Seth Cable Semantics and Generative Grammar Fall 2011 Ling610 1 The Semantics of Definite DPs 1 Thus far, our semantics is able to interpret common nouns that occupy predicate position (1a). …However, the most ‘common’ position for common nouns to occupy is internal to phrases that occupy argument positions, as in (1b). (1) The Possible Positions of Common Nouns a. Predicate Position: Obama is a politician. b. Argument Position: (i) [ A politician ] arrived from Washington. (ii) Joe likes [ the politician ]. (iii) [ Many politicians ] are corrupt. Much of our remaining work this term will be aimed at understanding how to interpret expressions like those in (1b), which are not proper nouns, but which are in argument position… We have to start somewhere, and so we’ll get things going by introducing a ‘rough-and-ready’ system for the interpretation of definite descriptions like “the politician”. (2) Crucial Syntactic Change, Part 1: DPs, Not ‘NPs’ • Thus far, we’ve assumed that expressions like Barack and a politician are ‘NPs’. • However, for several decades, syntacticians have held that their category is actually ‘DP’ (that they are projections of the determiner), as in the following structures: a. The ‘DP’ Structure of Nominal Arguments DP DP DP D NP D NP D NP a N the N many N politician politician politicians 1 These notes are based on the material in Heim & Kratzer (1998: 73-85).Seth Cable Semantics and Generative Grammar Fall 2011 Ling610 2 (3) Syntactic Change, Part 2: Proper Names as DPs • We will extend our ‘DP structure’ to proper names as well. • For proper names, we will assume that there is a phonologically empty determiner that combines with them (3a). • We will assume that this phonologically null D has the semantics in (3b). • Under this semantics, the extension of the null D is simply the identity function on individuals. Consequently, the extension of the DP will be identical to the extension of the NP name. a. The ‘DP’ Syntax of Proper Names DP D NP ∅ N Barack b. The Semantics of Null D [[ ∅D ]] = [ λx : x ∈ De . x ] (4) The Semantics of Definite DPs Although this remains a controversial view, we will assume that definite DPs are of type e. They denote entities, just like proper names. a. [[ the president ]] = Barack Obama b. [[ the professor for 610 ]] = Seth Cable c. [[ the capital of Massachusetts ]] = Boston d. [[ the drummer for Metallica ]] = Lars Ulrich With the extensions assumed in (4), let’s now work out what the semantics of the definite determiner itself must be…Seth Cable Semantics and Generative Grammar Fall 2011 Ling610 3 1. The Semantics of the Definite Article Let’s begin by figuring out what the semantic type of the definite article should be… (5) Deducing the Semantic Type of the Definite Article DP (type e) D NP (type <e,t>) the president • It seems that the extension of the takes a type <et> function as argument, and returns an entity as the extension of the entire DP. • Thus, the definite article the is of type <<e,t>e> OK… but what is this <<e,t >e > function? How is it defined, exactly? (6) Some Initial Observations, Part 1 [[ the ]] takes as argument: [[ president ]] and returns: Barack Obama [[ professor for 610 ]] Seth Cable [[ capital of MA ]] Boston [[ drummer for Metallica ]] Lars Ulrich (7) Some Initial Observations, Part 2 Following our assumptions about the semantics of NPs, the following equations hold: a. [[ president ]] = [ λxe : x is a president ] b. [[ professor for 610 ]] = [ λxe : x is a professor for 610 ] c. [[ capital of MA ]] = [ λxe : x is a capital of MA ] d. [[ drummer for Metallica ]] = [ λxe : x is a drummer for Metallica ]Seth Cable Semantics and Generative Grammar Fall 2011 Ling610 4 (8) Key Observation The functions in (7) above all share one rather salient property in common: Each assigns ‘true’ to only one entity a. In all of De, [ λxe : x is a president ] is (currently) T of only Barack Obama b. In all of De, [ λxe : x is a professor for 610 ] is (currently) T of only Seth Cable c. In all of De, [ λxe : x is a capital of MA ] is (currently) T of only Boston d. In all of De, [ λxe : x is a drummer for Metallica ] is T of only Lars Ulrich (9) Key Generalization [[ The ]] takes a function of type <et> as argument, and returns the one entity which the <et> function is (currently) true of! a. [[ the ]] ( [ λxe : x is a president ] ) = Barack Obama b. [[ the ]] ( [ λxe : x is a professor for 610 ] ) = Seth Cable c. [[ the ]] ( [ λxe : x is a capital of MA ] ) = Boston d. [[ the ]] ( [ λxe : x is a drummer for Metallica ] ) = Lars Ulrich (10) A Lexical Entry that Captures the Idea in (9) [[ the ]] = [ λf : f ∈ D<e,t> and there is exactly one x such that f(x) = T . the unique y such that f(y) = T ] The function whose domain is those <e,t> functions f which assign true to exactly one entity, and which for all such functions f, yields that unique entity y such that f(y) = T Technical Side-Note Note that the domain of [[the]] is not all of D<et>, but only a strict subset of it. Consequently, it’s not entirely accurate to say that [[the]] is of type <<et>, e> Technical Fix a. Partial Function: A function f is a partial function from X into Y iff the domain of f is a subset of X, and its range is a subset of Y b. < X , Y > = the type of partial functions from things of type X to things of type Y c. D<X,Y> = the set of all partial functions from DX into DYSeth Cable Semantics and Generative Grammar Fall 2011 Ling610 5 (11) Sample Derivation a. ‘ S ’ is T iff (by notation) DP VP D NP V The N smokes. president b. [[ S ]] = T c. Subproof (i) [[ VP ]] = (by NNx2, TN) (ii) [ λxe : x smokes ] d. Subproof (i) [[ NP