Logical System

Skills to be acquired in this tutorial:

To learn about the Uniqueness quantifier (a part of identity), and to be introduced to definite descriptions.

Why this is useful:

Uniqueness is central in mathematics, and definite descriptions is a core area in philosophical logic.

The Tutorial:

To review. We have rules for introducing and eliminating identity. The introduction rule allows you to bring in t=t for any term t that you like. And you use the elimination rule with one identity, say s=t, and one other formula, say F-- you are allowed to substitute chosen occurrences of s in F with t (and chosen occurrences of t in F with s). There is one restriction. No variable that occurs in s or t can be bound in F. So, with f(x)=g(y,z) and ∀m(Ff(x)m)∧∃z(g(y,z)=a) you are not allowed to substitute because the variable z occurs in the identity and is bound in the formula. Often the formula F is itself a statement of identity- perhaps one formula is a=b and the other b=c; in these cases there are lots of opportunities for substituting.

One often needs to prove uniqueness. A mathematicians might undertake to prove a theorem like 'a particular set has exactly one largest element'. This is done in two stages: proving that there is one, and proving that there is no more than one. That is, you prove that exactly one thing has a property by proving that something has the property and that all other things which also have that property are equal to the first one. That is, you want to prove something like

∃x(F(x)) and ∀x∀y((F(x)&F(y))→x=y))

This is so common that we introduce an abbreviation for it.

∃!xF(x) abbreviates ∃xF(x)&∀x∀y((F(x)&F(y))→x=y))

New tree rules for ∃!x and ~∃!x

These just 'unabbreviate' the quantifier. They also make a some extra steps for you. Expanding ∃!x will give an 'and' formula with an existential as its left conjunct. The tree rule will make the 'and' step and instantiate the quantifier. Expanding ~∃!x will give a 'not and' formula. The tree rule will make the 'not and' step. 

Definite descriptions

The logic here intersects with another area: that of definite descriptions. If a formula like (∃!x)Fx is true (ie there is exactly one x which is F), we might refer to the x in question by a phrase like 'the x which is F'. Phrases like 'the x which is F', or 'the F' for short, are known as definite descriptions. And we could introduce some logical symbolization for this, using the Greek letter iota which is ι, say

ιx(F(x)) which is read 'the x which is F'.


if ∃!xF(x) is true, we can use ιx(F(x)) to pick out that x which is F (note here that ∃!xF(x) is a statement, which is either true or false, whereas (ιx)(F(x)) is a 'term like' expression, it is a name of an individual, which either names an individual or it does not).

If, ∃!xF(x) is true, (ιx)(F(x)) names an individual. We have other names for individuals (eg 'a', 'b', 'c', etc). And it may be that (ιx)(F(x)) names one of those, so, perhaps, c=(ιx)(F(x)). Also it may be that c has the property G, in which case Gc will be true and so too will be G((ιx)(F(x))) (if, for the moment, we allow our logic to permit formulas like that). And, actually, even when c does not have the property G, the formula G((ιx))(F(x)) will still be perfectly good (but it will be false).

However, and this is where the difficulties start,

if ∃!xFx is false, then (ιx)(F(x)) does not pick any individual out. There is no 'the x which is F' -- there may be none of them or there may be two or more of them. But then think about a 'formula' like a G(ιx)(F(x)). It says 'the x which is F is G', but there is no such x. So this formula cannot be true, but nor can it be false (for then its negation would be true). It is just kind of floating there. [This is why, elsewhere, in our earlier semantics ,we have always insisted with formulas like Fa or ∼Fa that the Universe must contain an 'a'.]

So, there is an awkwardness to extending our logic, at this level of analysis, to iota operators. We will not make this extension.

The issue of definitive descriptions can be approached from philosophical logic or linguistics and, historically, this is its origin. Bertrand Russell in his 1905 paper 'On Denoting' discusses the sentence 'The present King of France is not bald' (where there was, or is, no present King of France so the 'the' word does not pick anything out). Russell, essentially, argued that this should be analysed as a pure quantificational expression, one that does not have any genuine referring expressions in it. His analysis was


which reads 'There is a King of France, and all other Kings of France equal that one, and he is not bald.' And this sentence is false. And so too is 'The present King of France is bald'


which reads 'There is a King of France, and all other Kings of France equal that one, and he is bald.' Both sentences are false because there is no present King of France.

Notice that this logical structure


is similar to but slightly different from our (∃!x) structure which, for (∃!x)(Kx&Bx) would be


The difference is in the scope of the existential quantifier.

Further optional reading

Holtz, Brian, Definite descriptions in one easy lesson [link good June 2012]
Russell, B [1905] 'On Denoting', Mind, N.S. XIV, 530-538.

Holtz refers to

Frege, Gottlob [1893]. “On Sense and Reference,” in Geach and Black, eds., Translations from the Philosophical Writings of Gottlob Frege (Oxford: Blackwell, 1952), 56-78.
Russell, Bertrand [1903]. Principles of Mathematics. London: George Allen and Unwin.
Russell, Bertrand [1905]. “On Denoting.” Mind 14, 479-93. Reprinted in Ostertag [1998].
Strawson, P. F. [1950]. “On Referring.” Mind 59, 320-44. Reprinted in Ostertag [1998].
A good anthology: Ostertag, Gary, ed. [1998]. Definite Descriptions: A Reader. Cambridge: MIT Press.
A nice modern defense of Russell: Neale, Stephen [1990]. Descriptions. Cambridge: MIT Press.


There are subtleties between something being unique and there being exactly one of them. Everyone has exactly one height (Fred may be 6ft. Jim 6ft, and Elsie 5ft), but the height that they have, while unique to them, is not unique period (note that Fred and Jim have the same height). Everyone in the US has exactly one Social Security Number, and the number that each person has is both unique to them and unique period. One way of seeing the difference between the two, is that if you are supplied with a person's height, you cannot in general infer who the person is who has that height, but if you are given a person's SSN you can always infer who the person is who has that SSN.

Exercise 1(of 1)

∴(∀y)(∃!x)(x=y) (*everything has exactly one thing identical to itself, namely itself*)