Truth, models and extension
For logical approaches to semantics, reference and truth are the principal semantic facts: the most important thing about the meaning of a word is what it refers to, and the most important thing about a sentence is whether or not it is true – whether or not things are as the sentence says they are. Meaning for a logical approach to semantics is thus principally truth-conditional (see 3.2.1). As discussed in Chapter 3, for a truth-conditional theory of meaning, knowing the meaning of a factual sentence is the same as knowing what the world would have to be like for that sentence to be true. This does not mean that truth conditions are all there is to meaning. It just means that, as Chierchia and McConnell-Ginet (2000: 72) put it, ‘if we ignore the conditions under which S [a sentence] is true, we cannot claim to know the meaning of S. Thus, knowing the truth conditions for S is at least necessary for knowing the meaning of S.’
Logical approaches to semantics deal with the question of truth and reference by providing a model for the sets of logical formulae used to represent meaning. The model of a set of logical formulae is a description of a possible world to which the formulae refer, a set of statements showing what each individual constant and predicate refers to in some possible world. The model relates the logical language to this world, by assigning referents to each logical expression. The aim of this is ultimately to pro duce, for a given set of referents, a statement of the truth values of the logical formulae in which they are included. In other words, the logical formalism will tell us, given a particular world, which sentences describing this world are false and which are true. Given the assumption of the centrality of truth to meaning, this is an important part of describing the meanings of a language. If the logical formulae are identified with sentences of natural language, we will have obtained a logical characterization of the truth conditions of a subset of natural language. We will see a simple example of such a truth-value assignment below.
The referent of a logical expression is called its extension. We will consider the extension of both individual constants (singular terms) and of predicates. The extension of an individual constant is simply the individual entity which the constant picks out or refers to in the world. In a universe consisting simply of Tom, Dick, Harry and Jemima, the individual constants t, d, h and j have the following extensions:
What is the extension of a predicate? Predicates are interpreted as sets of entities: a one-place predicate like ‘tall’ will have as its extension the entire set of tall entities. Imagine that Tom, Dick and Jemima are all tall, but that Harry isn’t. In this possible world the extension of ‘tall’ will be the set of entities {Tom, Dick, Jemima}. The extension of a two-place predicate like ‘respect’ will be the set of all pairs of individuals such that the first respects the second. Consider for example a universe where Jack respects Jill, Hank respects Mark, Holly respects René, and Adrian respects money. The extension of the predicate ‘respect’ will thus be the following:
The extension of a three-place predicate will be an ordered triple of entities. The predicate ‘give’, for instance, might have the following extension:
This extension describes a universe in which Don gives David wine, Briony gives Tom a present, and Judge Judy gives Selina a fine.
In general, we can say that the extension of an n-place predicate is a set of ordered n-tuples of entities. It’s important to realize how extensions differ from senses. When describing the sense (meaning, definition) of a verb like respect or give, we would not usually bother specifying the participants involved in different giving or respecting events. Instead, we would try to specify what seems to be essential to the event itself (for respect, something like ‘esteem’, and for give something roughly like ‘freely transfer to’). In logical approaches to semantics, this sort of definitional information is called the intension of a predicate. Its extension, on the other hand, is a purely external matter, the set of ordered n-tuples to which the predicate applies. Consequently, it is possible for two predicates to differ in intension but to have identical extensions. Say that in our universe in (50) the things which are respected are also disliked: Jack respects Jill but dislikes her, Hank respects Mark but dislikes him, and so on. In this case, the predicates ‘respect’ and ‘dislike’ will have the same extensions, while differing in intension or meaning. In a different universe, of course, there is no reason that the extensions of the two predicates should be identical.
We can now sketch the way in which the truth values of the sentences of the logical formula can be specified. Take a possible world with the following components:
We will now provide a model for these terms, in other words a statement showing their extensions. The individual constants have the following extensions:
(Note how (53) differs from (52): (52) shows how the single-letter constant abbreviations are to be translated; (53) shows the actual individuals to which they refer.)
The one-place predicates have the following extensions:
Together, these assignments of referents constitute the model of the language in (52).
Our aim here will be to determine the truth or falsity of sentences involving the constants and predicates we have just introduced. In order to do this, we need a statement of what it is for a sentence to be true in a given model. To accomplish this, let’s use the following definitions (adapted from Allwood et al. 1977: 74):
Let’s use (52) to construct some arbitrary sentences, which we will give both in their logical formulation, and in English translation:
Note especially (59 g). ‘Dick’ and ‘Briony’ do, it is true, individually figure among the constants described by the predicate T ‘is teaching’. But the ordered pair of constants which constitutes the extension of this predicate in (55) is (b, d), not (d, b). Hence, the formula Td, b is false in this model. Taking these truth assignments, we can now use the truth tables given in 6.1 to read off the truth values of compound sentences. Let’s start with (60):
We can symbolize this propositionally as p & q, where p stands for T t, j ‘Tom is teaching Jemima’ and q for C d, e ‘Dick is climbing Everest’. The truth table for & (Table 6.2) shows us that this compound proposition is true in the current model: both its conjuncts are true.
Now consider (61) in light of the truth table for Table 6.4)
T t, j and C d, e are both true. The truth-table for V propositions involving tells us that complex V are true when both disjuncts are true. For this reason, (61) is true in the current model.
In (62), the definitions in (54) tell us that the antecedent Mb is false and the consequent Sh is true:
This corresponds to the fourth line of the truth-table in Table 6.5, and is therefore true.
The truth tables allow us to work out the truth-values of some quite complex sentences. Consider the following:
We can determine the truth-value of this proposition by starting from its individual components and working upwards. We start by assigning truth-values to the individual propositions in accordance with the model:
We then assign truth values to the complex propositions, i.e. the propositions obtained by combining the simple propositions into complex propositions with the propositional operators. Let’s start with the proposition ¬Td, b. We always start with the basic proposition without any preceding operator: in this case, Td, b. Td, b is false. The truth-table tells us that the negation of a false proposition is true. ¬T d, b is therefore true. Now for T t, j Ct, k. The first disjunct, Tt, j, is true, the second, Ct, k, is false. According to the truth table for, a proposition with one true and one false disjunct is true. This means that the disjunction Tt, j C t, k as a whole is true. For ease of memory, let’s label the truth values we have determined so far:
We have therefore reduced the entire complex proposition to a material conditional in which the antecedent (T t, j V C t, k) and consequent ¬T d, b are both true. The truth-table given in Table 6.5 tells us that a material conditional is true when its antecedent and consequent are both true; as a result, (63) as a whole is true.
In this way, it is possible to specify the truth-values for arbitrarily long sentences, as long as a model is given showing the extensions of the basic terms.
