Semantic interpretation and matching
Of course, the translation of a sentence from object language to metalanguage does not in itself tell us anything about what the sentence means. To accomplish this, the symbols in the metalanguage must be assigned a semantic interpretation or value, at which point the formula, which represents the proposition expressed by the original natural language sentence, must be matched with the state of affairs it describes. The process of assigning values and matching the proposition to the state of affairs it describes can be divided into four steps.
Assigning
values The first step is to assign the symbols of predicate calculus a semantic interpretation. This idea was implicit in the previous section, where we assigned the symbols a semantic value. For example, predicates expressed by eat and love are represented by E, L and so on, and constants expressed by proper nouns like Jane and Tom are represented by j, t and so on. Because natural language connectives and operators are closed-class expressions, these correspond to fixed logical symbols. In contrast, predicates and constants can be expressed by upper- or lower-case letters of the alphabet, with the exception of x, y and z, which by convention are reserved for variables.
Establishing a model of the world
The second step is the establishment of some model of the world against which the symbols in the metalanguage can be matched. Within formal semantics, models are typically represented in terms of set theory. For example, in a model of the world in which all women love chocolate, the sentence All women love chocolate would be true. However, in a model in which only a subset of women love chocolate, a further subset love chips and an intersection of these two subsets love both, the sentence all women love chocolate would be false, whereas the sentences Some women love chocolate, Some women love chips, Some women love chocolate and chips and Not all women love chocolate would be true. It is because the symbols are matched with a model of the world that this type of approach is also known as model-theoretic semantics. This idea is illustrated by Figure 13.2.
Matching formula with model
The third step is a matching operation in which the symbols are matched with appropriate entities in the model. This is called denotation: expressions in the metalanguage denote or represent elements in the model, and the meaning of the sentence is equivalent to its denotatum, or the sum of what it corresponds to in the model. Matching of predicates establishes the extension of individuals over which the predicate holds, which is represented in terms of sets. For example, in the sentence All women love chocolate, the predicate love represents a relation between the set of all entities described as women and the set of all entities described as chocolate. Once this matching operation has taken place, then the truth value of the sentence can be calculated.
Calculating truth values
The fourth step involves the calculation of truth values. If the formula matches the model, then the sentence is true. If it does not, then the sentence is false. These steps are summarised in Table 13.2. As this brief overview shows, in truth-conditional semantics the meaning of a sentence is equivalent to the conditions that hold for that sentence to be true, relative to a model of the world. Central to this approach is the correspondence theory of truth that we considered earlier (section 13.1.1): meaning is defined in terms of the truth of a sentence, understood as conditions in the world (or a model of the world) to which the sentence corresponds.
We illustrate each of these steps with example (15), which is repeated here.
Once the sentence is translated into predicate calculus (15b), values are assigned to the logical symbols (e.g. j = Jane; t = Tom) and a model is established that identifies the entities corresponding to the linguistic expressions Jane and Tom. This model might represent the set of all people {Bill, Fred, Jane, Mary, Susan, Tom. . .}. Within this model is a domain or subset of entities who stand in the relation expressed by the predicate love (L). This is rep resented by (17), in which each ordered pair (inside angled brackets) stands in the relevant relation.
Next, the formula is matched with the model so that constants and predicates are matched with entities and relations in the model. As (17) shows, this set contains an ordered pair, which means that Jane loves Tom. Finally, the truth condition of the proposition expressed by (15) is evaluated relative to this model. The rule for this evaluation process is shown in (18).
In this rule, the number ‘1’ represents ‘true’ (as opposed to ‘0’, which represents ‘false’). This rule says ‘Jane loves Tom is true if and only if the ordered pair is a member of the set L’. Since the set L contains the ordered pair in the model, the sentence is true. Table 13.3 completes this brief overview of the truth-conditional approach to sentence meaning in formal semantics by summarising the properties that characterise this approach as it is conceived by generatively oriented semanticists.
