Related papers: Hard Provability Logics

The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$.…

For the Heyting Arithmetic HA, HA* is defined as the theory $\{A\mid {\sf HA}\vdash A^{\Box}\}$, where $A^{\Box}$ is called the box translation of $A$. We characterize the $\Sigma_1$-provability logic of HA* as a modal theory ${\sf…

We determine the strictly positive fragment $\mathsf{QPL}^+(\mathsf{HA})$ of the quantified provability logic $\mathsf{QPL}(\mathsf{HA})$ of Heyting Arithmetic. We show that $\mathsf{QPL}^+(\mathsf{HA})$ is decidable and that it coincides…

In this paper we introduce a modal theory $H_{\sigma}$, which is sound and complete for arithmetical $\Sigma$_1 substitutions in ${\bf HA}$, in other words, we will show that $H_{\sigma}$ is the $\Sigma$_1-provability logic of ${\bf HA}$.…

Defeasible statements are statements that are likely, or probable, or usually true, but may occasionally be false. Plausible reasoning makes conclusions from statements that are either facts or defeasible statements without using numbers.…

We axiomatize the provability logic of $\HA$ and prove its decidability. Furthermore, we axiomatize the preservativity and relative admissibility relations for several modal logics extending iK4. A principal technical tool is the…

We investigate several consequences of inclusion relations between quantified provability logics. Moreover, we give a necessary and sufficient condition for the inclusion relation between quantified provability logics with respect to…

This paper studies relative unification and admissibility in the intuitionistic logic. We generalize results of [Ghilardi, 1999; Iemhoff, 2001a] and prove them relative in NNIL(par) propositions, the class of propositions with No Nested…

In this paper (propositional) probability logic ($PL$) is investigated from model theoretic point of view. First of all, the ultraproduct construction is adapted for $\sigma$-additive probability models, and subsequently when this class of…

The branch of provability logic investigates the provability-based behavior of the mathematical theories. In a more precise way, it studies the relation between a mathematical theory $T$ and a modal logic $L$ via the provability…

Let L be some extension of classical propositional logic. The non-iterated probabilistic logic over L, is the logic PL that is defined by adding non-nested probabilistic operators in the language of L. For example in PL we can express a…

In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates $\Box$ and $\triangle$ that…

In this article, the decidability and computability issues of dynamic probability logic (DPL) are addressed. Firstly, a proof system $\mathcal{H}_{DPL}$ is introduced for DPL and shown that it is weakly complete. Furthermore, this logic has…

Plausible reasoning concerns situations whose inherent lack of precision is not quantified; that is, there are no degrees or levels of precision, and hence no use of numbers like probabilities. A hopefully comprehensive set of principles…

By Solovay's celebrated completeness result on formal provability we know that the provability logic $\mathrm GL$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable…

We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczy\'nski's pure logic of necessitation $\mathsf{N}$. Moreover, we introduce three extensions $\mathsf{N4}$, $\mathsf{NR}$, and…

The provability logic of a theory $T$ captures the structural behavior of formalized provability in $T$ as provable in $T$ itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability…

Vardanyan's Theorems state that $\mathsf{QPL}(\mathsf{PA})$ - the quantified provability logic of Peano Arithmetic - is $\Pi^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary…

Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable…

Possibilistic logic, an extension of first-order logic, deals with uncertainty that can be estimated in terms of possibility and necessity measures. Syntactically, this means that a first-order formula is equipped with a possibility degree…