Linguistics Encyclopedia.pdf

or false, and this is exploited in the conditions for explaining the truth conditions for
complex sentences. We must now turn to a language which does not share this feature.
QUANTIFIERS
L2 is obtained from L1 by adding universal and existential quantifiers (‘ ’ and ‘ ’)
together with a stock of individual variables, ‘x’, ‘y’ ‘z’, etc., as in formal logic (see
FORMAL LOGIC AND MODAL LOGIC). The grammatical sentences of L2 include all
the grammatical sentences of L1 together with such expressions as:
The tree diagram in Figure 2 displays the structure
Figure 2
A-Z 185
of the last of these. Such sentences are less straightforward than those discussed on page
137. First, it is unclear what the semantic properties of variables are: they do not refer to
specific objects as names do. Second, the expressions ‘Hz’, ‘Rzx’ ‘ x Rzx’ and ‘Hz &
x Rzx’ contain free variables, variables which are not bound by quantifiers. It is hard
to see how such expressions can be understood as having definite truth values. If that is
the case, then we need a different vocabulary for explaining the semantic properties of
some of the intermediate expressions in the tree. Furthermore, if these expressions do
lack truth values, the condition we specified for ‘&’, which was cast in terms of ‘truth’,
cannot be correct: ‘Hz & x Rzx’ is built out of such expressions and, indeed, is one
itself.
First, we can specify a set D: this is the domain or universe of discourse—it contains
everything that we are talking about when we use the language. The intuitive approach to
quantification is clear. ‘ xFx’ is a true sentence of L2 if at least one object in D belongs