Boolean Skeletons of MV-algebras and ?-groups

Let ?? be Mundici??s functor from the category ${\mathcal{LG}}$ whose objects are the lattice-ordered abelian groups (?-groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category ${\mathcal{MV}}$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an ?-group G, the Boolean skeleton of the MV-algebra ??(G, u) is isomorphic to the Boolean algebra of factor congruences of G.     

Some Normal Extensions of K4.3

This paper proves the finite model property and the finite axiomatizability of a class of normal modal logics extending K4.3. The frames for these logics are those for K4.3, in each of which every point has a bounded number of irreflexive successors if it is after an infinite ascending chain of (not necessarily distinct) points.     

BK-lattices. Algebraic Semantics for Belnapian Modal Logics

Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK-lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK-lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK. Finally, we describe invariants determining a twist-structure over a modal algebra.     

Fatal Heyting Algebras and Forcing Persistent Sentences

Hamkins and L?we proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H _ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.     

Interpolation properties of superintuitionistic logics

A family of prepositional logics is considered to be intermediate between the intuitionistic and classical ones. The generalized interpolation property is defined and proved is the following. Theorem on interpolation. For every intermediate logic L the following statements are equivalent:

1. Craig's interpolation theorem holds in L,
2. L possesses the generalized interpolation property,
3. Robinson's consistency statement is true in L.

There are just 7 intermediate logics in which Craig's theorem holds. Besides, Craig's interpolation theorem holds in L iff all the modal companions of L possess Craig's interpolation property restricted to those formulas in which every variable is proceeded by necessity symbol.     

The d-Logic of the Rational Numbers: A Fruitful Construction

We present a geometric construction that yields completeness results for modal logics including K4, KD4, GL and GL _n with respect to certain subspaces of the rational numbers. These completeness results are extended to the bimodal case with the universal modality.     

Dynamic Modalities

A new modal logic containing four dynamic modalities with the following informal reading is introduced: ${\square^\forall}$ – always necessary, ${\square^\exists}$ – sometimes necessary, and their duals – ${\diamondsuit^\forall}$ – always possibly, and ${\diamondsuit^\exists}$ – sometimes possibly. We present a complete axiomatization with respect to the intended formal semantics and prove decidability via fmp.     

Quantification in Some Non-normal Modal Logics

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems by Stolpe (Logic Journal of the IGPL 11(5), 557–575, 2003), but unfortunately that work used only models and not frames.     

Kripke Sheaf Completeness of some Superintuitionistic Predicate Logics with a Weakened Constant Domains Principle

The completeness w.r.t. Kripke frames with equality (or, equivalently, w.r.t. Kripke sheaves, [8] or [4, Sect. 3.6]) is established for three superintuitionistic predicate logics: (Q-H + D*), (Q-H + D*&K), (Q-H + D*&K&J). Here Q-H is intuitionistic predicate logic, J is the principle of the weak excluded middle, K is Kuroda’s axiom, and D* (cf. [12]) is a weakened version of the well-known constant domains principle D. Namely, the formula D states that any individual has ancestors in earlier worlds, and D* states that any individual has ${\neg\neg}$ -ancestors (i.e., ancestors up to ${\neg\neg}$ -equality) in earlier worlds. In particular, the logic (Q-H + D*&K&J) is the Kripke sheaf completion of (Q-H + E&K&J), where E is a version of Markov’s principle (cf. [12]). On the other hand, we show that the logic (Q-H + D*&J) is incomplete w.r.t. Kripke sheaves.     

Effective Cut-elimination for a Fragment of Modal mu-calculus

A non-effective cut-elimination proof for modal mu-calculus has been given by G. J?ger, M. Kretz and T. Studer. Later an effective proof has been given for a subsystem M ₁ with non-iterated fixpoints and positive endsequents. Using a new device we give an effective cut-elimination proof for M ₁ without restriction to positive sequents.     

Some theorems on structural entailment relations

The classesMatr( \( \subseteq \) ) of all matrices (models) for structural finitistic entailments \( \subseteq \) are investigated. The purpose of the paper is to prove three theorems: Theorem I.7, being the counterpart of the main theorem from Czelakowski [3], and Theorems II.2 and III.2 being the entailment counterparts of Bloom's results [1]. Theorem I.7 states that if a classK of matrices is adequate for \( \subseteq \) , thenMatr( \( \subseteq \) ) is the least class of matrices containingK and closed under the formation of ultraproducts, submatrices, strict homomorphisms and strict homomorphic pre-images. Theorem II.2 in Section II gives sufficient and necessary conditions for a structural entailment to be finitistic. Section III contains theorems which characterize finitely based entailments.     

Information closure and the sceptical objection

In this article, I define and then defend the principle of information closure (pic) against a sceptical objection similar to the one discussed by Dretske in relation to the principle of epistemic closure. If I am successful, given that pic is equivalent to the axiom of distribution and that the latter is one of the conditions that discriminate between normal and non-normal modal logics, a main result of such a defence is that one potentially good reason to look for a formalization of the logic of “ $S$ is informed that $p$ ” among the non-normal modal logics, which reject the axiom, is also removed. This is not to argue that the logic of “ $S$ is informed that $p$ ” should be a normal modal logic, but that it could still be insofar as the objection that it could not be, based on the sceptical objection against pic, has been removed. In other word, I shall argue that the sceptical objection against pic fails, so such an objection provides no ground to abandon the normal modal logic B (also known as KTB) as a formalization of “ $S$ is informed that $p$ ”, which remains plausible insofar as this specific obstacle is concerned.     

Axiomatic extensions of the constructive logic with strong negation and the disjunction property

We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B εV, thenA×B is a homomorphic image of some well-connected algebra ofV. We prove:

• each varietyV of Nelson algebras with PQWC lies in the fibre σ^?1(W) for some varietyW of Heyting algebras having PQWC,
• for any varietyW of Heyting algebras with PQWC the least and the greatest varieties in σ^?1(W) have PQWC,
• there exist varietiesW of Heyting algebras having PQWC such that σ^?1(W) contains infinitely many varieties (of Nelson algebras) with PQWC.

     

On the axiomatization of finiteK-frames

We find a short way to construct a formula which axiomatizes a given finite frame of the modal logicK, in the sense that for each finite frameA, we construct a formula ωA which holds in those and only those frames in which every formula true inA holds. To obtain this result we find, for each finite model \(\mathfrak{A}\) and each natural numbern, a formula ω \(\mathfrak{A}\) which holds in those and only those models in which every formula true in \(\mathfrak{A}\) , and involving the firstn propositional letters, holds.     

An Inconsistency-Adaptive Deontic Logic for Normative Conflicts

We present the inconsistency-adaptive deontic logic DP ^r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A?∧?O ～A, O A?∧?P ～A or even O A?∧?～O A. On the other hand, DP ^r still verifies all intuitively reliable inferences valid in Standard Deontic Logic (SDL). DP ^r interprets a given premise set ‘as normally as possible’ with respect to SDL. Whereas some SDL-rules are verified unconditionally by DP ^r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP ^r .     

The Church–Fitch knowability paradox in the light of structural proof theory

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$ , by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$ (KP). The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$ (OP). A Gentzen-style reconstruction of the Church–Fitch paradox is presented following a labelled approach to sequent calculi. First, a cut-free system for classical (resp. intuitionistic) bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism.