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(see SET THEORY for the meaning of ). It is easy to see that the following
specifications of truth conditions follow from these statements:
Fa is TL1 if, and only if, Caesar is a Roman. Rbc is TL1 if, and only if,
Brutus killed Cassius.
and so on. We have constructed an elementary semantic theory for part of our elementary
language.
It is easy to extend this to include sentential connectives:
(3) A sentence of the form A&B is TL1 if, and only if, A is TL1 and B is TL1. A sentence of the form is TL 1 if, and only if, A is not TL 1.
and so on. Relying upon such axioms, we can derive a statement of the TL1 conditions of
any sentence of our simple language.
The conditions listed under (1) specify semantic properties of subsentential
expressions: names and predicates. Those under (2) explain the truth conditions of the
simplest sentences in terms of the semantic properties of these subsentential expressions.
Finally, those in (3) concern the semantic roles of expressions which are used to construct
complex sentences out of these simple
Figure 1
ones. I mentioned that L1 was a rather simple language, and we can now notice an
important aspect of this simplicity. Consider the sentence: ‘Fa & (Rac Gb)’. We can
represent the way in which this sentence is built out of its elements with a tree diagram
(Figure 1).
The conditions in (1) state the semantic properties of expressions in the bottom nodes
of the tree: those in (2) concern how the truth conditions of the next higher nodes are
determined by these bottom semantic properties. All the higher nodes are explained by
the conditions in (3). It is a feature of this language that, apart from the subsentential
expressions at the bottom level, every expression of the tree has a truth value. It is true