Formal Semantics in Modern Type Theories: Is It Model-Theoretic, Proof-Theoretic, or Both?
In this talk, we contend that, for NLs, the divide between model-theoretic semantics and proof-theoretic semantics has not been well-understood. In particular, the formal semantics based on modern type theories (MTTs) may be seen as both model-theoretic and prooftheoretic. To be more precise, it may be seen both ways in the sense that the NL semantics can ﬁrst be represented in an MTT in a modeltheoretic way and then the semantic representations can be understood inferentially in a proof-theoretic way. Considered in this way, MTTs arguably have unique advantages when employed for formal semantics.
Continuations and Derivational Ambiguity
In categorial grammar, compositional interpretation is implemented along the lines of Montague’s ‘Universal Grammar’ program, taking the form of a structurepreserving mapping (a homomorphism) relating a source logic to a target logic. The homomorphism requirement implies that for an expression to be associated with multiple ‘readings’, there must be derivational ambiguity at the source end of the compositional mapping.
In this talk, I study the kind of derivational ambiguity that arises in some recent continuation-based categorial approaches. In Bastenhof (2013) and Moortgat and Moot (2013), the source derivations for compositional interpretation are focused proofs of a polarized sequent calculus; diﬀerent notions of derivational ambiguity result from the choices one can make in ﬁxing the polarity of atomic formulas. I compare this approach to the handling of continuations in terms of the combinatory and multimodal sequent calculi one ﬁnds in Barker and Shan (2014).
Combining Frame Semantics and Phrasal Semantics: A Proposal
Frame theory in the sense of Barsalou and others has very interesting things to say about various cognitive functions such as perception, categorisation, proprioception, and introspection. In fact, frames are supposed to implement, in Barsalou’s words, ‘a fully functional conceptual system’. Applying the theory to word meaning, as Löbner, Petersen & Osswald, and others have done, is highly attractive, as this results in a form of lexical semantics that is well embedded in a more general theory of cognitive functions. But while frames in general can convincingly represent content words and even certain simple sentences, it is, pace Barsalou, much less clear that they can also deal with logical notions, such as negation, disjunction and (generalised) quantiﬁcation. In this talk I will therefore propose a two-level system, in which a base component of meaning given by frames is extended with a phrasal semantics in Montague’s way. I will borrow certain insights from Frank Veltman’s Data Semantics and will let incompatibility of frames be a central notion, more central in fact than negation, which will be explained in its terms. Frames will play a double role, as carriers of lexical meaning, but also as standing proxy for possible worlds or situations. Since frames oﬀer a lot more structure than is usually provided for in phrasal semantics, more operations become available to the compositional system and we shall consider some of these.