We have been trying to build a semantic theory from the ground up. Our theory at each step was intended to capture some salient aspect of the inference patterns in English, and at each successive step to improve so as to capture a range of inferences that we couldn’t previously. While we have now dealt with the meanings of sentences involving quantifiers like everyone, noone, and someone, these three quantifiers are just the tip of the quantifier iceberg. Most quantifiers, however, are syntactically complex, often being composed of a determiner and a noun phrase.

By decomposing atomic sentences into subject and predicates we were able to provide a theoretical account of more empirical data; entailment judgements like the following became explicable. “John laughed and cried” ⊢ “John cried” “John either laughed or cried”, “John didn’t laugh” ⊢ “John cried” “John either laughed or cried” ⊢ “Either John laughed or John cried” “John both laughed and cried” \(\vdash\) “John laughed and John cried” “John didn’t laugh” ⊢ “It is not the case that John laughed” But not all subjects behave the same way.

Note: This is just for fun.

We have seen that the logical operators and, or, and not can be given a uniform proof-theoretic treatment in the context of tableaux. However, we have had to give them independent interpretations in denotational semantics; they are interpreted truth functionally in a sentential context, but set theoretically in a predicate context. Anticipating somewhat, these same connectives occur in other contexts as well: DPs Some student and most teachers Ds Some but not all PPs On the shelf or near the bookcase Ps On or near Advs quickly and accurately TVs praise and criticize DTVs show but not give In each context in which logical operators occur, they license the same inferences (i.

Having extended the range of entailments we are able to correctly predict (by deepening our syntactic analysis of sentences so as to view previously ‘atomic’ sentences as being composed of subects and predicates), we should ask whether, and if so, how, we can derive these predictions proof-theoretically (i.e. syntactically). We would like to extend the tableau method so as to treat sentences involving subjects and predicates. As our (meta-)language is modular, in the sense of having independent pieces for: