Logic and philosophy

.. While logically proper names (words such as "this" or "that" which refer to sensations of which an agent is immediately aware) do have referents associated with them, descriptive phrases (such as "the smallest number less than pi") should be viewed as a collection of quantifiers (such as "all" and "some") and propositional functions (such as "x is a number"). As such, they are not to be viewed as referring terms but, rather, as "incomplete symbols." In other words, they should be viewed as symbols that take on meaning within appropriate contexts, but that are meaningless in isolation. Thus, in the sentence
(1) The present King of France is bald, the definite description "The present King of France" plays a role quite different from that of a proper name such as "Scott" in the sentence
(2) Scott is bald.
Letting K abbreviate the predicate "is a present King of France" and B abbreviate the predicate "is bald," Russell assigns sentence (1) the logical form
(1′) There is an x such that (i) Kx, (ii) for any y, if Ky then y=x, and (iii) Bx.
Alternatively, in the notation of the predicate calculus, we have
(1″) ∃x[(Kx & ∀y(Ky → y=x)) & Bx].
In contrast, by allowing s to abbreviate the name "Scott," Russell assigns sentence (2) the very different logical form
(2′) Bs.
This distinction between various logical forms allows Russell to explain three important puzzles. The first concerns the operation of the Law of Excluded Middle and how this law relates to denoting terms. According to one reading of the Law of Excluded Middle, it must be the case that either "The present King of France is bald" is true or "The present King of France is not bald" is true. But if so, both sentences appear to entail the existence of a present King of France, clearly an undesirable result. Russell's analysis shows how this conclusion can be avoided. By appealing to analysis (1′), it follows that there is a way to deny (1) without being committed to the existence of a present King of France, namely by accepting that "It is not the case that there exists a present King of France who is bald" is true.
The second puzzle concerns the Law of Identity as it operates in (so-called) opaque contexts. Even though "Scott is the author of Waverley" is true, it does not follow that the two referring terms "Scott" and "the author of Waverley" are interchangeable in every situation. Thus although "George IV wanted to know whether Scott was the the author of Waverley" is true, "George IV wanted to know whether Scott was Scott" is, presumably, false. Russell's distinction between the logical forms associated with the use of proper names and definite descriptions shows why this is so.
To see this we once again let s abbreviate the name "Scott." We also let w abbreviate "Waverley" and A abbreviate the two-place predicate "is the author of." It then follows that the sentence
(3) s=s is not at all equivalent to the sentence
(4) ∃x[Axw & ∀y(Ayw → y=x) & x=s].
The third puzzle relates to true negative existential claims, such as the claim "The golden mountain does not exist." Here, once again, by treating definite descriptions as having a logical form distinct from that of proper names, Russell is able to give an account of how a speaker may be committed to the truth of a negative existential without also being committed to the belief that the subject term has reference. That is, the claim that Scott does not exist is false since
(5) ~∃x(x=s) is self-contradictory. (After all, there must exist at least one thing that is identical to s since it is a logical truth that s is identical to itself!) In contrast, the claim that a golden mountain does not exist may be true since, assuming that G abbreviates the predicate "is golden" and M abbreviates the predicate "is a mountain," there is nothing contradictory about
(6) ~∃x(Gx & Mx).
Russell's emphasis upon logical analysis also had consequences for his metaphysics. ...