Identity and Definite Descriptions

The language built in the previous article can say that something is $F$, that everything is, that some things stand in a relation to others, and a great deal more besides. It cannot say that there is exactly one $F$. It cannot say that nothing but Socrates is wise. And it cannot say what "the" contributes to a sentence, which turns out to be the largest of the three gaps.

All three are closed by adding a single two-place predicate.

One more predicate, and not like the others

That the interpretation is fixed is what distinguishes identity from every other predicate in the language, and it is why identity counts as part of the logic rather than part of the subject matter. An interpretation is free to make $F$ apply to whatever it likes, and no argument turning on the meaning of $F$ can be valid on logical grounds alone. Nothing is free about $=$, so arguments turning on its meaning are valid in the same way that arguments turning on the meaning of negation are.

Three properties follow immediately from the fixed interpretation. Every object is identical to itself, so $\forall x\, x = x$ is valid. If $a$ is identical to $b$ then $b$ is identical to $a$. And if $a$ is identical to $b$ and $b$ to $c$, then $a$ is identical to $c$.

The fourth property is the one that does the work in arguments. If $a$ and $b$ are the same object, then whatever is true of one is true of the other, so any formula containing $a$ remains true when $b$ is written in its place. This licenses substitution, it is what a proof system for identity is built around, and it is usually called the indiscernibility of identicals or Leibniz's Law. Its converse, the claim that objects sharing all their properties are identical, is a separate and much more contested thesis, and the article on identity in the ontology subject is where the difference is worked through.

One consequence deserves stating because it is easily assumed away. Identity cannot be defined from the rest of the language. It is tempting to try defining $a = b$ as the claim that $a$ and $b$ satisfy exactly the same predicates, but a language contains only finitely many predicates, and two distinct objects may be alike with respect to every one of them. What such a definition captures is indiscernibility relative to a particular vocabulary, which is a weaker relation. Identity has to be added, not extracted.

Counting

With identity in hand the language can count, which is worth seeing done because the pattern is not obvious in advance.

"At least one thing is $F$" needed no identity: $\exists x Fx$. "At least two things are $F$" needs it, and the reason is that two existential quantifiers by themselves do not guarantee two objects. The formula $\exists x \exists y (Fx \land Fy)$ is satisfied by a single object playing both roles. What is wanted is

$\exists x \exists y (Fx \land Fy \land \neg\, x = y)$

with the final conjunct forcing the two to be distinct. "At least three" extends the pattern with a third quantifier and three distinctness clauses, and the number of clauses grows quickly, which is a nuisance rather than a difficulty.

"At most one thing is $F$" is a universal claim rather than an existential one, since it says nothing about anything existing. It says that any two things that are both $F$ are the same thing:

$\forall x \forall y ((Fx \land Fy) \rightarrow x = y)$

If nothing is $F$ this comes out true, which is correct, since at most one is satisfied by none.

"Exactly one thing is $F$" is then the conjunction of the two, and it can be written more compactly by putting the uniqueness clause inside the existential quantifier:

$\exists x (Fx \land \forall y (Fy \rightarrow y = x))$

There is something that is $F$, and anything that is $F$ is that thing. This formula is worth reading until it is comfortable, because the rest of the article is built on it.

Only, except, and superlatives

The same predicate handles a group of English constructions that quantifiers alone cannot touch.

"Only Socrates is wise" says that Socrates is wise and that nothing else is, which is $Ws \land \forall x (Wx \rightarrow x = s)$. The second conjunct is the uniqueness clause again, now anchored to a name rather than to a variable.

"Every philosopher except Socrates is wise" says that the philosophers other than Socrates are wise and that Socrates is not: $\forall x ((Px \land \neg\, x = s) \rightarrow Wx) \land \neg Ws$. Dropping the second conjunct would leave a formula that says nothing about Socrates at all, and "except" plainly does say something about him.

Superlatives work the same way. "Socrates is the wisest philosopher" becomes a claim that Socrates is a philosopher and that every philosopher other than Socrates is less wise than he is, which requires a two-place predicate for the comparison but no new machinery. What all these have in common is that they are claims about how many things, or which things other than a given one, and that is exactly what identity supplies.

The definite article

Treating a description as a name is the obvious first move, and it produces trouble at once.

Suppose "the present King of France" is a name, and let it be $k$. Then "the present King of France is bald" is $Bk$, and it is true or false according to whether the object named is bald. But there is no present King of France, so $k$ names nothing, and a name that names nothing leaves the sentence with no truth value at all. That is uncomfortable, since the sentence is perfectly meaningful and most people asked would call it false.

It gets worse. If $Bk$ has no truth value then neither does $\neg Bk$, so both "the present King of France is bald" and "the present King of France is not bald" fail to be true, which looks like a violation of excluded middle. And a third difficulty arises with sentences that deny existence outright. "The present King of France does not exist" is true. Read as a claim about the object named by $k$, it says of that object that it does not exist, which is either false or unintelligible depending on how charitable one is feeling.

Russell's analysis

Written out, "the present King of France is bald" becomes

$\exists x (Kx \land \forall y (Ky \rightarrow y = x) \land Bx)$

which is the "exactly one" formula from above with a further conjunct attached. There is a present King of France; there is at most one; and he is bald.

Every one of the three difficulties dissolves. The sentence has a truth value, and the value is false, because the first conjunct is false. No name in it fails to refer, because there is no name in it: the whole analysis contains one predicate letter for "King of France", one for "bald", two quantifiers, and identity. And "the present King of France does not exist" becomes the straightforward denial that anything uniquely answers to the description, which is true and involves no object doing the strange work of not existing.

The formula also explains why the analysis was thought worth the trouble. A definite description looked like a referring expression and turned out, on analysis, to be a quantifier phrase in disguise. The grammatical form of the English sentence and the logical form of what it says come apart, and the notation is what makes the gap visible.

Scope, once more

The apparent violation of excluded middle has not been dodged. It has been located.

Once "the" has been analysed as a quantifier, a negation can attach in two places, and the two produce different claims. Putting the negation inside gives

$\exists x (Kx \land \forall y (Ky \rightarrow y = x) \land \neg Bx)$

which says that there is a unique present King of France and he is not bald. That is false, since the first conjunct fails. Putting the negation outside gives

$\neg \exists x (Kx \land \forall y (Ky \rightarrow y = x) \land Bx)$

which denies that there is a unique bald present King of France. That is true.

So "the present King of France is not bald" is ambiguous, and excluded middle is untouched: the false reading is the contradictory of nothing, since the true reading is the actual contradictory of the original sentence. English gives no way of marking which reading is meant, and the notation gives no way of failing to mark it. Russell's terms for the two cases were secondary and primary occurrence of the description, and later writers usually speak instead of the description taking narrow or wide scope, which is the same distinction under the name inherited from the previous article.

The ambiguity is not confined to negation. Any operator the description can sit inside or outside of produces one, which is why scope distinctions involving descriptions recur throughout the philosophy of language and the advanced articles on modal logic.

Next

The language is now complete for present purposes, and the subject turns from using it to proving things about it. Those proofs are conducted in a background theory of sets, and the next article supplies as much of that theory as the metatheory needs and no more.