Chiemsee Meeting: Semantic approaches to Truth and their
analysis
Slides here: likely not final. chiemsee-slides
Please check back
Reading: For the more advanced parts of the tutorials: the
philosophical aspects arising from Field's theory (from [7] or see [8]) are discussed in [1], [9]
(talks delivered at the APA Dec 09) whose mathematical counterpart on
hierarchies is in [2]
below.
Handout
for the talks: (perhaps not final) chiemsee-handout
The thrust of these tutorials is the
interaction between low level set theory, descriptive set theory, and semantical models of truth theories. In particular the
implications from this analysis for answering questions raised by philosophical
logicians such as: Is there an axiomatisation for the
Revision Theory of Truth? What are the lengths of path independent hierarchies
in Field's model of his theory with a conditional? What
are the implications for the Gupta-Belnap Revision
theory of the complexity of the stable truth sets?
We maintain that a full perspective on the resolution of
questions such as these are really only obtained by an analysis of the
underlying descriptive set theory, its interactions with Gödel's universe of
constructible sets, and the theory of (monotone) operators.
If time permits, we hope to mention
other connections with proof theory, reverse mathematics and transfinite Turing
machine models, that whilst not directly affecting the
philosophical viewpoint on the semantical theories,
nevertheless contribute to illuminating the conceptual structure at this level
of definability and the strengths of the notions being employed.
Part
I
will consist of a brief recapitulation of two older semantical models: Kripke's
Strong Kleene minimal fixed point model; Herzberger Revision Theory; and a very brief sketch of
Field's construction, and will raise the questions mentioned above.
Part
II
starts by introducing the Gödel hierarchy, Kripke-Platek set theory. It then gives some features of
the Herzbergerian, and Fieldian
hierarchies of determinateness, looking at how to develop an ``ineffable
liar'' sentence that escapes being captured by Field's path-independent
determinateness hierarchies. This is achieved by identifying sets, or ordinals,
that can be considered as internal to
the Kripke/Herzberger/Field
models. We see how a notation system extending Kleene's
O for the ordinals needed for the latter two models can be defined using
transfinite acting Turing machines; we further look at a possible proof-theoretic
argument at or below Π13-CA
which connects with this theory.
Reference
List
A. Gupta & N. Belnap
The Revision Theory of Truth, MIT
Press, 1975.
S. Kripke
An outline of a theory of truth, http://philo.ruc.edu.cn/logic/reading/Kripke_%20Theory%20of%20Truth.pdf
J. of Philosophy, vol 72, 1975.
H. Field
7 A Revenge-Immune Solution to the Semantic Paradoxes
Journal of Philosophical Logic April 2003, Volume 32, pp 139-177.
8
Saving Truth from Paradox, O.U.P. New
York, Oxford 2008.
Solving the Paradoxes, escaping
revenge, in "The Revenge of the Liar" Ed. J.C. Beall,
OUP, 2008.
V. Halbach
Axiomatic Theories of Truth, OUP,
20111.
H. Herzberger
Naive
semantics and the Liar paradox. Journal of
Philosophy, 79:479–497, 1982. http://www.jstor.org/stable/2026380
Notes on naive semantics. Journal of Philosophical Logic, 11:61–102, 1982. http://www.jstor.org/stable/30226237
L. Horsten
The Tarskian Turn: Deflationism and Axiomatic Truth,
MIT Press, 2011.
D.A. Martin
9
Saving Truth from Paradox: some things it does not do, in
Review of Symbolic Logic, 4, No.3, Sept. 2011.
V. McGee
Truth, Vagueness, Paradox, an essay on
the logic of truth, Hackett, 1991.
P.D. Welch