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.
We began without analysing the internal structure of sentences at all. We observed that our semantic analysis was insufficient, as there were semantic relationships between sentences that were not reflected in our analysis; [john_sings_and_john_dances] :- john_dances was not a valid entailment according to our analysis; it was analyzed as the claim that one atomic proposition entailed another, different one. To get a better fit between theory and data, we observed that some sentences were syntactically related to others in that they were boolean (and, or and not) combinations thereof.
Although we represented tableaux as trees in the previous post, the
way we interacted with these trees makes clear that the really crucial
part of a tableau-tree is that it is a way to represent a set of
branches. In building these tableaux, we operated exclusively on
branches - we chose a particular branch, and expanded it. Expanding a
branch amounts to replacing it with new branches (sometimes just
one, sometimes two). The (unfinished) tableau below has three
branches.
When we consider the internal structure of sentences, in particular
those involving the sentential connectives and, or and not, we
see that the truth conditions of these complex sentences are
predictable from the atomic sentences they contain and the ways in
which these are combined with the connectives.
We start our semantic journey at the beginning with the observation
that certain sentences are true, and others are false.