In the ordinary modal language, KD is the modal logic determined by the class of all serial frames. In this paper, we demonstrate that KD is nullary.

The problem of unification in a normal modal logic $L$ can be defined as follows: given a formula $\varphi$, determine whether there exists a substitution $\sigma$ such that $\sigma $ is in $L$. In this paper, we prove that for several non-symmetric non-transitive modal logics, there exists unifiable formulas that possess no minimal complete set of unifiers.

We study unification problem and problem of admissibility for inference rules in minimal Johanssonsʼ logic J and positive intuitionistic logic IPC+. This paper proves that the problem of admissibility for inference rules with coefficients is decidable for the paraconsistent minimal Johanssonsʼ logic J and the positive intuitionistic logic IPC+. Using obtained technique we show also that the unification problem for these logics is also decidable: we offer algorithms which compute complete sets of unifiers for any unifiable formula. Checking just unifiability (...) of formulas with coefficients also works via verification of admissibility. (shrink)

While specifications and verifications of concurrent systems employ Linear Temporal Logic , it is increasingly likely that logical consequence in image will be used in the description of computations and parallel reasoning. Our paper considers logical consequence in the standard image with temporal operations image and image . The prime result is an algorithm recognizing consecutions admissible in image, so we prove that image is decidable w.r.t. admissible inference rules. As a consequence we obtain algorithms verifying the validity of consecutions (...) in image and solving the satisfiability problem. We start by a simple reduction of logical consecutions of image to equivalent ones in the reduced normal form . Then we apply a semantic technique based on image-Kripke structures with formula definable subsets. This yields necessary and sufficient conditions for a consecution to be not admissible in image. These conditions lead to an algorithm which recognizes consecutions admissible in image by verifying the validity of consecutions in special finite Kripke structures of size square polynomial in reduced normal forms of the consecutions. As a consequence, this also solves the satisfiability problem for image. (shrink)

We investigate logical consequence in temporal logics in terms of logical consecutions. i.e., inference rules. First, we discuss the question: what does it mean for a logical consecution to be 'correct' in a propositional logic. We consider both valid and admissible consecutions in linear temporal logics and discuss the distinction between these two notions. The linear temporal logic LDTL, consisting of all formulas valid in the frame 〈L, ≤, ≥〉 of all integer numbers, is the prime object of our investigation. (...) We describe consecutions admissible LDTL in a semantic way—via consecutions valid in special temporal Kripke/Hintikka models. Then we state that any temporal inference rule has a reduced normal form which is given in terms of uniform formulas of temporal degree 1. Using these facts and enhanced semantic techniques we construct an algorithm, which recognizes consecutions admissible in LDTL. Also, we note that using the same technique it follows that the linear temporal logic L (N) of all natural numbers is also decidable w.r.t. inference rules. So, we prove that both logics LDTL and L (N) are decidable w.r.t. admissible consecutions. In particular, as a consequence, they both are decidable (Known fact), and the given deciding algorithms are explicit. (shrink)

We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.

Our paper aims to investigate inference rules for Nelson’s logics and to discuss possible ways to determine admissibility of inference rules in such logics. We will use the technique offered originally for intuitionistic logic and paraconsistent minimal Johannson’s logic. However, the adaptation is not an easy and evident task since Nelson’s logics do not enjoy replacement of equivalences rule. Therefore we consider and compare standard admissibility and weak admissibility. Our paper founds algorithms for recognizing weak admissibility and admissibility itself – (...) for restricted cases, to show the problems arising in the course of study. (shrink)

The paper studies the logic TL(NBox+-wC) – logic of discrete linear time with current time point clusters. Its language uses modalities Diamond+ (possible in future) and Diamond- (possible in past) and special temporal operations, – Box+w (weakly necessary in future) and Box-w (weakly necessary in past). We proceed by developing an algorithm recognizing theorems of TL(NBox+-wC), so we prove that TL(NBox+-wC) is decidable. The algorithm is based on reduction of formulas to inference rules and converting the rules in special reduced (...) normal form, and, then, on checking validity of such rules in models of singleexponential size in the rules. Also we consider the admissibility problem for TL(NBox+-wC) and show how to reduce the problem for admissibility to the decidability of TL(NBox+-wC) itself using definable universal modalities. (shrink)