn John seek a unicorn a unicorn seek a unicorn

18. Second analysis ! John seeks a unicorn S1 : seek VI/T, he 1 T S5 : F 5 (seek, he 1 ) = seek him 1 VI S4 : F 4 (John, seek him 1 ) = John seeks him 1 t S2 : F 2 (a, unicorn) = a unicorn T S14 : F 14,1 (a unicorn, John seeks him 1 ) = John seeks a unicorn t

19. John seeks a unicorn John John seeks him 1 seek a unicorn aunicorn seek him 1 him 1

20. remark In a « modern» grammar (cf. GPSG in the eighties), syntagmatic rules are put in correspondance with some semantic counterpart, In a « logical » grammar (eg. Lambek grammars), the correspondance automatically follows from a known isomorphism between logical derivations and -terms (Curry-Howard)

21. Syntagmatic grammar S SN SV SN Det N SN Np SV Vi SV Vt SN SV Vp que S SV Vint SV (S) = ( (SN) (SV)) (SN) = ( (Det) (N)) (SN) = (Np) (SV) = (Vi) (SV) = (SN) o (Vt) (SV) = ( (Vp) (S)) (SV) = (SV) o (Vint)

22. Det chaque | tout Det un N enfant | ballon Np stéphane Vi joue Vt cherche Vp dit Vint essaie (tout) = Q. P.[ x (Q(x) P(x))] (un) = Q. P.[ x (Q(x) P(x))] (enfant) = x.enfant(x) (stéphane) = P.P(stéphane) (joue) = x.joue(x) (cherche) = x. y.cherche(x, y) (dit) = P. x. dit(x,P) (essaie) = x. P.essaie(x, P) lexical rules

23. Example : stéphane cherche un ballon SN Det N unballon x. ballon(x) Q. P. x[Q(x) P(x)] ( Q. P. x[Q(x) P(x)] x. ballon(x)) P. x[( x. ballon(x) x) P(x)] P. x[ballon(x) P(x)]

24. Example : stéphane cherche un ballon SN Det N unballon P. x[ballon(x) P(x)] SV Vt x. y. chercher(x,y)

25. Example : stéphane cherche un ballon SN Det N unballon P. x[ballon(x) P(x)] SV Vt x. y. chercher(x,y) Composition : ( x.f(x)) o ( y.g(y)) = z. ( x.f(x), ( y.g(y), z)) z. ( P. x[ballon(x) P(x)],( x. y. chercher(x,y) z)) z. ( P. x[ballon(x) P(x)], y. chercher(z,y)) z. x[ballon(x) ( y. chercher(z,y), x)], z. x[ballon(x) chercher(z,x)]

26. Example : stéphane cherche un ballon SN Det N unballon P. x[ballon(x) P(x)] SV Vt x. y. chercher(x,y) z. x[ballon(x) chercher(z,x)] S SN Np Stéphane P. P(stéphane)

27. Example : stéphane cherche un ballon SN Det N unballon P. x[ballon(x) P(x)] SV Vt x. y. chercher(x,y) z. x[ballon(x) chercher(z,x)] S SN Np Stéphane P. P(stéphane) ( P. P(stéphane) z. x[ballon(x) chercher(z,x)]) ( z. x[ballon(x) chercher(z,x)] stéphane) x[ballon(x) chercher(stéphane,x)]

28. Example : stéphane cherche un ballon SN Det N unballon P. x[ballon(x) P(x)] SV Vt x. y. chercher(x,y) z. x[ballon(x) chercher(z,x)] S SN Np Stéphane P. P(stéphane) x[ballon(x) chercher(stéphane,x)]