The diagonal argument and the Liar (Q914657)

The paper gives the following as the abstract structure of a diagonal argument, which is first-order valid. Let \(M=(m_{ij}\); \(i\in I\), \(j\in J)\) be a matrix (possibly infinite); let F be a bijection from I to J; then if H is any sequence \((h_ i\); \(i\in I)\) such that \(h_ i\neq m_{iF(i)}\) (i\(\in I)\) then H is not a row of M. A good diagonal argument is one in which we have reason to suppose two of the following: M exists; F exists; and there is an H which is a row of M. The negation of the third can then be inferred. A diagonal paradox occurs when we have reason to believe all three. The paper then shows that a number of traditional diagonal arguments (e.g., Cantor's Theorem, the Halting Theorem) are good diagonal arguments in this sense; and that a number of paradoxes of self-reference (e.g., Russell's, the Liar) are diagonal paradoxes. (It does not observe, though it is perhaps worth noting, that not all such paradoxes are diagonal paradoxes, e.g., Berry's.) Using this characterisation, the paper then demonstrates that a number of modern theories proposing solutions to the Liar Paradox fail by being insufficiently general: even if they give reason to suppose that some of the relevant M, F and H's do not exist, there are others whose existence is untouched by these theories. Typically, they are those to be found in ``extended'' or ``strengthened'' paradoxes. See the reviewer: ``Semantic closure'' [Stud. Logica 43, 117-129 (1984; Zbl 0575.03002)] and Ch. 1 of: In contradiction (1987; Zbl 0682.03002)].