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We augment the first-order language of Peano Arithmetic with a unary
predicate $T$ with $T(x)$ intended to mean ``$x$ is the G"odel
number of a true sentence of arithmetic''. In (1987) Friedman and
Sheard constructed a list of twelve axioms and rules of inference
concerning the predicate $T$, each expressing some desirable property
of truth, and classified all subsets of the list as either consistent
or inconsistent. This gave rise to a collection of nine theories of
truth, two of which have been treated in the literature (see Friedman
and Sheard 1987, Halbach (1994), Sheard (2001) and Cantini (1990) for
more details). We uncover the proof-theoretic strength of the
remaining seven and in the process construct a proof-theory of truth
allowing the systems to be subject to an ordinal analysis.
(1987) H. Friedman and M. Sheard, An axiomatic approach to
self-referential truth, Annals of Pure and Applied Logic, 33 1--21.
(1990) A. Cantini, A theory of formal truth arithmetically equivalent
to ID1, Journel of Symbolic Logic, 55, 1, 244--259.
(1994) V. Halbach, A system of complete and consistent truth, Notre
Dame Journel of Formal Logic, vol. 35, 3.
(2001) M. Sheard, Weak and strong theories of truth, Studia Logica,