About Axiomatic Theories of Truth
Axiomatic truth is an interdisciplinary subject of mathematical logic and philosophy which has been rapidly developing in recent two decades. In contrast to the attempts to define truth in terms of correspondence, coherence, or other notions, the axiomatic approach focuses on deductive theories with axioms and/or rules for some truth predicate(s) taken as primitive symbol(s) of a formal language. The logical properties of axiomatic theories of truth are relevant to various questions about Gödel’s theorems, truth-theoretic deflationism, or the theory of meaning.
Abstracts and Slides
Volker Halbach
Axiomatic Approaches to Truth
In philosophy truth is usually conceived as a notion that can be defined, possibly in terms of correspondence with reality or in terms of coherence or still other notions. However, there are good reasons not to assume that a definition of truth is possible and to treat truth as a primitive undefined notion. The semantic paradoxes impose limitations on which axioms or postulates for the meaning of the truth predicate can be consistently combined. I will present the motivations for axiomatic theories of truth and survey some of the most popular axiomatisations.
Sebastian Eberhard
Unfolding Theories of Feasible Arithmetic
Feferman's unfolding program describes a method to extend a base theory by the functions and predicates which are implicit in its acceptance. This extension will be carried out in a combinatory algebra using a truth predicate. Unfoldings of finitist and non-finitist arithmetics have been analysed by Feferman and Strahm producing natural theories of logical strength between Peano Arithmetic and Predicative Analysis. In this talk, we will show that the unfolding program also yields interesting results when it is applied to a feasible base theory. In this setting the unfolding program produces a new theory of polynomial strength from a very restricted base theory. The fact that the unfolded theory has polynomial strength demonstrates the computational power and expressive strength the truth predicate is able to add to weak theories of combinatorial algebra.
Leon Horsten
Reflection, Trust, Entitlement
It has been argued by Feferman and others that when we accept a mathematical theory, we implicitly commit ourselves to reflection principles for this theory. When we reflect on this implicit commitment, we come to explicitly believe certain reflection principles. In my talk I will discuss our epistemic warrant for this resulting explicit belief in reflection principles.
Philip Welch
Some Interactions Between Set Theory and Theories of Truth
Semantic theories of truth involve some mathematical construction over a first order model. Typically such a model is taken as the standard model of the natural numbers to exemplify a paradigm case. The construction may be more, or less, involved, but for the more complicated ones set theoretical features begin to loom large. We review Kripke's theory of minimal fixed points using supervaluations, and Herzberger's Revision Theory. We consider possibilities of axiomatizing various levels of the truth sets the Herzberger sequence throws up. This brings out relationships between Cantini's VF (which he invented to axiomatise the supervaluation fixed point theory) and strong admissibility theory.