Definite descriptions Identity and existence
Whenever the Australian radio news carries an item about the doings of the Prime Minister, they refer to him as "the Prime Minister, Scott Morrison". This may be to make it clear which Scott Morrison they mean, or to inform Australians that Morrison is (still) their Prime Minister, but what is interesting from the logical point of view is that they use two referring expressions, "Scott Morrison" and "the Prime Minister" to denote the same individual. One refers to him by name; the other picks him out as the object satisfying a description. Expressions such as "the Prime Minister", "the man with the golden gun", "the natural satellite of the Earth" or "the square root of 36" are called definite descriptions, and their logic raises interesting questions.
Syntactically, a definite description behaves like a name, so it might seem quite natural to introduce a symbol and use this to create terms out of formulae. On this account, x Fx could be seen as a term able to occur in formulas just like a proper name, read as 'the thing, x, such that Fx' or more briefly as 'the F'. Using this notation, "The F is G" would be written
G(x Fx).
More generally, for any formula A containing a name m and for any variable v not occurring in it,
v A^{v}_{m}
would be a term. We might expect such terms to be semantically similar to names, since whether we use a name like "Scott Morrison" or a definite descrition like "the [first] Prime Minister of Australia in 2020" should make no difference to truth values, because they refer to the same person.
We shall see later that this way of forming terms is problematic, and suggest a better way of using the definite description operator, but for now we may continue to examine it, noting how useful it is to be able to express terms of this sort.
One of the main motivations for the early development of modern symbolic logic was the perceived need to provide logical foundations for mathematics. The logic of function symbols is clearly central to this enterprise, and a definite description operator seems to be exactly what we need to bring them within the scope of ordinary logic. Consider the addition function in ordinary arithmetic, for instance. Without using function symbols, we may represent the facts about addition by means of a 3place predicate symbol S, writing Sabc to mean that a and b sum to c. Thus the formula S(7,5,12) is true, for instance, while the formula SOMEx (S(7,x,12) AND NOTS(x,3,8)) is false. Of course, we really want to write "7 + 5 = 12" rather than "S(7,5,12)", but how are we to construe the term on the left of this equation? Well, one idea is simply to define t + u as x S(t,u,x) and rely on our logical account of definite descriptions to do the work.
Any such logical account must allow for the fact that definite descriptions are semantically unlike names in that a name is an arbitrary (unstructured) symbol which gets to refer to an object as a result of an act of naming, whereas a definite description refers to it as a result of the facts about which things satisfy certain predicates and which do not. You cannot understand a name or find out its reference without being made party to a convention about it, but you can investigate which thing is picked out by a definite description in much the same way as you would investigate any facts expressed by ordinary indefinite descriptions.
The classical theory of definite descriptions was proposed by Bertrand Russell (On Denoting, 1905, reprinted in [11]). He starts by noting the need for such a theory, and presents some puzzles which it needs to be able to solve. We focus on two of these:

Puzzle 1: Definite descriptions
threaten to violate laws of logic. Consider Russell's
example: George IV wished to know whether Walter Scott
was the author of the novel
Waverley. That is, George wanted to know
whether the identity statement "Scott = the author
of Waverley" was true. Now in fact Scott was
indeed the author of Waverley, so by the
logical rule =E we should be able to derive that
George wanted to know whether Scott was Scott. But this
is not true: the First Gentleman of Europe had no
interest in the natural deduction rule =I.
 Puzzle 2: Sentences containing definite descriptions can be perfectly meaningful in cases where those descriptions do not refer to anyone. "The President of Australia is in Brisbane" is just as meaningful as "The Prime Minister of Australia is in Brisbane", although Australia has no president. Are we to say that the President of Australia is a nonexistent object? Does that even make sense? Do we say that the sentence has no truth value? If we do, won't that contradict the law of the excluded middle? How can we even say (correctly) that the President does not exist without referring to someone who isn't there to be refered to?
Russell's idea is not to give a definition of the phrase "the F", but rather to give a systematic way of paraphrasing every sentence in which that phrase occurs into an equivalent sentence with only ordinary expressions of first order logic with identity. This kind of replacement is known as a contextual definition, because it allows the defined expression to be replaced only in context. Russell says that
The F exists
is really a conjunction. It means that there is one and only one F:
SOMEx Fx AND ALLxALLy ((Fx AND Fy) IMP x=y)
This formula asserts existence and uniqueness. It contains no part that directly translates "the F", so it does not get its truth value by referring to a possibly nonexistent object but rather by quantifying over the existing ones.
Similarly,
The F is G
is a conjunction of three claims. It means
SOMEx Fx AND ALLxALLy ((Fx AND Fy) IMP x=y) AND ALLx (Fx IMP Gx)
A shorter equivalent formula is
SOMEx (Fx AND Gx) AND SOMExALLy (Fy IMP x=y)
or even more succinctly
SOMEx (ALLy (Fy IFF x=y) AND Gx)
Russell thus succeeds in bringing definite descriptions, and hence function symbols, within the scope of ordinary logic with quantifiers and identity, accounting for the semantic difference between them and names. He is also able to answer his own motivating puzzles.

Puzzle 1: Let us write Wx to
mean that x wrote Waverley, and let s
be Scott. Then Russell's analysis of what George wanted
to know, i.e. the truth value of the identity
statement s = x
Wx,
is the formula
SOMEx (Wx AND s=x) AND SOMExALLy (Wy IMP x=y)
By elementary logical moves, this is equivalent toWs AND SOMExALLy (Wy IMP x=y)
Presumably George already knew the second conjunct of this, that Waverley was not written by a committee, so all he was really demanding was the truth value of Ws. That is, he just wanted to know whether Scott wrote Waverley, which is surely right. On analysis, according to Russell, what George wanted to know is not really an identity statement at all, so there is no failure of =E or any other rule of ordinary logic.
 Puzzle 2: The claim that the President of Australia is in Brisbane is a conjunction of three shorter claims: that a President exists, is unique and that whoever is a President is in Brisbane. It is false because its first conjunct is false, so the location of the nonexistent president is not an issue. Again, there is no failure of excluded middle or any other part of standard logic, and no need for a semantic theory involving nonexistent objects or other strange things.
Russell at the time thought of his theory as analysing what is really said when people use definite descriptions in languages like English. He saw it as allowing the termforming operator '' into logic. My preference is a slight modification of Russell's view, on which '' does not form terms at all, but instead is the quantity indicator of a quantifier. That is instead of writing
G(x Fx)
to represent the claim that the F is G, we write
(x: Fx) Gx.
This can count as a quantifier on our general theory of quantification. We could read it "For the thing x such that Fx, Gx". Note that it does not mean that exactly one F is G; it means that there is exactly one F and it is G. The introduction and elimination rules for it will not be specified here, as they are a little complicated, but the effect will be that the above formula is provably equivalent to Russell's threeway conjunction. A real advantage of this construal over the one taking definite descriptions to be terms is that it is able to represent the ambiguity of negation in the case of nonreferring descriptions. On the present view, as Russell points out, a sentence like "The President of Australia is in Brisbane" has two different negations: one could deny the whole sentence, or one could assert that the President is elsewhere than Brisbane. The two formulae
NOT (x: Fx) Gx
(x: Fx) NOTGx
are not equivalent, yet either could be used to deny that the F is G. This ambiguity is clear and important, but it cannot even be expressed in the vocabulary of as a termforming operator.
Part of the significance of Russell's contribution arose from the way in which it suggests that logical analysis can solve philosophical problems. It seems that the apparent or "grammatical" form of a sentence, the form that we see on the surface, may be different from its deeper "logical" form. The grammatical form may mislead us into philosophical problems or into mistaken world views like those postulating a universe of nonexistent objects with nonsensical properties. These philosophical errors disappear, says Russell, when logic shows us the true form of our language.