Is there a zande logic?

The development of non-classical logics brought to light the question of their philosophical interpretation. Given a non-classical logic L, what is the informal/philosophical interpretation of its vocabulary? We find in the literature arguments defending that logical systems do not have a canonical philosophical interpretation, so the same logical system L has different philosophical interpretations. Although the thesis that a logic can be philosophically interpreted in different ways is well known, the thesis that a specific philosophical interpretation can be captured by different logics has not been widely explored. In this paper, we argue that if Kreisel’s informal rigor method is adequate to show that formal notions of logical consequence capture informal notions of logical consequence, then a specific philosophical interpretation only corresponds to a unique logic.

The subject of this research is the scientific paradoxes and such means for its resolution as nonclassical logics. The author defends a thesis that paradoxes often stimulate the scientific development. It is demonstrated that most vividly the problem of paradoxes manifested in crises in the fundamentals of mathematics; the attempts for its resolution in many ways contributes to the emergence of nonclassical logics. It is substantiated that nonclassical logics helped to resolve and explain the paraded occurring in scientific cognition. Comparative analysis is conducted on the capabilities of  three-valued “quantum logics” of Garrett Birkhoff and John von Neumann and “logics of complementarity” of Hans Reichenbach. Potential of the three-valued logics of D. Bochvar and nonclassical systems of A. Zinoviev in resolution and explanation of logical paradoxes, as well as importance of temporary logics of G. H. Wright for the philosophy of science is revealed. Special attention is paid  to the paraconsistent logics. The author determines two points of view in understanding of their essence and value for science and philosophy, which juxtaposition shows that none of them fully complies with the actual state of affairs. The main conclusion consists in the statement that paradoxes of scientific cognition should not be assessed just negatively; they also carry a positive meaning: detection of paradoxes in the theory testifies to the need for their elimination, more detailed research and stricter approach towards development of the theory, which in solution of this task can be accomplished by nonclassical logics.

The past few decades have seen a resurgence of reasoning techniques in artificial intelligence involving both classical and non-classical logics. In his paper, ``Multi-valued Logics: A Uniform Approach to Reasoning in Artificial Intelligence'', Ginsberg has shown that through the use of bilattices, several reasoning techniques can be unified under a single framework. A bilattice is a structure that can be viewed as a class of truth values that can accommodate incomplete and inconsistent information and in certain cases default information. In bilattice theory, knowledge is ordered along two dimensions: truth/falsity and certainty/uncertainty. By defining the corresponding bilattices as truth spaces, Ginsberg has shown that the same theorem prover can be used to simulate reasoning in first order logic, default logic, prioritized default logic and assumption truth maintenance system. Although this is a significant contribution, Ginsberg's paper was lengthy and involved. This paper summarizes some of the essential concepts and foundations of bilattice theory. Furthermore, it discusses the connections of bilattice theory and several other existing multi-valued logics such as the various three-valued logics and Belnap's four-valued logic. It is noted that the set of four truth values in Belnap's logic form a lattice structure that is isomorphic to the simplest bilattice. Subsequently, Fitting proposed a conflation operation that can be used to select sub-sets of truth values from this and other bilattices. This method of selecting sub-sets of truth values provides a means for identifying sub-logic in a bilattice.

<p>Recibido: 18/06/2016 • Aceptado: 02/02/2017</p><p>Luego de una breve revisión de la noción de inferencia diagramática, mostramos en qué sentido VENN provee un marco lógico capaz de modelar cierta forma de inferencia diagramática no-monotónica. Este resultado indica que, así como existen sistemas lógicos sentenciales clásicos y no-clásicos, pueden existir sistemas lógicos diagramáticos clásicos y no-clásicos.</p>

Non-classical logic and ontological non-commitment, avoiding abstract objects through modal operators

The construction of many-valued logical systems is commonly believed to have been one of the major achievements of the Warsaw School, and specifically of Łukasiewicz.1 The first mention of many-valued (three-valued) logic is to be found in Łukasiewicz’s farewell speech at Warsaw University on March 7,1918. He said: In 1910 I published a book on the principle of contradiction in Aristotle’s work, in which I strove to demonstrate that that principle is not so evident as it is believed to be. Even then I strove to construct non-Aristotelian logic, but in vain. Now I believe I have succeeded in this. ... Examination of that issue was the subject-matter of my last lectures. I have proved that in addition to true and false propositions there are possible propositions, to which objective possibility corresponds as a third value in addition to being and non-being. This gave rise to a system of three-valued logic, which I worked out in detail last summer. That system is as coherent and self-consistent as Aristotle’s logic, and is much richer in laws and formulae. That new logic, by introducing the concept of objective possibility, destroyed the former concept of science, based on necessity. Possible phenomena have no cause, although they themselves can be the beginning of a causal sequence. An act of a creative individual can be free and at the same time affect the course of the world (1918, p. 86).

Substitution Frege and extended Frege proof systems in non-classical logics

We study some structural and numerical properties of varieties for determinative disjunctive normal forms, introduced here for given propositional formula. We consider for classical and non-classical propositional logics some proof systems, which are constructed on the base of determinative disjunctive normal forms. We investigate also the relation between the proof complexities in some well-known classical and non-classical proof systems (Resolution, Cut-free sequent, Gentzen refutation, Cutting planes etc.) and numerical properties of varieties for determinative disjunctive normal forms for classical and non-classical tautologies.

Ever since its inception 100 years back, multiple choice items have been widely used as a method of assessment. It has certain inherent limitations such as inability to test higher cognitive skills, element of guesswork while answering, and issues related with marking schemes. Various marking schemes have been proposed in the past but they are not balanced, skewed, and complex, which are based on mathematical calculations which are typically not within the grasp of medical personnel. Type X questions has many advantages being easy to construct, can test multiple concepts/application/facets of a topic, cognitive skill of various level of hierarchy can be tested, and unlike Type K items, they are free from complicated coding. In spite of these advantages, they are not in common use due to complicated marking schemes. This is the reason we explored the aspects of methods of evaluation of multiple correct options multiple choice questions and came up with the simple, practically applicable, nonstringent but logical scoring system for the same. The rationale of the illustrated marking scheme is that it takes into consideration the distracter recognition ability of the examinee rather than relying on the ability only to select the correct response. Thus, examinee's true knowledge is tested, and he is rewarded accordingly for selecting a correct answer and omitting a distracter. The scheme also penalizes for not recognizing a distracter thus controlling guessing behavior. It is emphasized that if the illustrated scoring scheme is adopted, then Type X questions would come in common practice.

Picture fuzzy sets, introduced by Cuong in 2014, extend intuitionistic fuzzy sets by assigning three independent degrees to each element-positive membership (μ), neutral membership (η), and negative membership (ν), together with an explicit refusal degree, thereby enabling a richer representation of uncertainty involving neutrality and abstention. In this paper, picture fuzzy structures are introduced into Sheffer stroke UP-algebras (SUP-algebras), an algebraic framework based on the functionally complete Sheffer stroke (NAND) operation. Picture fuzzy SUP-subalgebras and picture fuzzy SUP-ideals are defined, and several equivalent characterizations are established using level sets, complement functions, and characteristic picture fuzzy sets associated with crisp subalgebras and ideals. It is shown that every picture fuzzy SUP-ideal is a picture fuzzy SUP-subalgebra, while the converse does not hold in general, and that the family of all picture fuzzy SUP-subalgebras forms a complete distributive lattice. Preservation properties under direct products and surjective homomorphisms are also investigated. These results enrich the theory of SUP-algebras by incorporating neutrality andrefusal into their algebraic structure, thereby enhancing their applicability to uncertainty modeling, non-classical logical systems, and decision-making frameworks.

Initially proposed as rivals of classical logic, alternative logics have become increasingly important in areas such as computer science and arti ficial intelligence. Fuzzy logic, in particular, has motivated major technological developments in recent years. Susan Haack's Deviant Logic provided an extended examination of the philosophical consequences of alternative logics. In this volume, Haack includes the complete text of Deviant Logic, as well as five additional papers that expand and update it. Two of these essays critique fuzzy logic, while three augment Deviant Logic's treatment of deduction and logical truth. Haack also provides an extensive new foreword, brief introductions to the new essays, and an updated bibliography of current work in these areas.

This chapter explains in detail various methods to investigate which of several alternative logics and axioms for mathematics correspond most closely to the common and scientific patterns of reasoning. Some of these logics have sufficient power, while other logical systems lack the power, to decide whether certain logical formulae are theorems, and hence whether they are allowed (within such logics) in common or scientific arguments. Examples demonstrate that all the proofs within the implicational calculus (with implications, but without negation) do not suffice to prove certain implicational tautologies. In other words, the implicational calculus lacks the power to decide the status of certain of its own implicational formulae. The same examples also demonstrate how a computing language based on axioms and rules might fail to generate all sentences in the language.KeywordsClassical LogicPropositional CalculusTruth TableIntuitionistic LogicDouble NegationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter presents a proof-theoretical approach to the question ‘What is a logical constant?’ The approach starts with the assumption that logic is the science of formal deductions, and that basic formal deductions are structural deductions, i.e. deductions independent of any constant of the language to which the premises and conclusions belong. Logical constants, on which the remaining formal deductions are dependent, may be said to serve as ‘punctuation marks’ for some structural features of deductions. This punctuation function, exhibited in equivalences which amount to analyses of logical constants, is taken as a criterion for being a logical constant. The chapter presents an account of philosophical analysis which covers the proposed analyses of logical constants. Some related assumptions concerning logic are also considered. In particular, since a logical system is completely determined by its structural deductions, alternative logical systems arise by changing structural deductions while having constants with the same punctuation function. Some other approaches to the question ‘What is a logical constant?’, grammatical, model-theoretical, and proof- theoretical, are briefly considered.

In this paper we discuss the problem of termination of the deductive process in resolution proof systems for non-classical logics.

This chapter discusses the application of the ecclesiastic method to the study of causality. It argues that the ecclesiastic tradition coexists with empirical and empathic approaches to causal reasoning and that it provides a third alternative logic or method of causal reasoning. The term “religion” is generally used to refer to more formal, organized, and well-established belief systems that are shared by large groups of people. The phrase “spiritual beliefs” refers to more amorphous and usually less established systems of belief. These distinctions are important, but this chapter highlights their shared aspects. It also considers the features shared by ecclesiastic systems with the empiric and empathic methods, along with ecclesiastic reasoning as a distinct logic. The chapter concludes by analyzing causality in Hinduism and the Abrahamic religions.