Logical Constants as Punctuation Marks*

New computer systems of discovery create a research program for logic and philosophy of science. These systems consist of inference rules and control knowledge that guide the discovery process. Their paths of discovery are influenced by the available data and the discovery steps coincide with the justification of results. The discovery process can be described in terms of fundamental concepts of artificial intelligence such as heuristic search, and can also be interpreted in terms of logic. The traditional distinction that places studies of scientific discovery outside the philosophy of science, in psychology, sociology, or history, is no longer valid in view of the existence of computer systems of discovery. It becomes both reasonable and attractive to study the schemes of discovery in the same way as the criteria of justification were studied: empirically as facts, and logically as norms.

The aim of the tutorial is four-fold: 1 To present a very natural class of logical systems suitable for formalizing, generating and evaluating statements on dependences found in given data. The popular association rules form a particular, but by far not the only example. Technically, our logical systems are monadic observational predicate calculi, i.e. calculi with generalized quantifiers, only finite models and effective semantics. Be not shocked by these terms; no non-trivial knowledge of predicate logic will be assumed. Logic includes deductive properties, i.e. possibility to deduce truth of a sentence from other sentences already found true. Transparent deduction rules are very useful for systematic data mining. Special attention will be paid to sentences formalizing expressions of the form “many objects having a certain combination A of attributes have a also combination B” and, more generally, “combinations A,B of attributes are associated (dependent, correlated etc.) in a precisely defined manner”. 2 To show how suitable observational sentences are related to statistical hypothesis testing (this aspect appears sometimes unjustly neglected in data mining). A general pattern of statistical inference will be presented in logical terms (theoretical calculi and their relation to observational ones). In particular, statistical meaning of two variants of “associational rules” as well as of some “symmetric associations” will be explained. 3 To present short history of the GUHA method of automatic generation of hypotheses (General Unary Hypothesis Automaton). It is an original Czech method of exploratory data analysis, one of the oldest approaches to KDD or data mining. Its principle was formulated long before the advent of data mining. Its theoretical foundations (as presented in [5] and later publications) lead to the theory described in points (1), (2) above. 4 Finally, to show how modern fuzzy logic (in the narrow sense of the word, i.e. particular many-valued symbolic logic) may enter the domain of KDD and fruitfully generalize the field. This fourth point will concentrate to open problems and research directions. The tutorial will be complemented by demonstrations of two recent implementations of the GUHA method.

Introduction to Elementary LogicrnIntroduction to Elementary Logic, which serves as an introduction to the discipline of logic and explores both its traditional and modern aspects, consists of two main sections. The first section addresses the conceptual background of traditional logic, while the second section aims to acquaint readers with logic on both a conceptual and practical level through modern syntactic and semantic methods. The first section, titled “Traditional Logic”, provides readers with a solid conceptual foundation for the discipline of logic. This section compiles the Aristotelian system of logic and the unique contributions of the Islamic logical tradition. By focusing on the terms tasawwur and tasdiq, the section examines the fundamental elements that constitute conceptual and propositional knowledge, analyzing the building blocks of cognition. Thus, the section highlights the primary function of logic: to organize thought and ensure consistency, or in other words, to establish the logical basis for correct reasoning. Additionally, traditional (classical) topics in logic, such as the five universals, categories, and types of propositions, are enriched with examples to enhance the section’s content. The second section, titled “Propositional Logic”, complements the knowledge of traditional logic with modern approaches to propositional logic. The section begins with technical subjects such as the formalization of propositions, logical connectives, and well-formed formulas, continuing with a semantic presentation of propositional logic. This provides a comprehensive analysis of both the formal (syntactic) and semantic dimensions of modern logic. Within the section, logical connections between propositions are explained clearly and accessibly through examples using truth tables, analytic tableaux, and natural deduction methods. Introduction to Elementary Logic is the product of a concise yet comprehensive approach to the field of logic. It aims to serve as a fundamental reference for both newcomers to the discipline and those who wish to learn modern methods in the field of logic.rnKeywords: Traditional logic, propositional logic, conception, assent, inference

The article deals with the structural-semantic analysis of the linguistic terminology represented in J.-B. Rudnytsky’s scientific papers. The scientist’s approaches to term formation, ways of introducing the term into the text, interpretation of its meaning, description of situations of terminological instability, etc. have been outlined. The specific Ukrainian terms used by the linguist have been characterized on a semantic criterion. Among them are the names of sections of linguistics, linguistic phenomena, parts of speech and their types, grammatical categories and forms of their expression, types of names and word forms, names of different groups of words, affixes, and their varieties, types of sounds, graphic and punctuation marks. The structural features of the linguistic terms inherent in J.-B. Rudnytsky’s scientific idiolect (uniterms, composites, complex terms) has been defined. Terms with an adjective component are considered as a specific feature of the scientist’s idiolect. According to semantics, terms for denotation linguistic phenomena, types of language, grammatical forms, categories of tokens, sets, punctuation marks, etc. have been singled out among them. The influence of the Western Ukrainian version of the Ukrainian language on the style of Prof. J.-B. Rudnytsky’s scientific studies, in particular, linguistic terminology have been revealed. The most frequent linguistic terms of foreign origin from the researcher’s idiolexicon have been grouped on a structural criterion. The author outlines the word-forming models of term formation, characterizes the individual scientific style of J.-B. Rudnytsky, comparing them with modern trends in the development of the Ukrainian language and its terminology. The article analyzes the manifestation of terminological synonymy in the scientist’s linguistic explorations, identifies the most common terminological hyperonyms represented in his scientific texts on linguistics.

Modern formal logic, which is based on Kant’s logical project, interprets logical consequence as formal, which leads to substantive paradoxes that combine any thoughts at all and so to the loss of consequence as such. Beginning with A. Tarski, modern history of logic brings the problem of logical consequence into the realmof search for the relation of consequence, or grounding. In his doctoral dissertation on the nature of logical formality J. MacFarlane claims that the paradoxes of formal theories of logical consequence stem from the loss of grounding by the transcendental system of logic in the post­Kantian logical tradition. Arguably, analysis of logical terminology of consequence in Kant’s seminal works — Critique of Pure Reason, Critique of the Power of Judgment — in comparison with the terminology of earlier works, Prolegomena and lectures on logic attributed to him will clarify the question of the relation oflogical consequence in the formal and non­formal sense. The key concept of consequence in Kant’s terminology is Folgerung, which denotes ‘following’ in logical and non­logical contexts. I have also analysed related concepts: Folge, Abfolge, folglich etc., established dif- ferences between logical terms with similar meaning ‘inference’ (Schluss) and ‘conclusion’ (Konklusion). Finally, I make an attempt to formulate the problem of logical consequence in formal logic through the logical terms Schlussfolge, Folgerung and Konsequenz. On the strength of my analysis I propose to consider Kant’s consequence (Folgerung) to be a concept of transcendental logic that reflects the relation of consequence and grounds formal consequence.

We identify a pervasive contrast between implicit and explicit stances in logical analysis and system design. Implicit systems change received meanings of logical constants and sometimes also the notion of consequence, while explicit systems conservatively extend classical systems with new vocabulary. We illustrate the contrast for intuitionistic and epistemic logic, then take it further to information dynamics, default reasoning, and other areas, to show its wide scope. This gives a working understanding of the contrast, though we stop short of a formal definition, and acknowledge limitations and borderline cases. Throughout we show how awareness of the two stances suggests new logical systems and new issues about translations between implicit and explicit systems, linking up with foundational concerns about identity of logical systems. But we also show how a practical facility with these complementary working styles has philosophical consequences, as it throws doubt on strong philosophical claims made by just taking one design stance and ignoring alternative ones. We will illustrate the latter benefit for the case of logical pluralism and hyper-intensional semantics.

One of the central questions in 20th-century discussions within logic and philosophy of language is where logical constants, specifically propositional connectives, get their meaning from. What kind of determination/justification bond is found between the inference rules peculiar to a given connective and the meaning of that connective? Are these rules justified by it, or rather do they contribute to its construction? An observation made by G. Gentzen, who founded in the 1930s the proof-theoretical approach at large, triggered a view (called by some logical inferentialism) that gives a remarkable answer to the above question: the meaning of a logical constant in a logical language is provided, not by some sort of representational content, but by the inferential norms that govern its overall use. In 1960 A. N. Prior fictionalized as a counter-instance the connective tonk solely using a couple of inference rules, a connective capable of overthrowing the system of deduction; N. Belnap’s 1962 reply in the form of an analysis of the tonk problem opens the way to discussions in logic-cum-philosophy of language with important outcomes. The present little study can be read as some further deflation of the tonk problem with a relatively unconstrained inferentialistic view of the matter. The two main theses of the study are (i) that the problem posed by tonk-like connectives can be captured, more simply than in Belnap’s (otherwise correct) analysis, through inferential relations of a certain type which will be dubbed alethic relations; and (ii) that Prior’s challenge, brought to completion in whichever way, cannot give any result against the inferentialist conception.

In this paper, we propose a reconstruction of logic-based approaches to abductive reasoning in terms of ampliative adaptive logics. A main advantage of this reconstruction is that the resulting logics have a proof theory. As abductive reasoning is non-monotonic, the latter is necessarily dynamic (conclusions derived at some stage may at a later stage be rejected). The proof theory warrants, however, that the conclusions derived at a given stage are justified in view of the insight in the premises at that stage. Thus, it even leads to justified conclusions for undecidable fragments. Another advantage of the proposed logics is that they are much closer to natural reasoning than the existing systems. Usually, abduction is viewed as a form of “backward reasoning”. The search procedure by which this is realized (for instance, some form of linear linear resolution) is very different from the search procedures of human reasoners. The proposed logics treat abduction as a form of “forward reasoning” (Modus Ponens in the “wrong direction”). As a result, abductive steps are very natural, and are moreover nicely integrated with deductive steps. We present two new adaptive logics for abduction, and illustrate both with some examples from the history of the sciences (the discovery of Uranus and of Neptune). We also present some alternative systems that are better suited for non-creative forms of abductive reasoning.

The issue of what consequences to draw from the existence of non-classical logical systems has been the subject of an interesting debate across a diversity of fields. In this paper the matter of alternative logics is considered with reference to a specific belief system and its propositions :the Azande are said to maintain beliefs about witchcraft which, when expressed propositionally, appear to be inconsistent. When the Azande have been presented with such inconsistencies, they either fail to see them as such or else accept them as non-problematical. Is our knowledge of logical truths a relative and culturally determined phenomenon, or is there some (transcendent) criterion that allows us to adjudicate between alternative logical systems? The authors propose an approach for resolving disputes about the status of Azande reasoning which assumes a paraconsistent framework, thus providing a new perspective on this debate

A main goal of Formal Concept Analysis (FCA) from its very beginning has been the support of rational communication by formalizing and visualizing concepts. In the last years, this approach has been extended to traditional logic based on the doctrines of concepts, judgements and conclusions, leading to a framework called Contextual Logic. Much of the work on Contextual Logic has been inspired by the Existential Graphs invented by Charles S. Peirce at the end of the 19th century. While his graphical logic system is generally believed to be equivalent to first order logic, a proof in the strict mathematical sense cannot be given, as Peirce's description of Existential Graphs is vague and does not suit the requirements of contemporary mathematics. In his book 'A Peircean Reduction Thesis: The Foundations of topological Logic', Robert Burch presents the results of his project to reconstruct in an algebraic precise manner Peirce's logic system. The resulting system is called Peircean Algebraic Logic (PAL). He also provides a proof of the Peircean Reduction Thesis which states that all relations can be constructed from ternary relations in PAL, but not from unary and binary relations alone. Burch's proof relies on a major restriction on the allowed construction of graphs. Removing this restriction renders the proof much more complicated. In this paper, a new approach to represent an arbitrary graph by a relational normal form is introduced. This representation is then used to prove the thesis for infinite and two-element domains.

Logical Consequence in Modal Logic: Alternative Semantic Systems for Normal Modal Logics

This chapter deals with fuzzy logic in Zadeh’s ‘narrow sense’ [27], pointing to t-norm based truth functional logics, where the truth values model ‘degrees of truth’, identified with reals from the unit interval [0,1]. In particular, we are interested in Łukasiewicz logic Ł and some of its variants, but also in Godel logic G, and in Product logic P. The literature on formal deduction systems for these and related many-valued logics is vast and even more has been written about their algebraic background. We refer to the monographs [23], [22], and [10] for more information and references. Most authors take the usefulness of these logics in the context of approximate reasoning, i.e., reasoning with vague and imprecise notions for granted. However the corresponding proof systems are hardly ever explicitly related to models of correct reasoning with vague information. In other words: the challenge to derive particular fuzzy logics from first principles about approximate reasoning is not addressed explicitly. The reference to general models of reasoning and to theories of vagueness – a prolific discourse in contemporary analytic philosophy – is only implicit, if not simply missing, in most presentations of inference systems for fuzzy logics. Some notable exceptions, where an explicit semantic foundation for particular fuzzy logics is aimed at, are: Ruspini’s similarity semantics [35]; voting semantics [25]; ‘re-randomizing semantics’ [24]; measurement-theoretic justifications [7]; the Ulam-Renyi game based interpretation of D. Mundici [29]; and ‘acceptability semantics’ of J. Paris [31]. As we have argued elsewhere [12], these formal semantics for various t-norm based logics (in particular Łukasiewicz logic) should be placed in the wider discourse about adequate theories of vagueness, a prolific subfield of analytic philosophy.

In this article, the model of an inconsistent mind according to suggestions of Hilary Putnam and Alan Turing is presented from the perspective of the cognitive sciences and the evolutionary psychology. An attempt to reconcile the two versions of the modular model of mind by Jerry Fodor and Steven Pinker is undertaken followed by the discussion of the problem of evolutionary origin of mind. Next, the problem of the central module (interface) is considered which is supposed to integrate the individual and specialized modules of mind. The main thesis of this article states that the ‘global’ inconsistency of mind may result from the inconsistencies among ‘local’ computational modules of mind. Mind may be modeled as an inconsistent formal system which remains non-trivial. Consequently, it seems rational to postulate that the operation of mind is not based on the classical Aristotelian logic and is better described the systems of a paraconsistent logic. Best examples of such logical systems include the discussive logic by Stanisław Jaśkowski, the logic of formal inconsistency (LFI) by Newton da Costa and the many-valued logic by Jan Łukasiewicz and Graham Priest.

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a system of beliefsS (and seen to be inS), basedboth on some being inS and on some restraints not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.

The aim of this paper is to investigate a Curry-Howard interpretation of the intersection and union type inference system for Combinatory Logic. Types are interpreted as formulas of a Hilbert-style logic L, which turns out to be an extension of the intuitionistic logic with respect to provable disjunctive formulas (because of new equivalence relations on formulas), while the implicational-conjunctive fragment of L is still a fragment of intuitionistic logic. Moreover, typable terms are translated in a typed version, so that $\\vee$-$\\wedge$-typed combinatory logic terms are proved to completely codify the associated logical proofs.