A paradox about sets of properties is presented. The paradox, which invokes an impredicatively defined property, is formalized in a free third-order logic with lambda-abstraction, through a classically proof-theoretically valid deduction of a contradiction from a single premise to the effect that every property has a unit set. Something like a model is offered to establish that the premise is, although classically inconsistent, nevertheless consistent, so that the paradox discredits the logic employed. A resolution through the ramified theory of types (...) is considered. Finally, a general scheme that generates a family of analogous paradoxes and a generally applicable resolution are proposed. (shrink)

The paper argues that the liar paradox teaches us these lessons about English. First, the paradox-yielding sentence is a sentence of English that is neither true nor false in English. Second, there is no English name for any such thing as a set of all and only true sentences of English. Third, ‘is true in English’ does not satisfy the axiom of comprehension.

In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that there exists (...) a direct line leading from Cantor’s diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church’s theorem concerning the undecidability of formal arithmetic, Gödel’s theorem concerning the incopleteness of formal arithmetic, Tarski’s theorem concerning truth, and Turing’s theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor’s diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme. (shrink)

In a series of papers R. L. Martin propounds a theory for dealing with the semantical paradoxes. This paper is a criticism of that theory.