# Truth Value of Definite Descriptions

I'm currently studying definite descriptions in logic. My textbook postulates Bertrand Russell's view of definite descriptions, but I'm curious about other views as well (in the context of classical 2-valued logic).

Take the sentence, "The King of the United States treats his subjects well."

Obviously, there is no King of the United States, so calling this sentence true would be absurd. However, calling this statement false seems absurd to me as well because that may imply that there exists a King of the United States, but he doesn't treat his subjects well. My textbook states that Russell's view on syncategorematic definite descriptions stems down to his view of a definite description -- "there exists an X, there is no more than one such X, and X has quality Y." Using this definition, we can symbolize the above statement.

(Ǝx)(Kx ^ (y)(Ky → x=y) ^ Tx)

Where Kx ↔ x is a king

Tx ↔ x treats x's subjects well

x=y (identity -- x is y)

This symbolization seems odd to me. Would this be the proper way to symbolize the above sentence using Russell's view? If this is the correct way, are there other competing views that would give an alternate symbolization? If there are any popular alternate views, I would like them to remain within the realm of two-valued logic.

Thanks.

• Why does it seem odd? It literally follows Russell's verbal description. In some cases it is more natural to use an indefinite description (without specifying uniqueness), but not here. One might think of this sentence as a subjunctive ("if there was a King of the United States then he would treat his subjects well"), but that is a stretch. There are alternative approaches that introduce nonexistent objects and the existence predicate in addition to quantifiers,etc., but using them is more than a variation in "symbolization", you'll have to learn a new calculus. Nov 26, 2019 at 22:55
• @Conifold I'm not saying it doesn't follow his description. I'm merely saying that Russell's concept itself seems odd. Why would a definite description imply existence when it is merely referring to an individual? Also, which calculus specifically would provide an alternative approach? Nov 27, 2019 at 1:02
• How would one go about attributing properties to individuals that do not exist? Either one requires existence, or endorses a prolific ontology of non-existent objects with properties systematically made up somehow (Meinongianism). For Meinongian calculi see logics of nonexistent objects and references therein. If one wishes to "merely refer", using subjunctive mood seems more natural, and that is formalized in modal logic, which has its own calculi. Nov 27, 2019 at 1:18

The purpose of Russell’s Theory of Descriptions is precisely to give meaning (i.e. truth value) to a statement concerning a non-existent entity.

The basic assumtpion is that names of individuals must refer to existing objects (individuals).

Thus, what does it mean to assert something about a non-existing objects referring to it with a sort of "name" ?

The idea of Russell is that a definite description is not a "long name" but must be analyzed away through a corerct logical analysis of statements using it.

Thus, the well-known statement concerning "The present king of France" is different in logical form from the statement "Socrates is a philosopher".

While the second has the form : "philosopher (Socrates)", i.e. the statement predicates a property of an individual, the first one must not be analyzed with "bald (The present King of France)", exactly because there is no individual to whom predicate applies to.

The correct analysis is, as you said :

"some x is such that x is King of France, and that any y is currently King of France only if y = x, and that x is bald".

Now it is possible to evaluate the truth value of the statement, because it is a conjunction of statements of which the first one is false.

Reagrding competing views, see Descriptions.

• Why we need to say that we have at most one thing that is currently king of France when by definition king means "one man"? Aug 21, 2020 at 18:13

What might seem odd to you is that Russell treats the description operator in a syncategorematic way. That is, the operator itself is not associated with an explicitly defined operation, but formulas containing the operator are associated with satisfaction conditions. The problem with syncategorematic treatments is that the syntax of the formula interpreting the natural language sentence is often worlds away from that of the sentence. This is problematic if the interpretation is supposed to proceed compositionally.

But in higher-order logic it is routine to give a categorematic treatment of the description operator. Simply let the operation be that function f from pairs of sets to classical truth values such that f(X, Y) = 1 iff |X| = 1 & X is a subset of Y. This means that the description operator is treated as a higher-order relation between sets (some call such guys generalized quantifiers). So your sentence can be rendered THE(λx. Kx, λx.Tx), where THE is a higher-order predicate of arity 2 denoting the operation f. To mirror English syntax even more closely we can 'Curry' f to yield an equivalent function f* that is defined on sets and outputs functions from sets to truth values such that f*(X)(Y) = f(X, Y). Using the higher-order predicate THE* to denote f* we can represent your sentence as (THE*(λ.Kx))(λx.Tx).

Note: λx. Gx is of course just a fancy way of denoting the set of objects satisfying the formula Gx.