Up until now, we have been restricting our attention to a simplified syntax for predicates, whereby they are uniformly boolean compounds of atomic one-place predicates - in plain English, we have been restricting our attention to intransitive verbs.
A rationale for doing this is that very many interesting aspects of language can be studied with this restriction in place: we have discovered boolean algebras lurking everywhere language, allowing us to interpret boolean words uniformly across different denotational spaces; we have discovered universal properties of determiner denotations, and further sub-classified the space of determiners, observing that certain syntactic constructions seem to make reference to properly semantic properties of expressions; and we have seen that complicated inference patterns can be derived from monotonicity properties of lexical items.
Last time we saw that natural language determiners can be classified according to the information they require to determine whether their noun argument is appropriately related to their predicate argument, and that this classification seemed to have some broader linguistic reality.
This time, we will take a closer look at the partial order relation (\(\le\)) that is associated with any boolean domain. Recall that in the domain of truth values, the boolean order coincides with implication (with \(b \le c\) holding just in case \(b\) implies \(c\)), and that in the domain of sets, the boolean order coincides with the subset relation (with \(P \le Q\) holding just in case \(P\) is a subset of (or is equal to) \(Q\)).
Last time we learned that, viewing determiners as relations between sets, they only take into account their first argument (\(A\)) and the intersection of their first and second arguments (\(A\cap B\)) into consideration. This empirical fact drastically reduces the possible determiners - it is a very powerful language universal. As it turns out, not all determiners even take this information into account.
Here, I briefly introduce three (or six, depending on how you count) classes of determiners, which are individuated by the kind of information they make use of.
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.