Why the liar does not matter (Q1404392)

Tarski's approach to the liar paradox was to distinguish the sentences of the object language from those of a metalanguage, though he conceded that no rational ground could be given for forbidding the use of ``true sentence'' from the metalanguage in the object language. Berk takes a language \textsterling{} for Peano Arithmetic and splits the sentences into the arithmetic hierarchy, each equivalent to a \(\Sigma_n\) or \(\Pi_n\) formula. It follows that \(\text{Tr}(A)\leftrightarrow\varphi\) (where \(A\) is a structural description of \(\varphi\)) cannot be a formula of \textsterling{} for all sentences \(\varphi\). Deflationists cannot distinguish those instances which hold and those which do not, as correspondence theorists and Tarski himself can do, for deflationists hold that disquotation is all there is to truth. But because of what Russell once dismissed as ``merely medical limitations'', there is some (very large) number \(m\) such that every sentence we ever utter is equivalent to a formula \(\Sigma_n\) (or \(\Pi_n\)), \(n<m\); and a(n even larger) number \(k\) such that every sentence we might utter is equivalent to a \(\Sigma_k\) (or \(\Pi_k\)) formula, \(k< m\). The predicate \(\text{Tr}_k(A)\) will then be all we ever need as a truth predicate. Its fixed point, though it exists, is not among the sentences of \textsterling{} we might express. ``Every theorem of PA is true'' will be false if ``theorem'' is understood as ``grammatical sentence''; but if as ``expressible sentence'' or ``truth bearer'' (i.e., of rank \(< m\)), it will be true. Berk proceeds to the modal formulation.