Axiomatic proof/refutation systems for the paraconsistent modal logics: KN4 and KN4.D are presented. The completeness proofs boil down to showing that every sequent is either provable or refutable. By constructing finite tree-type countermodels from refutations, the refined characterizations of these logics by classes of finite tree-type frames are established. The axiom systems also provide decision procedures for these logics.

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction method (...) and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one. (shrink)

Sufism has two main dimensions: vertical (God’s pleasure) and horizontal (harmony with nature, society and local wisdom). In reality, many Sufis are considered less concerned about the balancing between vertical and horizontal dimensions. The research explores the concepts and practices of hybrid Sufism undertaken by Kyais (religious leaders) and their followers in improving quality of life. Ethnography was used for exploring the mindset and activities of Kyai and his followers. This study involved four Kyais in Java and Kalimantan, Indonesia. Research (...) data were obtained through participant observations, in-depth interviews and documentation. The data were analysed by Spadley’s ethnographic steps as follows: (1) domain analysis, (2) taxonomy analysis, (3) componential analysis and (4) cultural-thematic analysis. The results showed that hybrid Sufism could improve quality of life. Hybrid Sufism can better appreciate and interpret local wisdom with an attitude of preserving nature and a positive understanding of worldly wealth, increasing the hard work ethos to achieve material–spiritual qualities. Contribution: This article shows that hybrid Sufism encourages the life of Sufis in harmony between vertical and horizontal aspects. This understanding and lifestyle give rise to respect for others, being friendly to the environment and interpreting life and local wisdom. (shrink)

I introduce a novel hybrid calculus H for the validities and invalidities of classical propositional logic. The calculus H is different in nature from other hybrid calculi that can be found in the literature in that it does not include specific anti-sequent rules. Instead, I add to the sequent rules of classical propositional logic only two structural rules that allow us to introduce and eliminate anti-sequents in our derivations. The resultant system is much simpler than the existing systems in the (...) literature. For one thing, it does not suffer from a proliferation of rules. For another, it is structurally complete: its admissible rules are all derivable. (shrink)

A unified and modular falsification-aware single-succedent Gentzen-style framework is introduced for classical, paradefinite, paraconsistent, and paracomplete logics. This framework is composed of two special inference rules, referred to as the rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Similar to the cut rule in Gentzen’s LK for classical logic, these rules are admissible in cut-free LK. A falsification-aware single-succedent Gentzen-style sequent calculus fsCL for classical logic is formalized based (...) on the proposed framework. The calculus fsCL is obtained from the existing falsification-aware single-succedent Gentzen-style sequent calculus GN4 for Nelson’s paradefinite (or paraconsistent) four-valued logic N4 by adding the rules of explosion and excluded middle. A falsification-aware single-succedent Gentzen-style sequent calculus GN3 for Nelson’s paracomplete three-valued logic N3 is also obtained from GN4 by adding the rule of explosion. The cut-elimination theorems for fsCL, GN3, and some of their neighbors as well as the Glivenko theorem for fsCL are proved. (shrink)

Sufism has two main dimensions: vertical (God’s pleasure) and horizontal (harmony with nature, society and local wisdom). In reality, many Sufis are considered less concerned about the balancing between vertical and horizontal dimensions. The research explores the concepts and practices of hybrid Sufism undertaken by Kyais (religious leaders) and their followers in improving quality of life. Ethnography was used for exploring the mindset and activities of Kyai and his followers. This study involved four Kyais in Java and Kalimantan, Indonesia. Research (...) data were obtained through participant observations, in-depth interviews and documentation. The data were analysed by Spadley’s ethnographic steps as follows: (1) domain analysis, (2) taxonomy analysis, (3) componential analysis and (4) cultural-thematic analysis. The results showed that hybrid Sufism could improve quality of life. Hybrid Sufism can better appreciate and interpret local wisdom with an attitude of preserving nature and a positive understanding of worldly wealth, increasing the hard work ethos to achieve material–spiritual qualities. Contribution: This article shows that hybrid Sufism encourages the life of Sufis in harmony between vertical and horizontal aspects. This understanding and lifestyle give rise to respect for others, being friendly to the environment and interpreting life and local wisdom. (shrink)