8.6.1. The role of definite descriptions in entailment

If Russell’s analysis of definite descriptions is accepted, their logical properties follow from those of the logical constants used in the analysis; but the description operator is a new symbol, and studying its logical properties requires stating new principles for it. We looked at the logical properties of the description operator informally in 8.4.3. Now we will look at a way of stating a principle of entailment that captures these properties.

First, we must first find a place for the description operator in our semantic scheme. All our logical constants so far—whether the connectives, the quantifiers, or the identity predicate—have been ways of producing compound formulas. The description operator, on the other hand, yields a compound term when it is applied to a predicate. This means that the extension of the operator I will be a function from the extensions of one-place predicates to reference values. We can represent the extension of a one-place predicate by the set of reference values of which it is true, so the extension of the description operator can be seen as a function which takes a set of reference values as input and yields a single reference value as output.

According to the account of definite descriptions we are considering, a term Ix ρx formed using the description operator refers to the single value in the extension of ρ if there is just one value, and otherwise its reference value is the Nil. This means that the extension of the description operator is not settled until we identify the Nil as a specific value in the referential range. This identification must be considered a further component of a structure, a respect in which two structures may differ. So when we make the description operator a part of our language, we require that a structure distinguish a member of the referential range as the Nil. This will serve as the reference value of the constant individual term ∗ introduced in 8.4.3. Then, to find the semantic value given to Ix ρx by a structure, we find the extension the structure gives to the predicate ρ. If the extension of ρ has just one member, that reference value will be the extension of Ix ρx; otherwise, the extension of Ix ρx is the value the structure assigns to ∗.

A specification made regarding structures and the interpretation of logical vocabulary will typically result in some logical law. For example, the requirement that the referential range serve both as a source of extensions for terms and as the domain of unrestricted universals gives us the principle of universal instantiation. And even the simple requirement that a referential range be non-empty yields the law ∀x θx ⊨ ∃x θx, which assures us that universal predicates are exemplified. In the case of our specifications for definite descriptions and the Nil, we get a principle that identifies a certain sentence as a tautology.

This tautology is a disjunction whose two components express the two alternatives for the reference value of a definite description. Let us see how that works in a little more detail.

The existential quantifier in the first disjunct should be familiar as one way of writing the quantifier that Russell used to analyze definite descriptions. The whole first disjunct might be read as Something such that (ρ fits it and it is all that ρ fits) is such that (the thing that ρ fits is it) or, a little more idiomatically, as The thing that ρ fits is something that ρ fits uniquely.

The second disjunct of the sentence is a conjunction whose first conjunct says Anything that ρ fits is such that something ρ fits is different from it. This is a compact but somewhat roundabout way of saying that the extension of ρ does not have exactly one member—i.e., if we can find anything in it, we can find something else in it, too. The second conjunct of this part of the sentence can be read as The thing that ρ fits is the Nil.

Putting this all together, the law tells us that the following is a tautology:

The first disjunct specifies the reference of the definite description when this is determined by the description, and the second disjunct specifies the reference when the description does not succeed in determining it.

In 8.4.3 the content of an analysis using the description operator was expressed using a similar disjunction. On that account, a sentence θ(Ix ρx) says that either (i) ρ is true of exactly one thing and (∃x: ρx) θx is true or (ii) ρ is not true of exactly one thing and θ∗. Given the law for descriptions, the properties of identity will tell us that

and the right-hand side is a more formal version of the disjunction used in 8.4.3.