On Pretabular Logics in NExtK4 (Part I)

This paper partly answers the question “what a frame may be exactly like when it characterizes a pretabular logic in NExtK4”. We prove the pretabularity crieria for the logics of finite depth in NExtK4. In order to find out the criteria, we create two useful concepts—“pointwise reduction” and “invariance under pointwise reductions”, which will remain important in dealing with the case of infinite depth.     

On Modal Logics Characterized by Models with Relative Accessibility Relations: Part II

This work is divided in two papers (Part I and Part II). In Part I, we introduced the class of Rare-logics for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability were established by faithfully translating the Rare-logics into more standard modal logics (some of them contain the universal modal operator).In Part II, we push forward the results from Part I. For Rare-logics with nominals (present at the level of formulae and at the level of modal expressions), we show that the constructions from Part I can be extended although it is technically more involved. We also characterize a class of standard modal logics for which the universal modal operator can be eliminated as far as satifiability is concerned. Although the previous results have a semantic flavour, we are also able to define proof systems for Rare-logics from existing proof systems for the corresponding standard modal logics. Last, but not least, decidability results for Rare-logics are established uniformly, in particular for information logics derived from rough set theory.Since this paper is the continuation of Part I, we do not recall here the definitions of Part I although we refer to them.     

De Finettian Logics of Indicative Conditionals Part II: Proof Theory and Algebraic Semantics

Journal of Philosophical Logic - In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de...     

On Modal Logics Characterized by Models with Relative Accessibility Relations: Part I

This work is divided in two papers (Part I and Part II). In Part I, we study a class of polymodal logics (herein called the class of "Rare-logics") for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability are established by faithfully translating the Rare-logics into more standard modal logics. The main idea of the translation consists in eliminating the Boolean terms by taking advantage of the components construction and in using various properties of the classes of semilattices involved in the semantics. The novelty of our approach allows us to prove new decidability results (presented in Part II), in particular for information logics derived from rough set theory and we open new perspectives to define proof systems for such logics (presented also in Part II).     

Products of Modal Logics. Part 3: Products of Modal and Temporal Logics

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.     

On AGM for Non-Classical Logics

The AGM theory of belief revision provides a formal framework to represent the dynamics of epistemic states. In this framework, the beliefs of the agent are usually represented as logical formulas while the change operations are constrained by rationality postulates. In the original proposal, the logic underlying the reasoning was supposed to be supraclassical, among other properties. In this paper, we present some of the existing work in adapting the AGM theory for non-classical logics and discuss their interconnections and what is still missing for each approach.     

Myth & Math,Part II (preliminary draft)

It is argued that there can only be a small-finite number of mathematical objects; that these objects range from the very concrete to the very abstract; and that mathematics is essentially not concerned with objects but with concepts. This viewpoint is described as mentalist and is upheld over Platonism, intuitionism, and formalism.     

On Dynamic Topological and Metric Logics

We investigate computational properties of propositional logics for dynamical systems. First, we consider logics for dynamic topological systems (W.f), fi, where W is a topological space and f a homeomorphism on W. The logics come with ‘modal’ operators interpreted by the topological closure and interior, and temporal operators interpreted along the orbits {w, f(w), f² (w), ˙˙˙} of points w ε W. We show that for various classes of topological spaces the resulting logics are not recursively enumerable (and so not recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints. Second, we consider logics for dynamical systems (W, f), where W is a metric space and f and isometric function. The operators for topological interior/closure are replaced by distance operators of the form ‘everywhere/somewhere in the ball of radius a, ‘for a ε Q ⁺. In contrast to the topological case, the resulting logic turns out to be decidable, but not in time bounded by any elementary function.     

On Inclusions Between Quantified Provability Logics

Studia Logica - We investigate several consequences of inclusion relations between quantified provability logics. Moreover, we give a necessary and sufficient condition for the inclusion relation...     

On Conservative Extensions in Logics with Infinitary Predicates

If the language is extended by new individual variables, in classical first order logic, then the deduction system obtained is a conservative extension of the original one. This fails to be true for the logics with infinitary predicates. But it is shown that restricting the commutativity of quantifiers and the equality axioms in the extended system and supposing the merry-go-round property in the original system, the foregoing extension is already conservative. It is shown that these restrictions are crucial for an extension to be conservative. The origin of the results is algebraic logic. Presented by Daniele Mundici Supported by grant OTKA T43242.