Philip D. Welch

ULTIMATE TRUTH VIS - À - VIS STABLE TRUTH

We show that the set of ultimately true sentences in Hartry Field’s Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the semantic values of sentences in Field’s relative consistency proof of his theory over the ground model of the standard natural numbers: ∆3-CA0 (second order number theory with a ∆3-Comprehension Axiom scheme) is insufficient. We briefly consider his claim to have produced a “revenge-immune” solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a “determinately true” operator can be introduced in other settings.

Possible-Worlds Semantics for Modal Notions Conceived as Predicates

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □⌜A⌝ holds at a world w∈W if and only if A holds at every world v∈W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several ‘paradoxes’ (like Montagues Theorem, Gödels Second Incompleteness Theorem, McGees Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic.

Characteristics of discrete transfinite time Turing machine models: Halting times, stabilization times, and Normal Form theorems

We give an acount of the basic determinants of the courses of computation of the Infinite Time Turing Machine model of Hamkins and Kidder, a model of computation which allows for transfinitely many steps of computation, and therefore may accept and output infinite strings of bits. We provide, inter alia, a Normal form Theorem, and a characterisation of which ordinals start gaps in halting times of such machines.

The Extent of Computation in Malament–Hogarth Spacetimes

We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some ∀ ∃ relations on natural numbers that are neither universal nor co-universal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second-order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query ? n ∈ H ? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime, which is thus a universal constant of that spacetime. Theorem C. Assuming the (modest and standard) requirement that spacetime manifolds be paracompact and Hausdorff, for any spacetime there will be a countable ordinal upper bound, , on the complexity of questions in the Borel hierarchy computable in it. 1. Introduction1.1. History and preliminaries2. Hyperarithmetic Computations in MH Spacetimes2.1. Generalising SADn regions2.2. The complexity of questions decidable in Kerr spacetimes3. An Upper Bound on Computational Complexity for Each Spacetime Introduction History and preliminaries Hyperarithmetic Computations in MH Spacetimes Generalising SADn regions The complexity of questions decidable in Kerr spacetimes An Upper Bound on Computational Complexity for Each Spacetime

The undecidability of propositional adaptive logic

We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and can be

On the Possibility, or Otherwise, of Hypercomputation

\Sigma^0_3

Games for Truth

-complete. These classifications are exact. For first order theories even finite sets of premises can generate such consequence sets in either calculus.

Pf ≠ NPf for almost all f

We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their theory of circular definitions as those notions universally definable over the next stable set . We give a simplified (in terms of definitional complexity) account of varied revision sequences —as a generalised algorithmic theory of truth . This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.