Basic Categorial Semantics Gary Hardegree Department of Philosophy University of Massachusetts Amherst, MA 01003 1. Introduction......................................................................................................................................2 2. The Basic Picture of Semantic Processing.......................................................................................2 3. An Example Phrase..........................................................................................................................3 4. Isomorphism Thesis and Compositionality .....................................................................................3 5. Truth-Conditional Semantics – Denotations and Meanings............................................................4 6. What is a Situation? .........................................................................................................................5 7. Redrawing the Basic Picture of Semantic Processing .....................................................................5 8. Extensional Semantics.....................................................................................................................6 9. Denotation Typology.......................................................................................................................8 10. Comparison with Montague's Typology..........................................................................................9 11. Types and their Associated Domains...............................................................................................9 12. Types = Associated Domains ........................................................................................................10 13. The Type-Correspondence Principle .............................................................................................11 14. Simple Examples ...........................................................................................................................11 0. One-Place Connectives......................................................................................................11 0. Two-Place Connectives .....................................................................................................12 0. One-Place Predicates .........................................................................................................13 0. Two-Place Predicates and Transitive Verbs ......................................................................13 0. Quantifier Phrases..............................................................................................................14 0. Semantic Composition – Frege's Thesis........................................................................................15Hardegree, Basic Categorial Semantics (updated spring 2009) page 2 of 16 1. Introduction Having examined categorial syntax, we now turn our attention to categorial semantics. In this chapter, we examine Basic Categorial Semantics, which corresponds to Basic Categorial Syntax. In particular, we do not consider case-inflections or generalized-composition techniques. 2. The Basic Picture of Semantic Processing We begin with the following basic semantic flow-chart. step 1: syntactic de-composition: a given phrase is broken down into its component phrases, and ultimately into its fundamental component phrases (morphemes). step 2: lexical assignment of meanings: the fundamental component phrases are assigned meanings by consulting the lexicon. step 3: semantic re-composition: the meanings of the fundamental component phrases are combined to produce (compute) the meaning of the original phrase. This process can be diagrammed as follows, where ›αœ is the meaning of α. φ original phrase 1. de-compose phrase α1 α2 … αk-1 αk atomic components 2. assign meanings (based on lexicon) ›α1œ ›α2œ … ›αk-1œ ›αkœ meanings of atoms 3. re-compose meaning of original phrase ›φœ meaning of original phraseHardegree, Basic Categorial Semantics (updated spring 2009) page 3 of 16 3. An Example Phrase This diagram conceals important structural details in the de-composition and re-composition processes, since such details vary from phrase to phrase. By way of illustrating how the details might look, we consider a very simple example. the brown dog the brown dog brown dog ›brownœ ›dogœ ›theœ ›brown dogœ ›the brown dogœ 4. Isomorphism Thesis and Compositionality Notice in the above diagram that re-composition (∨) formally mirrors de-composition (∧). We will generally find it convenient to draw everything "right-side-up", in which case we can depict these processes in parallel, as follows. Syntactic De-Composition Semantic Re-Composition (nodes are phrases) (nodes are meanings) the brown dog ›the brown dogœ the brown dog ›theœ ›brown dogœ brown dog ›brownœ ›dogœ The two structures depicted above are isomorphic (i.e., structurally-identical). This observation is the basis of a fundamental thesis of categorial grammar – the Isomorphism Thesis – which may be summarized as follows. The semantics of a phrase is isomorphic to its syntax. A key feature of the Isomorphism Thesis yields one the central tenets of modern formal semantics– the Principle of (Local) Compositionality, which may be stated as follows. The meaning of a compound phrase is composed out of the meanings of its immediate constituents.Hardegree, Basic Categorial Semantics (updated spring 2009) page 4 of 16 5. Truth-Conditional Semantics – Denotations and Meanings An outstanding question remains – what are meanings? We pursue a truth-conditional model of semantics, which may be described as follows.1 (1) the most basic units of significance are denotations (extensions); (2) the meaning (intension) of a phrase α is what α denotes on each occasion of usage. Let's discuss denotations (extensions) first. By way of illustration, consider the phrase ‘the Eiffel Tower’; this phrase refers to a specific object – namely, the Eiffel Tower, which is the central architectural feature of Paris. We can formally describe this relation by saying either of the following. the phrase ‘the Eiffel Tower’ denotes the Eiffel Tower the denotation of the phrase ‘the Eiffel