Type-free truth

This book is a contribution to the flourishing field of formal and
philosophical work on truth and the semantic paradoxes. Our aim is
to present several theories of truth, to investigate some of their
model-theoretic, recursion-theoretic and proof-theoretic aspects,
and to evaluate their philosophical significance. In Part I we
first outline some motivations for studying formal theories of
truth, fix some terminology, provide some background on Tarski’s
and Kripke’s theories of truth, and then discuss the prospects of
classical type-free truth. In Chapter 4 we discuss some minimal
adequacy conditions on a satisfactory theory of truth based on the
function that the truth predicate is intended to fulfil on the
deflationist account. We cast doubt on the adequacy of some
non-classical theories of truth and argue in favor of classical
theories of truth. Part II is devoted to grounded truth. In chapter
5 we introduce a game-theoretic semantics for Kripke’s theory of
truth. Strategies in these games can be interpreted as
reference-graphs (or dependency-graphs) of the sentences in
question. Using that framework, we give a graph-theoretic analysis
of the Kripke-paradoxical sentences. In chapter 6 we provide
simultaneous axiomatizations of groundedness and truth, and analyze
the proof-theoretic strength of the resulting theories. These range
from conservative extensions of Peano arithmetic to theories that
have the full strength of the impredicative system ID1. Part III
investigates the relationship between truth and set-theoretic
comprehen- sion. In chapter 7 we canonically associate extensions
of the truth predicate with Henkin-models of second-order
arithmetic. This relationship will be employed to determine the
recursion-theoretic complexity of several theories of grounded
truth and to show the consistency of the latter with principles of
generalized induction. In chapter 8 it is shown that the sets
definable over the standard model of the Tarskian hierarchy are
exactly the hyperarithmetical sets. Finally, we try to apply a
certain solution to the set-theoretic paradoxes to the case of
truth, namely Quine’s idea of stratification. This will yield
classical disquotational theories that interpret full second-order
arithmetic without set parameters, Z2- (chapter 9). We also
indicate a method to recover the parameters. An appendix provides
some background on ordinal notations, recursion theory and graph
theory.