Iterated reflection principles over full disquotational truth

Iterated reflection principles have been employed extensively to unfold epistemic commitments that are incurred by accepting a mathematical theory. Recently this has been applied to theories of truth. The idea is to start with a collection of Tarski-biconditionals and arrive by finitely iterated reflection at strong compositional truth theories. In the context of classical logic it is incoherent to adopt an initial truth theory in which A and 'A is true' are inter-derivable. In this article we show how in the context of a weaker logic, which we call Basic De Morgan Logic, we can coherently start with such a fully disquotational truth theory and arrive at a strong compositional truth theory by applying a natural uniform reflection principle a finite number of times

Well-orders in the transfinite Japaridze algebra

This paper studies the transfinite propositional provability logics \glp_\Lambda and their corresponding algebras. These logics have for each ordinal $\xi< \Lambda$ a modality \la \alpha \ra. We will focus on the closed fragment of \glp_\Lambda (i.e., where no propositional variables occur) and \emph{worms} therein. Worms are iterated consistency expressions of the form \la \xi_n\ra \ldots \la \xi_1 \ra \top. Beklemishev has defined well-orderings $<_\xi$ on worms whose modalities are all at least $\xi$ and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original $<_\xi$ orderings and provide a calculus for the corresponding generalized order-types $o_\xi$. Our calculus is based on so-called {\em hyperations} which are transfinite iterations of normal functions. Finally, we give two different characterizations of those sequences of ordinals which are of the form \la {\formerOmega}_\xi (A) \ra_{\xi \in \ord} for some worm $A$. One of these characterizations is in terms of a second kind of transfinite iteration called {\em cohyperation.}Comment: Corrected a minor but confusing omission in the relation between Veblen progressions and hyperation

The omega-rule interpretation of transfinite provability logic

A

Models of transfinite provability logic

For any ordinal \Lambda, we can define a polymodal logic GLP(\Lambda), with a modality [\xi] for each \xi<\Lambda. These represent provability predicates of increasing strength. Although GLP(\Lambda) has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities. Later, Icard defined a topological model for the same fragment which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary \Lambda. More generally, for each \Theta,\Lambda we build a Kripke model I(\Theta,\Lambda) and a topological model T(\Theta,\Lambda), and show that the closed fragment of GLP(\Lambda) is sound for both of these structures, as well as complete, provided \Theta is large enough

Hypatia's silence. Truth, justification, and entitlement.

Hartry Field distinguished two concepts of type-free truth: scientific truth and disquotational truth. We argue that scientific type-free truth cannot do justificatory work in the foundations of mathematics. We also present an argument, based on Crispin Wright's theory of cognitive projects and entitlement, that disquotational truth can do justificatory work in the foundations of mathematics. The price to pay for this is that the concept of disquotational truth requires non-classical logical treatment

Ultimate approximations in nonmonotonic knowledge representation systems

We study fixpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order approximations by means of a precision ordering. We show that each lattice operator O has a unique most precise or ultimate approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We apply our theory to logic programming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their standard counterparts We argue that ultimate semantics for logic programming have attractive epistemological properties and that, while in general they are computationally more complex than the standard semantics, for many classes of theories, their complexity is no worse.Comment: This paper was published in Principles of Knowledge Representation and Reasoning, Proceedings of the Eighth International Conference (KR2002