# [FOM] Kripke's outline of a theory of truth

Timothy Y. Chow tchow at alum.mit.edu
Fri Feb 24 16:52:58 EST 2012

```Thanks to everyone for responding.  After doing some searching on my own,
I found a recent paper by Kentaro Fujimoto, "Autonomous progression and
transfinite iteration of self-applicable truth," J. Symbolic Logic Volume
76, Issue 3 (2011), 914-945.  The paper begins:

As far as the author knows, Jäger et al. [13] first introduced and
studied a transfinitely iterated self-applicable truth. They presented
the system of transfinitely iterated Kripke-Feferman truth and gave its
proof-theoretic analysis. Then, Strahm [18] determined the
proof-theoretic strength of the autonomous progression of
Kripke-Feferman truth. The present paper extends these studies to
systems of other kinds of self-applicable truths.

The two references cited are:

[13] Gerhard Jäger, Reinhard Kahle, Anton Setzer, and Thomas Strahm, The
proof-theoretic analysis of transfinitely iterated fixed point theories,
J. Symbolic Logic vol. 64 (1999), pp. 53-67.

[18] Thomas Strahm, Autonomous fixed point progressions and fixed point
transfinite recursion, Logic colloquium '98 (Samuel Buss, editor),
Lecture Notes in Logic, vol. 13, A K Peters, 2000, pp. 449-464.

Also helpful was the entry on axiomatic truth in the Stanford Encyclopedia
of Philosophy: http://plato.stanford.edu/entries/truth-axiomatic/#4.3
In the subsection on "the Kripke-Feferman theory," it gives an intuitive
description and then continues:

By making this idea precise, one obtains a variant of Kripke's (1975)
theory of truth with the so-called Strong Kleene valuation scheme (see
the entry on many-valued logic). If axiomatized it leads to the
following system, which is known as KF ("Kripke-Feferman"), of which
several variants appear in the literature:

After listing the axioms, it gives the following historical information:

Apart from the truth-theoretic axioms, KF comprises all axioms of PA and
all induction axioms involving the truth predicate. The system is
credited to Feferman on the basis of two lectures for the Association of
Symbolic Logic, one in 1979 and the second in 1983, as well as in
subsequent manuscripts. Feferman published his version of the system
under the label Ref(PA) ("weak reflective closure of PA") only in 1991,
after several other versions of KF had already appeared in print (e.g.,
Reinhardt 1986, Cantini 1989, who both refer to this unpublished work by
Feferman). Feferman's version does not contain the consistency axiom
(14), which does not contribute to the proof-theoretic strength of KF
anyway (see Cantini 1989 for more on the consistency axiom).

So it seems that the answer to my question is more complicated than I
expected, due in part to the numerous possible variations on the same
basic theme.

Tim
```