8.6.3. Consequences for adequacy

Our stipulations about the interpretation of definite descriptions insure that any interpretation of the vocabulary in a description Ix ρx will divide one of the two gaps that result from SD—that’s why there is no precondition for its application—so the rule is strict and its addition will not disturb the soundness of our system. It is also clearly safe since the new gaps it introduces differ from their parent only by having added resources. But the argument we had used to establish the completeness of the system of derivations—in particular, the argument used in 7.7.4 to show that any fully developing gap is divided by an interpretation—will no longer apply since this argument assumed that the reference values of all terms could be settled without considering the extensions of predicates, something that is not true in the case of definite descriptions.

We will not consider ways of reformulating that argument for a system including SD. Instead we will consider the completeness of a system of derivations for definite descriptions that employs not only SD but also certain uses of the rule LFR. The stipulations we have made concerning the interpretation of the description operator can be imposed on a structure simply by requiring that it make true every sentence of the form:

where we follow the form of the law for descriptions but apply a quantifier ∀w_i for each variable w_i that appears unbound in ρ. We will call this sentence a meaning postulate for the description Ix ρx. Making all these meaning postulates true comes to the same thing as making true all instances of that law for a language expanded by the range of the structure. When assessing the validity of a particular argument, all that is relevant is the interpretation of the definite descriptions actually appearing in the argument, provided we consider descriptions that contain variables that are not bound within the description. And the correct intepretation of these descriptions can be insured by the truth of the meaning postulates for them. That is, if Δ includes the meaning postulate for each description in an argument Γ / φ, this argument is valid given the interpretation of the description operator if and only if the argument Γ, Δ / φ is valid even without stipulating the interpretation of definite descriptions—i.e., even if we treat them as unanalyzed individual terms.

Now, any question of validity can be reduced to a question of the validity of a reductio argument, so let us limit consideration to such arguments. Given an argument Γ / ⊥, let δ be the conjunction of the meaning postulates for all descriptions appearing in the members of Γ. Now suppose that Γ / ⊥ is valid when we fix the interpretation of definite descriptions. We have seen that Γ, δ / ⊥ will be valid without fixing this interpretation. Therefore, a derivation for Γ, δ / ⊥ will close using only the basic system of previous chapters, so it will certainly close if we add the rules SD and LFR. And the rule SD will enable us to show the meaning postulate for any description is a tautology, so it will certainly enable us to show the validity of Γ / δ. Finally, the rule LFR lets us establish the validity of Γ / ⊥ if we can show both Γ ⊨ δ and Γ, δ ⊨ ⊥. In short, the system of derivations with SD and LFR is complete because SD enables us to establish any meaning postulate, and we can establish the validity of all arguments involving descriptions when we add their meaning postulates as further premises.

Since it introduces a new independent term, the rule SD introduces a new way that gaps can be prevented from reaching a dead end. It can be modified to search for finite structures in the way we have done for other rules using independent terms, and named, following the same pattern as with those rules, as Securing a Description Supplemented (SD+).

When we use this rule, we consider the possibility that one of the already existing alias sets provides names of an object that uniquely satisfies the description.

Notice that one of these alias sets will be the one including ∗. And that is to be expected since there are two different ways in which the nil value might end up as the reference of a definite description. This will happen not only when the description fails to be uniquely satisfied but also when the nil value is the one value satisfying it uniquely. Indeed, the reference of any term τ will uniquely satisfy the predicate [ _ = τ], so whether or not [ _ is a C] is not uniquely satisfied [ _ = the thing that is a C] will be—though, of course, perhaps only by the nil value.