Capturing k-ary existential second order logic with k-ary inclusion–exclusion logic

In this paper we analyze k-ary inclusion–exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX[k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO[k].We also introduce several useful operators that can be expressed in INEX[k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.     

The expressive power of k-ary exclusion logic

In this paper we study the expressive power of k-ary exclusion logic, EXC[k], that is obtained by extending first order logic with k-ary exclusion atoms. It is known that without arity bounds exclusion logic is equivalent with dependence logic. By observing the translations, we see that the expressive power of EXC[k] lies in between k-ary and ($k+1$)-ary dependence logics. We will show that, at least in the case when $k=1$, both of these inclusions are proper.In a recent work by the author it was shown that k-ary inclusion-exclusion logic is equivalent with k-ary existential second order logic, ESO[k]. We will show that, on the level of sentences, it is possible to simulate inclusion atoms with exclusion atoms, and in this way express ESO[k]-sentences by using only k-ary exclusion atoms. For this translation we also need to introduce a novel method for “unifying” the values of certain variables in a team. As a consequence, EXC[k] captures ESO[k] on the level of sentences, and we obtain a strict arity hierarchy for exclusion logic. It also follows that k-ary inclusion logic is strictly weaker than EXC[k].Finally we use similar techniques to formulate a translation from ESO[k] to k-ary inclusion logic with an alternative strict semantics. Consequently, for any arity fragment of inclusion logic, strict semantics is strictly more expressive than lax semantics.     

Axiomatizations of team logics

In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(～) of Väänänen's first-order team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion.As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-preserving elimination of modalities and quantifiers from formulas.     

Negation and partial axiomatizations of dependence and independence logic revisited

In this paper, we axiomatize the negatable consequences in dependence and independence logic by extending the systems of natural deduction of the logics given in [22] and [11]. We prove a characterization theorem for negatable formulas in independence logic and negatable sentences in dependence logic, and identify an interesting class of formulas that are negatable in independence logic. Dependence and independence atoms, first-order formulas belong to this class. We also demonstrate our extended system of independence logic by giving explicit derivations for Armstrong's Axioms and the Geiger-Paz-Pearl axioms of dependence and independence atoms.     

Incompleteness Results in Kripke Bundle Semantics

Kripke bundle and C-set semantics are known as semantics which generalize standard Kripke semantics. In [4] and in [1, 2] it is shown that Kripke bundle and C-set semantics are stronger than standard Kripke semantics. Also it is true that C-set semantics for superintuitionistic logics is stronger than Kripke bundle semantics ([6]). Modal predicate logic Q-S4.1 is not Kripke bundle complete ([3] - it is also yielded as a corollary to Theorem 6.1(a) of the present paper). This is shown by using difference of Kripke bundle semantics and C-set semantics. In this paper, by using the same idea we show that incompleteness results in Kripke bundle semantics which are extended versions of [2].     

QUANTIFIED MODAL LOGIC WITH NEIGHBORHOOD SEMANTICS

The paper presents a semantics for quantified modal logic which has a weaker axiomatization than the usual Kripke semantics. In particular, the Barcan Formula (BF) and its converse are not valid with the proposed semantics. Subclasses of models which validate BF and other interesting formulas are presented. A completeness theorem is proved, and the relation between this result and completeness with respect to Kripke models is investigated.     

Weakly Implicative (Fuzzy) Logics I: Basic Properties

This paper presents two classes of propositional logics (understood as a consequence relation). First we generalize the well-known class of implicative logics of Rasiowa and introduce the class of weakly implicative logics. This class is broad enough to contain many “usual” logics, yet easily manageable with nice logical properties. Then we introduce its subclass–the class of weakly implicative fuzzy logics. It contains the majority of logics studied in the literature under the name fuzzy logic. We present many general theorems for both classes, demonstrating their usefulness and importance.The work was supported by grant A100300503 of the Grant Agency of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AVOZ10300504.     

Sémantique de type Kripke d'un système logique basé sur un ensemble ordonné fini

In order to modelize the reasoning of intelligent agents represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems the use of a set of constants constitutes a fundamental tool. We have introduced in [8] a logic system called without this kind of constants but limited to the case that T is a finite poset. We have proved a completeness result for this system w.r.t. an algebraic semantics. We introduce in this paper a Kripke‐style semantics for a subsystem of for which there existes a deduction theorem. The set of “possible worldsr is enriched by a family of functions indexed by the elements of T and satisfying some conditions. We prove a completeness result for system with respect to this Kripke semantics and define a finite Kripke structure that characterizes the propositional fragment of logic . We introduce a reational semantics (found by E. Orlowska) which has the advantage to allow an interpretation of the propositionnal logic using only binary relations. We treat also the computational complexity of the satisfiability problem of the propositional fragment of logic .