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1. Type Shi+ing
1. Type Shi+ing
Classics
Montague (1973)
Montague (1973)
Montague (1970)
Montague (1970)
Type shi-s are injec1ve
Type shi-s preserve seman1c substance
Type shi-s do not add seman1c proper1es
Montague (1970)
Type shi-s are (true) embeddings
Läuchli (1970)
Type shi-s are total …
… and thus cannot be reversed
Partee (1987)
Partee (1987)
Type shi-s need not be unique
Montague (1973)
Montague (1973)
Type shi-s are global
Type shi-s are uniform
S1
S2
Type shi-s are uniform across domains
Type shi-s are purely formal … whatever that means:
definable in (fragments of type logic),…
permuta1on invariant, …
…
van Benthem (1991)
2. Types in dynamic seman4cs Indefinites as variables A farmer owns a donkey ≈ Sentences as predicates A farmer owns a donkey
Pardon my German
Indefinites as variables A farmer owns a donkey
≈
Cf. Zimmermann (1993)
Dynamic conjuncMon … A farmer owns a donkey. He beats it
≈
A farmer owns a donkey He beats it
Dynamic conjuncMoncon[An
[AND]
… as intersecMon
Dynamic conjuncMon … A farmer owns a donkey. He beats it
A farmer owns a donkey He beats it
Dynamic conjuncMoncon[An
[AND]
… as Cartesian producMon
Dynamic conjuncMon as type-‐shiTed conjuncMon Intersec1on: Product forma1on:
Dynamic conjuncMoncon[An
Cf. Quine (1960)
… vs. Bernays (1957)
Donkeys Every farmer who owns a donkey beats it
DenotaMons (simplified): every: farmer who owns a donkey: beats [it]: B
Donkeys Every farmer who owns a donkey beats it
Types: every: (et)((et)t) farmer who owns a donkey: (e2t) beats [it]: (e2t) every farmer who owns a donkey ((e2t)t) every (e2t)((e2t)t)
Need for type shiT from (et)((et)t) to (e2t)((e2t)t)
Need for type shiT from (et)((et)t) to (ent)((ent)t)
QUESTION
ARE THERE ANY?
ANSWER: SURE …
Need for type shiT from (et)((et)t) to (ent)((ent)t)
QUESTION
ARE THERE ANY?
ANSWER: SURE …
Asymmetric (weak) shiT: no proporMon problem, no dimes wasted …
Need for type shiT from (et)((et)t) to (ent)((ent)t)
QUESTION
ARE THERE ANY?
How about?
Lewis (1975), Kamp (1981), Heim (1982)
Need for type shi+
… not a domain shi+
Intended type shi+
… is not unique:
LHK type shi+
On a domain of n individuals we may have:
or:
… can be defined for invariant determiners on infinite domain:
LKH type shi+
… and extended by gerrymandering:
References van Benthem, Johan: Language in Ac1on. Amsterdam 1991. [2nd ed.,
Cambridge, Mass., 1995] Bernays, Paul: ‘Über eine natürliche Erweiterung des RelaMonenkalkuls’. In: A.
HeyMng (ed.), Construc1vity in Mathema1cs. Amsterdam 1957. 1-‐14. Heim, Irene: The Seman1cs of Definite and Indefinite Noun Phrases. UMass
1982. Kamp, Hans: ‘A Theory of Truth and SemanMc RepresentaMon’. In: J.
Groenendijk et al. (eds.), Formal Methods in the Study of Language. Part 1. Amsterdam 1981. 277-‐322.
Läuchli, H.: ‘An abstract noMon of realizability for which intuiMonisMc predicate calculus ist complete’. In: A. Kind et al. (eds.), Intui1onism and proof theory. Amsterdam and London 1970. 227-‐234.
Lewis, David K.: ‘Adverbs of QuanMficaMon’. In: E. L. Keenan (ed.), Formal Seman1cs of Natural Language. Cambridge 1975. 3-‐15.
Partee, Barbara: ‘Noun Phrase InterpretaMon and Type ShiTing Principles’. In: J. Groenendijk et al. (eds.), Studies in Discourse Representa1on Theory and the Theory of Generalized Quan1fiers. Dordrecht 1987. 115-‐143.
Montague, Richard: ‘Universal Grammar’. Theoria 36 (1970), 373-‐398. –: ‘The Proper Treatment of QuanMficaMon in Ordinary English’. In: J. HinMkka
et al. (eds.), Approaches to Natural Language. Dordrecht 1973. 221-‐242. Quine, Willard Van Orman: ‘Variables explained away’. Proceedings of the
American Philosophical Society 104 (1960), 343-‐347. Zimmermann, Thomas Ede: ‘On the Proper Treatment of Opacity in Certain
Verbs’. Natural Language Seman1cs 1 (1993), 149-‐179.
