Russell's Theory of Descriptions
My favourite example of an elegant and inspirational theory in philosophy is Russell's Theory of Descriptions. It did not prove definitive, but it prompted richly insightful trains of enquiry into the structure of language and thought.
In essence Russell's theory turns on the idea that there is logical structure beneath the surface forms of language, which analysis brings to light; and when this structure is revealed we see what we are actually saying, what beliefs we are committing ourselves to, and what conditions have to be satisfied for the truth or falsity of what is thus said and believed.
One example Russell used to illustrate the idea is the assertion 'the present King of France is bald,' said when there is no King of France. Is this assertion true or false? One response might be to say that it is neither, since there is no King of France at present. But Russell wished to find an explanation for the falsity of the assertion which did not dispense with bivalence in logic, that is, the exclusive alternative of truth and falsity as the only two truth-values.
He postulated that the underlying form of the assertion consists in the conjunction of three logically more basic assertions: (a) there is something that has the property of being King of France, (b) there is only one such thing (this takes care of the implication of the definite article 'the') (c) and that thing has the further property of being bald. In the symbolism of first-order predicate calculus, which Russell took to be the properly unambiguous rendering of the assertion's logical form (I omit strictly correct bracketing so as not to clutter):
(Ex)Kx & [(y)Ky— >y=x] & Bx
which is pronounced 'there is an x such that x is K; and for anything y, if y is K then y and x are identical (this deals logically with 'the' which implies uniqueness); and x is B,' where K stands for 'has the property of being King of France' and B stands for 'has the property of being bald.' 'E' is the existential quantifier 'there is...' or 'there is at least one...' and '(y)' stands for the universal quantifier 'for all' or 'any.'
One can now see that there are two ways in which the assertion can be false; one is if there is no x such that x is K, and the other is if there is an x but x is not bald. By preserving bivalence and stripping the assertion to its logical bones Russell has provided what Frank Ramsey wonderfully called 'a paradigm of philosophy.'
To the irredeemable sceptic about philosophy all this doubtless looks like 'drowning in two inches of water' as the Lebanese say; but in fact it is in itself an exemplary instance of philosophical analysis, and it has been very fruitful as the ancestor of work in a wide range of fields, ranging from the contributions of Wittgenstein and W. V. Quine to research in philosophy of language, linguistics, psychology, cognitive science, computing and artificial intelligence.