×

zbMATH — the first resource for mathematics

Axiomatizing Kripke’s theory of truth. (English) Zbl 1101.03005
Kripke’s fixed point models were formalized in classical logic by S. Feferman in a quite natural way [see “Reflecting on incompleteness”, J. Symb. Log. 56, 1–49 (1991; Zbl 0746.03046)]. That axiomatization is commonly known as \(\mathbf{KF}\) system and is equivalent to the system \(\mathbf{RA}_{<\varepsilon_0}\) of ramified analysis up to any ordinal level smaller than \(\varepsilon_0\). Reinhardt has posed the problem whether \(\mathbf{KF}\) can be viewed as a tool for producing theorems that would also be derivable in a direct formalization of Kripke’s original theory in partial logic, if one focuses on the sentences that are provably true in \(\mathbf{KF}\) [see W. Reinhardt, “Remarks on significance and meaningful applicability”, in: L. P. de Alcantara (ed.), Mathematical logic and formal systems, Coll. Pap. Hon. N. C. A. da Costa, Lect. Notes Pure Appl. Math. 94, 227–242 (1985; Zbl 0611.03004), and “Some remarks on extending and interpreting theories with a partial predicate for truth”, J. Philos. Logic 15, 219–251 (1986; Zbl 0629.03002)].
The authors solve Reinhardt’s problem negatively and present an axiomatization \(\mathbf{PKF}\) of Kripke’s theory of truth in partial logic. They claim that any natural axiomatization of Kripke’s theory in Strong Kleene logic will be equivalent to their system \(\mathbf{PKF}\). The proof-theoretic strength of \(\mathbf{PKF}\) is determined as that of \(\mathbf{RA}_{<\omega^\omega}\) of ramified analysis up to \(\omega^\omega\) in contrast to the much stronger \(\mathbf{KF}\). This result shows that axiomatizing Kripke’s theory in the most natural way leads to a system that is much weaker than the classical system \(\mathbf{KF}\). In particular, the arithmetical content of both theories is far from identical. This proof-theoretic analysis sheds some light on the classification of axiomatizations of Kripke’s theory with respect to other theories of truth as well.

MSC:
03A05 Philosophical and critical aspects of logic and foundations
03F03 Proof theory in general (including proof-theoretic semantics)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Basic proof theory (1997)
[2] Systems of predicative analysis 29 pp 1– (1964) · Zbl 0134.01101
[3] Proof theory (1987)
[4] Understanding truth (1999)
[5] DOI: 10.2307/2182187
[6] DOI: 10.1002/malq.19740201903 · Zbl 0299.02015
[7] Principles of truth pp 169– (2002)
[8] Logical frameworks for truth and abstraction. An axiomatic study 135 (1996) · Zbl 0860.03015
[9] A theory of formal truth arithmetically equivalent to ID1 55 pp 244– (1990)
[10] {\(\lambda\)}-calculus and computer science pp 1– (1975)
[11] DOI: 10.1002/malq.19890350202 · Zbl 0661.03043
[12] Handbook of mathematical logic pp 867– (1977)
[13] Handbook of philosophical logic 5 pp 261– (2002)
[14] DOI: 10.1007/BF00305492 · Zbl 0629.03002
[15] The revision theory of truth (1993) · Zbl 0858.03010
[16] Mathematical logic and formal systems. A collection of papers in honor of Professor Newton C.A. Da Costa 94 pp 227– (1985)
[17] Deflationism and paradox (2005)
[18] Proof theory. An introduction (1989)
[19] DOI: 10.1305/ndjfl/1040511343 · Zbl 0822.03014
[20] On n-quantifier induction 37 pp 466– (1972) · Zbl 0264.02027
[21] Elementary induction on abstract structures (1974)
[22] Formal philosophy: Selected papers of Richard Montague (1974)
[23] Acta Philosophica Fennica 16 pp 153– (1963)
[24] Truth, vagueness, and paradox: An essay on the logic of truth (1991) · Zbl 0734.03001
[25] Truth and paradox, Solving the riddles (2004) · Zbl 1083.03002
[26] DOI: 10.1007/BF02379018
[27] Recent essays on truth and the Liar paradox (1984)
[28] DOI: 10.2307/2024634 · Zbl 0952.03513
[29] Journal of Philosophical Logic 17 pp 225– (1988)
[30] DOI: 10.1002/malq.19790250307 · Zbl 0407.03030
[31] Minnesota studies in the philosophy of science, II pp 37– (1958)
[32] Stanford Encyclopedia of Philosophy (2006)
[33] DOI: 10.1023/A:1005662017962 · Zbl 0971.03008
[34] Axiomatische Wahrheitstheorien (1996)
[35] DOI: 10.1305/ndjfl/1040511340 · Zbl 0828.03030
[36] DOI: 10.1016/0168-0072(88)90038-3 · Zbl 0669.03026
[37] On supervaluations in free logic 49 pp 943– (1984)
[38] DOI: 10.1016/0168-0072(87)90073-X · Zbl 0634.03058
[39] DOI: 10.1002/malq.19610071705 · Zbl 0124.24604
[40] Reflecting on incompleteness 56 pp 1– (1991)
[41] Handbook of philosophical logic 4 pp 617– (1989)
[42] Transfinite recursive progressions of axiomatic theories 27 pp 259– (1962)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.