This paper presents Rasiowa–Sikorski deduction systems for logics \, \, \ and \. For each of the logics two systems are developed: an R–S system that can be supplemented with admissible cut rule, and a \-version of R–S system in which the non-admissible rule of cut is the only branching rule. The systems are presented in a Smullyan-like uniform notation, extended and adjusted to the aims of this paper. Completeness is proved by the use of (...) abstract refutability properties which are dual to consistency properties used by Fitting. Also the notion of admissibility of a rule in an R–S-system is analysed. (shrink)

The non-Fregean logic SCI is obtained from the classical sentential calculus by adding a new identity connective = and axioms which say ?a = ß' means ?a is identical to ß'. We present complete and sound proof system for SCI in the style of Rasiowa-Sikorski. It provides a natural deduction-style method of reasoning for the non-Fregean sentential logic SCI.

This paper presents a monotonic system of Post algebras of order +* whose chain of Post constans is isomorphic with 012 ... -3-2-1. Besides monotonic operations, other unary operations are considered; namely, disjoint operations, the quasi-complement, succesor, and predecessor operations. The successor and predecessor operations are basic for number theory.

The Ramsey Test is considered to be the default test for the acceptability of indicative conditionals. I will argue that it is incompatible with some of the recent developments in conceptualizing conditionals, namely the growing empirical evidence for the _Relevance Hypothesis_. According to the hypothesis, one of the necessary conditions of acceptability for an indicative conditional is its antecedent being positively probabilistically relevant for the consequent. The source of the idea is _Evidential Support Theory_ presented in Douven (2008). I will (...) defend the hypothesis against alleged counterexamples, and show that it is supported by growing empirical evidence. Finally, I will present a version of the Ramsey test which incorporates the relevance condition and therefore is consistent with growing empirical evidence for the relevance hypothesis. (shrink)

Extended algorithmic logic (EAL) as introduced in [18] is a modified version of extended +-valued algorithmic logic. Only two-valued predicates and two-valued propositional variables occur in EAL. The role of the +-valued logic is restricted to construct control systems (stacks) of pushdown algorithms whereas their actions are described by means of the two-valued logic. Thus EAL formalizes a programming theory with recursive procedures but without the instruction CASE.The aim of this paper is to discuss EAL and prove the completeness (...) theorem. A complete formalization of EAL was announced in [20] but no proof of the completeness theorem was given. (shrink)

An algebraic approach to programs called recursive coroutines — due to Janicki [3] — is based on the idea to consider certain complex algorithms as algebraics models of those programs. Complex algorithms are generalizations of pushdown algorithms being algebraic models of recursive procedures (see Mazurkiewicz [4]). LCA — logic of complex algorithms — was formulated in [11]. It formalizes algorithmic properties of a class of deterministic programs called here complex recursive ones or interacting stacks-programs, for which complex algorithms constitute mathematical (...) models. LCA is in a sense an extension of algorithmic logic as initiated by Salwicki [14] and of extended algorithmic logic EAL as formulated and examined by the present author in [8], [9], [10]. In LCA — similarly as in EAL-ω + -valued logic is applied as a tool to construct control systems (stacks) occurring in corresponding algorithms. The aim of this paper is to give a complete axiomatization. of LCA and to prove a completeness theorem. (shrink)

We present relational proof systems for the four groups of theories of spatial reasoning: contact relation algebras, Boolean algebras with a contact relation, lattice-based spatial theories, spatial theories based on a proximity relation.

The main goal of the paper is to suggest some analytic proof systems for LC and its finite-valued counterparts which are suitable for proof-search. This goal is achieved through following the general Rasiowa-Sikorski methodology for constructing analytic proof systems for semantically-defined logics. All the systems presented here are terminating, contraction-free, and based on invertible rules, which have a local character and at most two premises.

The paper discusses a four-valued propositional logic FOUR≤, similar to Belnap's logic, which can be used to describe incomplete or inconsistent knowledge. In addition to the two classical logical values tt, ff, FOUR≤ features also two nonclassical values: ⊥, representing incomplete information, and ⊤, representing inconsistency. The nonclassical values are incomparable, and together with the classical ones they form a diamond-shaped lattice L4 known from Belnap's logic, which underlies the semantics of FOUR≤. The set of connectives contains those of Belnap's (...) logic, together with an additional operator representing the lattice order, which decisively increases the expressive power of the logic. In the paper we develop a complete decompositional deduction system in Rasiowa-Sikorski style for FOUR≤. (shrink)

We describe a relational framework that uniformly supports formalization and automated reasoning in varied propositional modal logics. The proof system we propose is a relational variant of the classical Rasiowa-Sikorski proof system. We introduce a compact graph-based representation of formulae and proofs supporting an efficient implementation of the basic inference engine, as well as of a number of refinements. Completeness and soundness results are shown and a Prolog implementation is described.

We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.

Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa and Sikorski (1963) for relation algebras generated by a contact relation.

The paper is concerned with the old conjecture that there are infinitely many twin primes. In the paper we show that this conjecture is true, that is, it is true in the standard model of arithmetic. The proof is based on Rasiowa–Sikorski Lemma. The key role are played by the derived notion of a Rasiowa–Sikorski set and the method of forcing adjusted to arbitrary first–order languages. This approach was developed in the papers Czelakowski [ 4, 5 (...) ]. The central idea consists in constructing an appropriate countable model \(\textbf{A}\) of Peano arithmetic by means of a Rasiowa–Sikorski set. This model validates the twin prime conjecture. Since \(\textbf{A}\) is elementarily equivalent to the standard model, the conjecture follows. Thus the standard model validates the twin primes conjecture. More generally, it is shown that de Polignac’s conjecture has a positive solution. The paper employs methods borrowed from the contemporary mathematical logic. Such a ’logical’ approach may be viewed as a useful addition to the dominant methodology in number theory based on mathematical analysis. (shrink)

The paper is concerned with the old conjecture that there are infinitely many Mersenne primes. It is shown in the work that this conjecture is true in the standard model of arithmetic. The proof refers to the general approach to first–order logic based on Rasiowa-Sikorski Lemma and the derived notion of a Rasiowa–Sikorski set. This approach was developed in the papers [ 2 – 4 ]. This work is a companion piece to [ 4 ].

This paper aims at formulating a condition neccssary and sufficient for the existing of a prime filter preserving enumerable infinite joins and meets in a distributive lattice.

In order to modelize the reasoning of an intelligent agent represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this papers, we consider logic systems called L′T without this kind of constants but limited to the case where T is a finite poset. We prove a weak deduction theorem. We introduce also an algebraic semantics using Hey ting algebra with operators. (...) To prove the completeness theorem of the L′T system with respect to the algebraic semantics, we use the method of H. Rasiowa and R. Sikorski for first order logic. In the propositional case, a corollary allows us to assert that it is decidable to know “if a propositional formula is valid”. We study also certain relations between the L′T logic and the intuitionistic and classical logics. (shrink)

We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that (...) the two systems are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other. (shrink)

Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations (...) between intervals and/or various time orderings. (shrink)

We present several classes of logics for reasoning with information stored in information systems. The logics enable us to cope with the phenomena of incompleteness of information and uncertainty of knowledge derived from such an information. Relational inference systems for these logics are developed in the style of dual tableaux.

Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a b a for all b, then a a* a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style representation theorem for (...) complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic. (shrink)

Non-deterministic matrices are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one . We use the Rasiowa-Sikorski decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such structures with either (...) of the above types of semantics. Later we demonstrate how these systems can be converted into cut-free ordinary Gentzen calculi which are also sound and complete for the corresponding non-deterministic semantics. As a by-product, we get new semantic characterizations for some well-known logics. (shrink)

ABSTRACT Two objects o1, o2 of an information system are said to be complementary with respect to attribute a if α(o1) = -α(o2), where α(o) is the set of values of attribute a assigned to o. They are said to be complementary with respect to a set of attributes A if they are complementary with respect to each attribute α ε A. A multi-modal logical language for reasoning about complementarity relations is presented, with modalities [A] and ?A? parameterised by subsets (...) of a given set ATTR of attributes. Two complete deduction system for the language are presented: a Rasiowa-Sikorski style system using signed formulae, and an equivalent sequent calculus system in Gentzen style complete for theories. (shrink)

The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Čech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the "Henkin method" of proving deductive completeness of logical systems. The Rasiowa-Sikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces.

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and its dual (...) tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object. (shrink)

We prove an extension of the Lemma of Rasiowa and Sikorski and give some applications. Moreover, we analyze the relationship to corresponding results on the omission of types.

We show that there is a purely proof-theoretic proof of the Rasiowa–Harrop disjunction property for the full intuitionistic propositional calculus ), via natural deduction, in which commuting conversions are not needed. Such proof is based on a sound and faithful embedding of \ into an atomic polymorphic system. This result strengthens a homologous result for the disjunction property of \ and answers a question then posed by Pierluigi Minari.

Children’s movies and animated cartoons today increasingly include homosexual characters, which can be welcomed from an equal-rights perspective. Yet, an intensive public debate has been initiated regarding the (age) appropriateness of such depictions. So far, it is unclear how heterosexual adults react to the presence of gay characters in children’s animated cartoons. Drawing from social identity theory, we conducted an experiment in Germany. Using the Powtoon animation software, we created two versions of a trailer of a fictitious animated cartoon based (...) on two almost identical storylines – one involving a heterosexual couple and the other a lesbian couple. Participants were exposed to only one version. Results of the experiment revealed that adults gave higher age ratings to the version that featured the lesbian couple. Yet, this effect was moderated, with only center- to right-leaning persons being affected. No effects were detected for left-leaning individuals. An exploratory analysis further revealed a moderation effect for individuals who believed in protection myths (i. e., protecting children from exposure to homosexuality), resulting in them giving higher age ratings compared to people who did not believe in protection myths. Implications are discussed, and age-rating measures are proposed for examining reactions to gay characters in communication research. (shrink)

Antisemitism is on the rise. Recently, discussions have considered so-called “no-go zones for Jews” (city areas Jews should avoid to reduce the likelihood of being attacked). In this context and drawing from attribution theory, we examined if news consumers perceive a Jewish hate crime victim as partly responsible for being attacked when news coverage explicitly emphasizes that the victim displayed religious symbols (kippah) in a certain inner-city location. We conducted a quota-based survey experiment (N = 392) in Germany (4 groups, (...) between-subjects design) and randomly exposed participants to news coverage about an anti-Semitic attack. The article either emphasized that the Jewish victim displayed religious symbols (kippah) or not, and highlighted the specific location of the attack (Berlin synagogue vs. deprived inner-city district of Berlin). Moderated mediation analysis suggested that participants perceived the victim’s behavior to be more provocative when the news article highlighted that the victim displayed religious symbols and when the attack occurred in a deprived Berlin district. Yet, effects were only detected for individuals with low (vs. high) levels of education. Perceived provocativeness in turn increased victim blaming indicating that some individuals indirectly regarded the Jewish victim to be partly responsible for being attacked. (shrink)

Provability, Computability and Reflection.

In the last decade, many problematic cases of scientific conduct have been diagnosed; some of which involve outright fraud others are more subtle. These and similar problems can be interpreted as caused by lack of scientific objectivity. The current philosophical theories of objectivity do not provide scientists with conceptualizations that can be effectively put into practice in remedying these issues. We propose a novel way of thinking about objectivity for individual scientists; a negative and dynamic approach.We provide a philosophical conceptualization (...) of objectivity that is informed by empirical research. In particular, it is our intention to take the first steps in providing an empirically and methodologically informed inventory of factors that impair the scientific practice. The inventory will be compiled into a negative conceptualization, which could in principle be used by individual scientists to assess objectivity of scientific practice. We propose a preliminary outline of a usable and testable instrument for indicating the objectivity of scientific practice. (shrink)