8.6.2. Derivations for the description operator

Although, in stating the tautologousness of a single long sentence, the law for the description operator takes a somewhat different form than those we considered for other logical constants, the real novelty in handling this constant lies in the fact that it is used to form terms rather than sentences. This means that what we must account for is not the role of a premise or conclusion. Instead, we need to account for what a definite description refers to.

The law for the description operator provides a way to draw conclusions about what a definite description refers to. We will implement this law in a rule that amounts to a couple steps in the exploitation of the sentence the law asserts to be a tautology. In particular, our rule will lead us directly to what we would get as the result of using a proof by cases to exploit the disjunctive law (restated using unrestricted quantifiers) and then using proof by choice and extraction for its existential first disjunct. The remaining non-atomic sentences in the law are universals so we cannot expect to go further in a single step. We will call this rule Securing a Description (SD).

There are really no preconditions for the use of this rule, but it is relevant only when the definite description in question actually appears in the gap being developed. The description is displayed above the derivation (perhaps among a list of other definite descriptions) and the stage number of the development is listed after it to show that it has been handled—we will say secured—at that stage in developing some gap. The description may need to be secured in a number of different gaps at different stages, so this stage is perhaps only the latest of a long list.

The term secure was used in 7.8.1 in connection with the rule ST, which was intended to provide a way to locate finite structures when the normal development of a gap would introduce ever more complex compound terms. Our aim in the rule SD is different but the consequences are similar. When a definite description is secured, it will be in the same alias set as some simple term, either the independent term introduced in the first gap or the term ∗. However, SD is not designed to search for finite structures, and we are as interested in the other assumptions introduced in each of the two gaps as in the equations that actually secure the description.

The occurrence of the definite description operator in an argument forces us to consider ∗ among the terms which will be grouped into alias sets. This is not merely because this term will be introduced into one of the gaps that is the result of applying SD. The semantics of the description operator require that range of reference values contain a distinguished nil value. This value must be included among the reference values of the terms for which we exploit universals if we are to insure that the universal is true. And that means we will need to exploit a universal for ∗ (or a co-alias) before a gap reaches a dead end. This is true even if ∗ does not actually appear in the gap (in which case it will be its only co-alias). Thus the terms to be considered when forming alias sets are not merely the terms appearing the resources and goals of the gap and its ancestors but also ∗ if any definite description appears among the terms we need to consider. Although it is possible for new definite descriptions to appear in the course of a derivation, this will happen only if one appears in the initial argument, so the requirement for including ∗ is that it must be counted among the terms whenever a definite description appears in the argument whose validity we are considering.

As an example of the use of SD, here is a derivation showing that if have the premise There was at most one winner, we can conclude The winner won if anything did.

Ix Wx: 3 │¬ ∃x ∃y (¬ y = x ∧ (Wx ∧ Wy)) (14) ├─ │ⓐ │││Wa (6), (10) ││├─ │││ⓑ ││││Wb (4) ││││∀x (Wx → b = x) ││││Ix Wx = b a, b—IxWx, ∗ │││├─ ││││● │││├─ 4 QED= ││││W(Ix Wx) 3 │││ ││││∀x (Wx → ∃y (Wy ∧ ¬ x = y)) a:5 ││││Ix Wx = ∗ a, c, IxWx—∗ │││├─ 5 UI ││││Wa → ∃y (Wy ∧ ¬ a = y) 6 6 MPP ││││∃y (Wy ∧ ¬ a = y) 7 ││││ ││││ⓒ │││││Wc ∧ ¬ a = c 8 ││││├─ 8 Ext │││││Wc (10) 8 Ext │││││¬ a = c (11) │││││ ││││││¬ W(Ix Wx) │││││├─ 10 Adj ││││││Wc ∧ Wa X, (11) 11 Adj ││││││¬ a = c ∧ (Wc ∧ Wa) X, (12) 12 EG ││││││∃y (¬ y = c ∧ (Wc ∧ Wy)) X, (13) 13 EG ││││││∃x ∃y (¬ y = x ∧ (Wx ∧ Wy)) X, (14) ││││││● │││││├─ 14 Nc ││││││⊥ 9 ││││├─ 9 IP │││││W(Ix Wx) 7 │││├─ 7 PCh ││││W(Ix Wx) 3 ││├─ 3 SD │││W(Ix Wx) 2 │├─ 2 CP ││Wa → W(Ix Wx)) 1 ├─ 1 UG │∀y (Wy → W(Ix Wx))

Notice that the list of alias sets in the first gap includes ∗ even though that term does not appear in either resources or goals of that gap because a definite description does appear in the initial conclusion.

Notice also that both the premise and the hedge if anything did in the conclusion played a role in closing the second gap in the derivation above. Since both are required to insure the existence and uniqueness of a winner, it is to be expected that the absence of either would keep us from ruling out the possibility that the definite description is undefined (which is the possibility explored by the second gap).

It may seem odd that The winner won is not a tautology. But on both of the accounts of definite descriptions that we have considered, it entails Something won, and that is not a tautology. It follows that The winner didn’t win is not absurd if it is contradictory to The winner won, and a derivation showing this when the sentence is interpreted using the description operator provides another example of the use of SD.

Ix Wx: 1 │¬ W(Ix Wx) (2) ├─ │ⓐ ││Wa (2) ││∀y (Wx → a = x) ││Ix Wx = a (Ix Wx)—a │├─ ││● │├─ 2 Nc= ││⊥ │ ││∀y (Wx → ∃y (Wy ∧ ¬ x = y)) ∗:3 ││Ix Wx = ∗ (Ix Wx)—∗ │├─ 3 UI ││W∗ → ∃y (Wy ∧ ¬ ∗ = y) 4 ││ ││││¬ W∗ │││├─ ││││○ ¬ W(Ix Wx), Ix Wx = ∗, ││││ ¬ W∗ ⊭ ⊥ │││├─ ││││⊥ 5 ││├─ 5 IP │││W∗ 4 ││ │││∃y (Wy ∧ ¬ ∗ = y) ││├─ │││(unfinished) ││├─ │││⊥ 4 │├─ 4 RC ││⊥ 1 ├─ 1 SD │⊥

The definite description Ix Wx does not appear in the diagram of the counterexample because, as a compound expression, its value is determined by the values shown there. In particular, the fact that the extension of W is empty insures that Ix Wx has the same reference value as ∗, and that would be true even if the referential range contained more than this value.

The sentence The winner didn’t win is contingent also on Russell’s analysis provided we interpret it as the denial of The winner won. For the latter sentence will be contingent according to Russell’s analysis since it is true on that analysis if and only if there is exactly one winner. However, on Russell’s analysis, an interpretation giving the winner widest scope—that is, an interpretation of the sentence as The winner is such that (he or she didn’t win)—is absurd since it implies Some winner didn’t win and thus that something has both the property of winning and the property of not winning. This is another consequence of the ambiguity that can arise with definite descriptions on Russell’s account. The sentence The winner won is definitely contingent on his way of analyzing it, but The winner didn’t win may be either contingent or absurd, depending on whether the negation or definite description is understood to have wider scope.