Formal Semantics

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1 Type-shifting Recall these natural deduction rules for implication elimination and introduction shown: (1) B : A-B a : A (2) [a : Ajl B(a) : B B : B -I,1 Aa. B : A - B For each of the type shifts below, show how the logic of implication derives the more complex type (right side) from the simpler type (left side), using.' 1. at ((a, b), b) Lift (Montague) 10 points 2. (a, b), (b, c) + (a, c) Composition (Geach) 10 points 3. (((a, b), b), c) + (a,c) Argument Lowering (Partee & Rooth) 10 points 2 Type-shifting in action For each of the shifts in $1: 4. provide a natural language example that shows a use of the shift in action; and 15 points 5. explain how the example exemplifies the shift. 15 points Note: For #5, you can choose to explain it in words or to show a proof (or a combination). References Partee, Barbara H. 1986. Noun Phrase Interpretation and Type-Shifting Principles. In Jeroen Groenendijk, Dick de Jongh, and Martin Stokhof, eds., Studies in Discourse Representation Theory and the Theory of Generalized Quantifiers, vol. 8 of GRASS, 115-143. Dordrecht: Foris. Reprinted in Partee (2004: 203-230). . 2004. Compositionality in Formal Semantics: Selected Papers by Barbara H. Partee. Oxford: Blackwell. van Benthem, Johan. 1991. Language in Action: Categories, Lambdas, and Dynamic Logic. Amsterdam: North-Holland. Reprinted in 1995 by the MIT Press, Cambridge, MA. The nomenclature is based on Partee (1986) and van Benthem (1991)