Abstract. In Béziau (Log Log Philos 15:99-111, 2006) a logic Z was defined with the help of the modal logic S5. In it, the negation operator is understood as meaning 'it is not necessary that'. The strong soundnesscompleteness result for Z with respect to a version of Kripke semantics was also given there. Following the formulation of Z we can talk about Z-like logics or Beziau-style logics if we consider other modal logics instead of S5-such a possibility has been mentioned in [1]. The correspondence result between modal logics and respective Beziau-style logics has been generalised for the case of normal logics naturally leading to soundness-completeness results [see Marcos :279-300, 2005) and Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 34(4): [229][230][231][232][233][234][235][236][237][238][239][240][241][242][243][244][245][246][247][248] 2005)]. In Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 37(3-4): [185][186][187][188][189][190][191][192][193][194][195][196] 2008), (Bull Sect Log 38(3-4):189-203, 2009) some partial results for non-normal cases are given. In the present paper we try to give similar but more general correspondence results for the non-normal-worlds case. To achieve this aim we have to enrich original Beziau's language with an additional negation operator understood as 'it is necessary that not'.
In Jaśkowski’s model of discussion, discussive connectives represent certain interactions that can hold between debaters. However, it is not possible within the model for participants to use explicit modal operators. In the paper we present a modal extension of the discussive logic $\textbf{D}_{\textbf{2}}$ that formally corresponds to an extended version of Jaśkowski’s model of discussion that permits such a use. This logic is denoted by $\textbf{m}\textbf{D}_{\textbf{2}}$. We present philosophical motivations for the formulation of this logic. We also give syntactic characterizations of the logic and propose a comparison with certain other modal systems. In particular, we prove that $\textbf{m}\textbf{D}_{\textbf{2}}$ is neither normal nor regular. On the basis of the axiomatization of $\textbf{D}_{\textbf{2}}$, we give an axiomatization of $\textbf{m}\textbf{D}_{\textbf{2}}$. We also give another axiomatization which is not based on the axiomatization of $\textbf{D}_{\textbf{2}}$. Furthermore, we give a natural Kripke-style semantics for $\textbf{m}\textbf{D}_{\textbf{2}}$ and prove the respective adequacy theorems.
In the present paper we give syntactical and semantical characterization of the class of algebras defined by P-compatible identities of modular ortholattices. We also describe the lattice of some subvarieties of the variety MOL Ex defined by so called externally compatible identities of modular ortholattices.
Abstract. The equational theories were studied in many works (see [4], [5], [6], [7]). Let r be a type of Abelian groups. In this paper we consider the extentions of the equational theory Ex(Q n ) defined by so called externally compatible identities of Abelian groups and the identity x n « y n . The equational base of this theory was found in [3]. We prove that each equational theory Cn(Ex(Q n ) U {φ « Φ}), where φ « φ is an identity of type r, is equal to the extension of the equational theory Cn(Ex(Q n ) U E), where E is a finite set of one variable identities of type r.The notation in this paper are the same as in [1]. PreliminariesLet r : {·, -1 }->N be a type of Abelian groups where τ(·) = 2, r( _1 ) = l. By Q n we denote the class of all Abelian groups satisfying the identityThe identity of type τ is externally compatible (see [2]) if it is one of the form χ « χ or of the form φχ · >2 w φι · Φ2, Φΐ is the equational theory. Let Id(r) be a set of all identities of type r. By Cn(E), where Σ Ç Id(r), we denote the deductive closure of Σ.It is well known fact, that the lattice of all equational theories extendingis dually isomorphic to the lattice of all natural divisors of η with divisibility relation. It implies that Cn{Id{G n ) The algorithm presented above neglects the structure of identities, and that is why it is useless in the case of extensions of the theory Ex(Q n ).Using the Galois connection between algebras and identities we have that the lattice of all equational theories of type τ is dually isomorphicto the lattice of all varieties of the same type. So, if we know all theories, where φ and ψ are terms of type r, we can describe all subvarieties of the variety defined by all externally compatible identities of the variety Q n . The extension of the theory Ex(Q n )In this paper, as in [3], by x° we denote χ · χ -1 . Let us consider the following identities: Sets of identities satisfied in Abelian groups 449Let us consider the identity (1). The following lemma is obvious. Proof. Without losing generality we can assume that j = 1. Let S\ = Cn(Ex(G n )U{ (2) ,k s ).Putting Xj = x\' for j G {2,..., s} in the identity (1) we get, that (xi « Xi · g Si and thus (xi · x? « Xl · x? · € S u so we have (χχ « ιξ · xf +1 ) G Si. Finally, we have S 2 Ç Si. To prove the opposite inclusion let us note, that from the condition (χι « x? · xf +1 ) G S 2 it follows that (x° ~ x? · xf) G S 2 . The immediate consequence of these conditions is (χι « χξ • χχ) G S 2 . The definition of d implying that for each j from the set {2,..., s} a number d is a divisor of kj. Hence there exist elementsp 2 ,...,p s in the set Z n such that kj = pj • d. As a result of the condition (x° « x° · xf ) G S 2 we have that for each je {2,..., s} the identity x° « x° · x^y d belongs to S 2 . From the fact that where d = (ki,..., kj-1, kj+i,..., ks). (fci,..., kj-1, kj -1, kj+i,..., ks).Proof. The proof of this lemma is analogously to the proof of the Lemma 2.• Let we study the identity (4).
In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the smallest one, this logic can be treated as a basis which could be extended to richer discussive logics that are obtained by varying accessibility relation and resulting in a lattice of discussive logics. One has to remember that while formulating discussive logics there is no one-to-one determination of discussive logics by modal logics. For example, it is proved that Jaśkowski’s logic D 2 can be expressed by other than S 5 modal logics. In this paper we consider a deductive system for the sketchily described minimal logic. While formulating the deductive system, we apply a method of Kotas that was used to axiomatize D 2 . The obtained system determines a logic D 0 as a set of theses that is contained in D 2 . Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain D 0 , so it is the smallest discussive logic.
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