Tarski's two theories of truth: Why and How?

Tarski's theory of truth is nowadays usually presented as the notion of
truth in a structure. Wilfrid Hodges has pointed out that the first
definition of that notion that actually appears in the literature is by
Tarski and Vaught in 1957, though it is usually ascribed to Tarski in his
famous Wahrheitsbegriff article of 1935. Instead, the latter is (in my
view) a version of absolute truth or truth simpliciter. The technical
underpinning is the same for both, via the inductive definition of a
notion of satisfaction. Since the informal notion of truth in a structure
was in common use by a number of Tarski's contemporaries, including
Skolem, Gödel and even Tarski himself, it is of historical interest to
determine what motivated him to first define truth instead in the absolute
sense. A second question has to do with his choice of a set-theoretical
metalanguage in which to formulate his semantical notions. Finally, why
did it take so long to publish a definition of truth in a structure? This
talk is based on the following article; it will consist of both expository
and historical sections.

S. Feferman, `Tarski¹s conceptual analysis of semantical notions',
available at
http://math.stanford.edu/~feferman/papers/conceptanalysis.pdf

What Truth Depends On

We owe to Tarski both a positive and a negative result on truth: the positive one is that the extension of the truth predicate "Tr" can be defined in a formally correct and materially adequate manner if the vocabulary of the language for which it is to be defined does not contain "Tr". The negative one is that if the vocabulary of a language does contain "Tr" and if the language is also sufficiently expressive, then the extension of "Tr" cannot be defined in a manner such that all T-biconditionals could be derived from the definition. However, neither of these results tells us anything about whether it is possible to define truth adequately for a set of sentences which lies somewhere "in between" complete languages with and complete languages without truth predicate. This leads us to the question: What kinds of sentences with truth predicate may be inserted plausibly and consistently into the T-scheme?

We state an answer in terms of dependence: those sentences which depend directly or indirectly on non-semantic states of affairs. In order to make this precise we introduce a theory of dependence for sentences with or without truth predicate and we show that there is an extension of Tarski's definition of truth which is adequate with respect to all sentences that depend on non-semantic states of affairs.

Formal Properties of Meaning

Being (1) a mathematician and (2) completely unclear what meanings are, I read some of the founding fathers of semantics from Jurjani to Tarski, formalised some things and drew some consequences. The results have attracted some interest. I hope to give at least informal descriptions of recent applications and extensions in various fields of logic and semantics by Bradfield, Parikh, Vaananen, Werning, Westerstahl and maybe others.

Logicality and Constancy of Logical Constants

The model-theoretic approach has proved to be particularly suited for
an account of quantification in natural languages. Distinguish
(generalized) quantifier expressions from predicate expressions
(denoting relations between individuals). Are quantifier expressions
logical constants whereas predicate expressions are not? How do you
in general identify the logical constants, in any type? While there
may be no adequate definition of this notion, I look at two necessary
conditions. The first - topic-neutrality - is a familiar
model-theoretic requirement (Isomorphism Closure) with precise
consequences. Some stronger proposals (including one by Feferman)
will be briefly examined. But I focus on the second - constancy -
which is less precise but no less significant. The idea is to use
basic intuitions about form and entailment as data, concluding that
(most) quantifier expressions are constants whereas (most) predicate
expressions are not. More precisely, given a Bolzano style notion of
appropriate replacement, look at preservation of entailment under
such replacements (Bolzano, and later Tarski, rather considered
preservation of truth). These facts about constancy are rather
trivial (perhaps too trivial to be noticed), but allow the conclusion
that quantifier expressions such as most or few should not be
interpreted in models, contrary to proposals by e.g. Barwise & Cooper
(1981). The ensuing use of first-order models (not first-order
logic!) for the semantics of quantification is also consonant with
another crucial but somewhat elusive property of quantifiers in
language and logic: that they express the 'same (second-order)
relation' on every universe.

Most of the talk is based on chapter 9 in a forthcoming book with
Stanley Peters.

Towards a Grammar Formalism for Dialogue

Standard grammar formalisms are defined without reflection of the incremental, serial and context-dependent nature of language processing; any incrementality must therefore be reflected by independently defined parsing and/or generation techniques, and context-dependence by separate pragmatic modules. This leads to a poor setup for modelling dialogue, with its rich speaker-hearer interaction and high proportion of apparently grammatically ill-formed utterances. This talk will introduce an inherently incremental grammar formalism, Dynamic Syntax (Kempson et al. 2001), in which grammaticality is defined via the successful incremental projection of a semantic interpretation, represented as a decorated tree structure defined in a modal tree logic, LOFT (Blackburn & Meyer-Viol 1994). A context-based extension will then be proposed, together with corresponding parsing and generation models. These are shown to allow a straightforward model of otherwise problematic dialogue phenomena such as shared utterances, ellipsis and alignment, and to allow a natural definition of context-dependent well-formedness.

This work is a collaboration with Ruth Kempson (King's College, London) and Ronnie Cann (University of Edinburgh).