In this paper we introduce non-normal modal extensions of the sub-classical logics CLoN, CluN and CLaN, in the same way that S0.5 0 extends classical logic. The first modal system is both paraconsistent and paracomplete, while the second one is paraconsistent and the third is paracomplete. Despite being non-normal, these systems are sound and complete for a suitable Kripke semantics. We also show that these systems are appropriate for interpreting □ as “is provable in (...) class='Hi'>classical logic”. This allows us to recover the theorems of propositional classical logic within three sub-classical modal systems. (shrink)

The application of paraconsistent logics to theological contradictions is a fascinating move. Jc Beall’s (J Anal Theol, 7(1): 400–439, 2019) paper entitled ‘Christ—A Contradiction: A Defense of ‘Contradictory Christology’ is a notable example. Beall proposes a solution to the fundamental problem of Christology. His solution aims at making the case, and defending the viability of, what he has termed, ‘Contradictory Christology’. There are at least two essential components of Beall’s ‘Contradictory Christology’. These include the dogmatic statements of Chalcedon and FDE (...) logic. The first is the theological contradiction in question. The second is the type of paraconsistent logic. Both components are integral to a contradictory theology in general. I argue that there can be no such thing as an Islamic contradictory theology. I make the case by establishing two points. These points correspond to both integral components of a contradictory theology in general. The first is that an Islamic theological contradiction does not entail an actual (logical) contradiction. The second is that FDE logic, including alternative sub-classical systems of logic, are not adequate in tolerating an Islamic theological contradiction. (shrink)

The paper studies first order extensions of classical systems of modal logic (see (Chellas, 1980, part III)). We focus on the role of the Barcan formulas. It is shown that these formulas correspond to fundamental properties of neighborhood frames. The results have interesting applications in epistemic logic. In particular we suggest that the proposed models can be used in order to study monadic operators of probability (Kyburg, 1990) and likelihood (Halpern-Rabin, 1987).

The article focuses on representing different forms of non-adjunctive inference as sub-Kripkean systems of classical modal logic, where the inference from □A and □B to □A ∧ B fails. In particular we prove a completeness result showing that the modal system that Schotch and Jennings derive from a form of non-adjunctive inference in (Schotch and Jennings, 1980) is a classical system strictly stronger than EMN and weaker than K (following the notation for classical modalities presented (...) in Chellas, 1980). The unified semantical characterization in terms of neighborhoods permits comparisons between different forms of non-adjunctive inference. For example, we show that the non-adjunctive logic proposed in (Schotch and Jennings, 1980) is not adequate in general for representing the logic of high probability operators. An alternative interpretation of the forcing relation of Schotch and Jennings is derived from the proposed unified semantics and utilized in order to propose a more fine-grained measure of epistemic coherence than the one presented in (Schotch and Jennings, 1980). Finally we propose a syntactic translation of the purely implicative part of Jaśkowski's system D₂ into a classical system preserving all the theorems (and non-theorems) explicilty mentioned in (Jaśkowski, 1969). The translation method can be used in order to develop epistemic semantics for a larger class of non-adjunctive (discursive) logics than the ones historically investigated by Jaśkowski. (shrink)

The application of paraconsistent logics to theological contradictions is a fascinating move. Jc Beall’s (J Anal Theol, 7(1): 400–439, 2019) paper entitled ‘Christ—A Contradiction: A Defense of ‘Contradictory Christology’ is a notable example. Beall proposes a solution to the fundamental problem of Christology. His solution aims at making the case, and defending the viability of, what he has termed, ‘Contradictory Christology’. There are at least two essential components of Beall’s ‘Contradictory Christology’. These include the dogmatic statements of Chalcedon and FDE (...) logic. The first is the theological contradiction in question. The second is the type of paraconsistent logic. Both components are integral to a contradictory theology in general. I argue that there can be no such thing as an Islamic contradictory theology. I make the case by establishing two points. These points correspond to both integral components of a contradictory theology in general. The first is that an Islamic theological contradiction does not entail an actual (logical) contradiction. The second is that FDE logic, including alternative sub-classical systems of logic, are not adequate in tolerating an Islamic theological contradiction. (shrink)

A rapidly growing body of scholarship shows that we can gain new insights into theories and policies by understanding and increasing their systemic structure. This paper will present an overview of this expanding field and discuss how concepts of structure are being applied in a variety of contexts to support collaboration, decision making, learning, prediction, and results. Next, it will delve into the underlying structures of logic that may be found within those theories and policies. Here, we will go (...) beyond Toulmin’s logics of claim and proof that have not proven useful for advancing the social sciences and focus on five structures of “causal logic.” The results suggest a useful and more comprehensive approach to developing deeper understanding of our conceptual systems such as theory and policy. (shrink)

This is an advanced study of systems of propositional logic which offers a comprehensive account of a wide variety of logical systems and which encourages students to take a critical stance towards the subject. A great variety of systems and subsystems are defined and compared as regards their deductive power and relation to their model theory. Interesting features include a more refined treatment of modal logic and the special attention given to the weakenings of (...) class='Hi'>classical logic. Useful appendices provide a topical bibliography and review of basic set theory. (shrink)

Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives (...) that may be used for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics. (shrink)

Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually (...) corollaries of theorems with much wider scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction. (shrink)

Where is logic heading today? There is a general feeling that the discipline is broadening its scope and agenda beyond classical foundational issues, and maybe even a concern that, like Stephen Leacock’s famous horseman, it is ‘riding off madly in all directions’. So, what is the resultant vector? There seem to be two broad answers in circulation today. One is logical pluralism, locating the new scope of logic in charting a wide variety of reasoning styles, often marked (...) by non-classical structural rules of inference. This is the new program that I subscribed to in my work on sub-structural logics around 1990, and it is a powerful movement today. But gradually, I have changed my mind about the crux of what logic should become. I would now say that the main issue is not variety of reasoning styles and notions of consequence, but the variety of informational tasks performed by intelligent interacting agents, of which inference is only one among many, involving observation, memory, questions and answers, dialogue, or general communication. And logical systems should deal with a wide variety of these, making information-carrying events first-class citizens in their set-up. The purpose of this brief paper is to contrast and compare the two approaches, drawing freely on some insights from earlier published papers. In particular, I will argue that logical dynamics sets itself the more ambitious diagnostic goal of explaining why substructural phenomena occur, by ‘deconstructing’ them into classical logic plus an explicit account of the relevant informational events. (shrink)

ABSTRACTMonoidal logics were introduced as a foundational framework to analyze the proof theory of logical systems. Inspired by Lambek's seminal work in categorical logic, the objective is to defin...

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and (...) the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

I extract several common assumptions in the Classical Theory of Mind (CTM) - mainly of Locke and Descartes - and work out a partial formalisation of the logic implicit in CTM. I then define the modal (logical) properties and relations of propositions, including the modality of conditional propositions and the validity of argument, according to the principles of CTM: that is, in terms of clear and distinct ideas, and without any reference to either possible worlds, or deducibility in (...) an axiomatic system, or linguistic convention. (shrink)

The first systematic exposition of all the central topics in the philosophy of logic, Susan Haack's book has established an international reputation for its accessibility, clarity, conciseness, orderliness, and range as well as for its thorough scholarship and careful analyses. Haack discusses the scope and purpose of logic, validity, truth-functions, quantification and ontology, names, descriptions, truth, truth-bearers, the set-theoretical and semantic paradoxes, and modality. She also explores the motivations for a whole range of non-classical systems of (...) logic, including many-valued logics, fuzzy logic, moddal and tense logics, and relevance logics. Persupposing only an elementary knowledge of formal logic, this book includes many useful summary tables and diagrams, as well as a helpful glossary of technical terms. Wide-ranging, informative, and eminently readable, this book has proven a valuable resource for generations of students and scholars in a variety of disciplines outside philosophy needing guidance on the philosophy of logic. (shrink)

Tagmemic theory as a semiotic theory can be used to analyze multiple systems of logic and to assess their strengths and weaknesses. This analysis constitutes an application of semiotics and also a contribution to understanding of the nature of logic within the context of human meaning. Each system of logic is best adapted to represent one portion of human rationality. Acknowledging this correlation between systems and their targets helps explain the usefulness of more than one (...) system. Among these systems, the two-valued system of classical logic takes its place. All the systems of logic can be incorporated into a complex mathematical model that has a place for each system and that represents a larger whole in human reasoning. The model can represent why tight formal systems of logic can be applied in some contexts with great success, but in other contexts are not directly applicable. The result suggests that human reasoning is innately richer than any one formal system of logic. (shrink)

This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, 'a if b' or 'a given b', ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is (...) due to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum concepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a|b) is said to be "inapplicable" (or "undefined") in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3-valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b = d, their conditions are equivalent. Since all aspects of quantum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the 'influence-at-a-distance', quantum entanglement phenomena. A quantum realist must still embrace non-local influences to hold that "hidden variables" are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics. (shrink)

We introduce a basic intuitionistic conditional logic \(\textsf{IntCK}\) that we show to be complete both relative to a special type of Kripke models and relative to a standard translation into first-order intuitionistic logic. We show that \(\textsf{IntCK}\) stands in a very natural relation to other similar logics, like the basic classical conditional logic \(\textsf{CK}\) and the basic intuitionistic modal logic \(\textsf{IK}\). As for the basic intuitionistic conditional logic \(\textsf{ICK}\) proposed in Weiss (_Journal of Philosophical (...) Logic_, _48_, 447–469, 2019 ), \(\textsf{IntCK}\) extends its language with a diamond-like conditional modality \(\Diamond \hspace{-4.0pt}\rightarrow \), but its ( \(\Diamond \hspace{-4.0pt}\rightarrow \) )-free fragment is also a proper extension of \(\textsf{ICK}\). We briefly discuss the resulting gap between the two candidate systems of basic intuitionistic conditional logic and the possible pros and cons of both candidates. (shrink)

This note corrects an error in my paper 'Normalisation and Subformula Property for a System of Classical Logic with Tarski's Rule' (Archive for Mathematical Logic 61 (2022): 105-129, DOI 10.1007/s00153-021-00775-6): Theorem 2 is mistaken, and so is a corollary drawn from it as well as a corollary that was concluded by the same mistake. Luckily this does not affect the main result of the paper.

Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka _The Journal of Philosophy_, _56_, 125–137 1959 ; Lambert _Notre Dame Journal of Formal Logic_, _8_, 133–144 1967, 1997, 2001 ). What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential (...) import. Free logics reject the claim that names need to denote in (ii). _Positive_ free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, _negative_ free logic treats them as uniformly false, and _neutral_ free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli (_Bulletin of the Section of Logic_, _48_(2), 137–158 2019 ) as well as classical logic in Pavlović and Gratzl (_Journal of Philosophical Logic_, _50_, 117–148 2021 ), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad (_Studia Logica_, _105_(1), 93–119 2017, _Journal of Applied Non-Classical Logics_, _30_(3), 272–289 2020 ), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties. (shrink)

The aim of this paper is to emphasize the fact that for all finitely-many-valued logics there is a completely systematic relation between sequent calculi and tableau systems. More importantly, we show that for both of these systems there are al- ways two dual proof sytems (not just only two ways to interpret the calculi). This phenomenon may easily escape one’s attention since in the classical (two-valued) case the two systems coincide. (In two-valued logic the assignment (...) of a truth value and the exclusion of the opposite truth value describe the same situation.). (shrink)

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking (...) just one syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence. (shrink)

The purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments ofk-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number$k \ge 2$, Ramsey’s Theorem for pairs and recursive assignments ofkcolors is equivalent to the Limited Lesser Principle of Omniscience for${\rm{\Sigma }}_3^0$formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinitek-ary tree there is some$i < k$and some branch with infinitely many (...) children of indexi. (shrink)

EI objetivo de este artículo es presentar los principios de la programación lógica borrosa y de sus principales variantes, ilustrándolas a través de un conjunto de aproximaciones que, a nuestro entender, son representativas de los avances en esta área. También incluimos la descripción de otros sistemas de programación lógica que se sustentan en lógicas de la incertidumbre diferentes de la lógica borrosa. En esta presentación presuponemos que la mayoría de los lectores no son expertos en programación lógica; para seguirla sólo (...) se requiere un conocimiento básico de la lógica de predicados.The purpose of this paper is to present the principles of the fuzzy logic programming, exemplifying them by a couple of proposals that we think are representatives of the advances in this field. We include also the description of another systems of logic programming with uncertain information that are based on other logics of uncertainty which are different from fuzzy logic. This article only presuppose anelementary knowledge of the classical first-order logic. (shrink)

Nyana is the most rational and logical of all the classical Indian philosophical systems. In the study of Nyana philosophy, Karikavali with its commentary Muktavali, both by Visvanatha Nyayapancanana, with the commentaries Dinakari and Ramarudri, have been of decisive significance for the last few centuries as advanced introductions to this subject. The present work concentrates on inference in Karikavali, Muktavali and Dinakari, carefully divided into significant units according to the subject, and translates and interprets them. Its commentary makes (...) use of the primary interpretation in Sanskrit contained especially in the Ramarudri and Subodhini. The book begins with the Sanskrit texts of Karikavali and Muktavali; followed by English translation of these texts. Next is given the Sanskrit text of Dinakari which comments on the first two texts, followed by its English translation. Lastly, the book contains a commentary on all the texts included. (shrink)

Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative" de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives (...) to classical, set-theoretically based mathematics until the 1960s. Now we have a massive amount of information, to which this lecture will constitute an introduction, about what can be done by what means, and about the theoretical interrelationships between various formal systems for constructive, predicative and classical analysis. (shrink)

Systems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they (...) are not constructive. (shrink)

This paper aims at clarifying the nature of Frege's system of logic, as presented in the first volume of the Grundgesetze. We undertake a rational reconstruction of this system, by distinguishing its propositional and predicate fragments. This allows us to emphasise the differences and similarities between this system and a modern system of classical second-order logic.

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule (...) for implication. The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic. (shrink)

This paper presents two systems of natural deduction for the rejection of non-tautologies of classical propositional logic. The first system is sound and complete with respect to the body of all non-tautologies, the second system is sound and complete with respect to the body of all contradictions. The second system is a subsystem of the first. Starting with Jan Łukasiewicz's work, we describe the historical development of theories of rejection for classical propositional logic. Subsequently, we (...) present the two systems of natural deduction and prove them to be sound and complete. We conclude with a ‘Theorem of Inversion’. (shrink)

Along the line of Hirst‐Mummert and Dorais, we analyze the relationship between the classical provability of uniform versions Uni(S) of Π2‐statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for every Π2‐statement S of some syntactical form, if its uniform version derives the uniform variant of over a classical system of arithmetic in all finite types with weak extensionality, then S is not provable in strong (...) semi‐intuitionistic systems including bar induction in all finite types but also nonconstructive principles such as Kőnig's lemma and uniform weak Kőnig's lemma. Our result is applicable to many mathematical principles whose sequential versions imply. (shrink)

Change of logic is typically taken as requiring that the meanings of the connectives change too. As a result, it has been argued that legitimate rivalry between logics is under threat. This is, in a nutshell, the meaning‐variance argument, traditionally attributed to Quine. In this paper, we present a semantic framework that allows us to resist the meaning‐variance claim for an important class of systems: classical logic, the logic of paradox and strong Kleene logic. (...) The major feature of the semantics is that the connectives have the same meanings in these systems, so that the meaning‐variance argument is straightforwardly blocked. We discuss the effects of this semantics for two uses of the argument of meaning variance, and also consider its impact on related issues. (shrink)

Combining non-classical (or sub-classical) logics is not easy, but it is very interesting. In this paper, we combine nonclassical logics of negation and possibility (in the presence of conjunction and disjunction), and then we combine the resulting systems with intuitionistic logic. We will find that Kracht's results on the undecidability of classical modal logics generalise to a non-classical setting. We will also see conditions under which intuitionistic logic can be combined with a non-intuitionistic (...) negation without corrupting the intuitionistic fragment of the logic. (shrink)

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on: admissible or permissible inference rules the derivability of the admissible inference rules the structural completeness of logics the bases for admissible and valid inference rules. There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered. The book is basically self-contained (...) and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included. The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook. (shrink)

A system of logic usually comprises a language for which a model-theory and a proof-theory are defined. The model-theory defines the semantic notion of model-theoretic logical consequence (⊨), while the proof-theory defines the proof- theoretic notion of logical consequence (or logical derivability, ⊢). If the system in question is sound and complete, then the two notions of logical consequence are extensionally equivalent. The concept of full formalization is a more restrictive one and requires in addition the preservation of the (...) standard meanings of the logical terms in all the admissible interpretations of the logical calculus, as it is proof-theoretically defined. Although classical first-order logic is sound and complete, its standard formalizations fall short to be full formalizations since they allow non-intended interpretations. This fact poses a challenge for the logical inferentialism program, whose main tenet is that the meanings of the logical terms are uniquely determined by the formal axioms or rules of inference that govern their use in a logical calculus, i.e., logical inferentialism requires a categorical calculus. This paper is the first part of a more elaborated study which will analyze the categoricity problem from its beginning until the most recent approaches. I will first start by describing the problem of a full formalization in the general framework in which Carnap (1934/1937, 1943) formulated it for classical logic. Then, in sections IV and V, I shall discuss the way in which the mathematicians B.A. Bernstein (1932) and E.V. Huntington (1933) have previously formulated and analyzed it in algebraic terms for propositional logic and, finally, I shall discuss some critical reactions Nagel (1943), Hempel (1943), Fitch (1944), and Church (1944) formulated to these approaches. (shrink)

In Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics—multi-valued, (...) intuitionistic, paraconsistent and quantum logics. Its purpose is to show that there is no particular domain or reason that demands the use of a non-classical logic; the particular reasons given for the non-classical logic can also be handled within classical logic. The result of Sect. 2 is substantiated in Sect. 3, where it is shown (referring to other work) that all four kinds of non-classical logics can be translated into classical logic in a meaning-preserving way. Based on this fact a justification of classical logic is developed in Sect. 4 that is based on its representational optimality. It is pointed out that not many but a few non-classical logics can be likewise representationally optimal. However, the situation is not symmetric: classical logic has ceteris paribus advantages as a unifying metalogic, while non-classical logics can have local simplicity advantages. (shrink)

We define a Kripke semantics for a conditional logic based on the propositional logic $$\textsf{N4}$$ N 4, the paraconsistent variant of Nelson’s logic of strong negation; we axiomatize the minimal system induced by this semantics. The resulting logic, which we call $$\textsf{N4CK}$$ N 4 CK, shows strong connections both with the basic intuitionistic logic of conditionals $$\textsf{IntCK}$$ IntCK introduced earlier in (Olkhovikov, 2023) and with the $$\textsf{N4}$$ N 4 -based modal logic $$\textsf{FSK}^d$$ FSK d (...) introduced in (Odintsov and Wansing, 2004) as one of the possible counterparts to the classical modal system $$\textsf{K}$$ K. We map these connections by looking into the embeddings which obtain between the aforementioned systems. (shrink)

Any "classical" theory of truth will satisfy tarski's criterion ("p" is true if and only if p), And the principle of bivalence (every proposition is either true or false). A non-Classical theory may be obtained by rejecting these principles: - in fact it is shown that rejection of the second entails rejection of the first. If the resulting non-Classical theory is formalized, A system structurally isomorphic to either s4 or s5 is obtained. An attempt is made to (...) show that the essential insights of intuitionist logic may be preserved if we replace their complex and in many respects "unintuitive" propositional logic by a theory of truth which is non-Classical in the sense described above. (shrink)

Logical and philosophical literature provides different classifications of reasoning. In the Polish literature on the subject, for instance, there are three popular ones accepted by representatives of the Lvov-Warsaw School: Jan Łukasiewicz, Tadeusz Czeżowski and Kazimierz Ajdukiewicz (Ajdukiewicz in Logika pragmatyczna [Pragmatic Logic]. PWN, Warsaw (1965, 2nd ed. 1974). Translated as: Pragmatic Logic. Reidel & PWN, Dordrecht, 1975). The author of this paper, having modified those classifications, distinguished the following types of reasoning: (1) deductive and (2) non-deductive, and (...) additionally two types of them in each of the two, depending on the manner of combining their premises with the conclusion through the relation of classical logical entailment. Consequently, the four types of reasoning: unilateral deductive (incl. its sub-types: deductive inference and proof), bilateral deductive (incl. complete induction), and reductive (incl. the sub-types: explanation and verification), logically nonvaluable (incl. inference by analogy, statistic inference), correspond to four operators of derivability. They are defined formally on the ground of Tarski’s axiomatic theory of deductive systems, by means of the consequence operation _Cn_ (Tarski in Monatshefte Math Phys 37:361–404, 1930a, C R Soc Sci Lett Vars 23:22–29, 1930b). Also, certain metalogical properties of these operators are given, as well as their relations with Tarski’s consequence operations \(Cn^+\) ( \(Cn^+ = Cn\) ) and dual consequences \(Cn^{-1}\) (Słupecki in Zeszyty Naukowe Uniwersytetu Wrocławskiego Seria B Nr 3:33–40, 1959, Słupecki et al. in Stud Log 29:76–123, 1971, Wybraniec-Skardowska, in: Wybraniec-Skardowska, Bryll (eds) Z badań nad teorią zdań odrzuconych [Studies in the Theory of Rejected Propositions], Series B, Studia i Monografie, Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Opole, 1969), and \(Cn^-\) (Wójcicki in Bull Sect Log 2(2):54–57, 1973)). (shrink)

Although many modern philosophers of law describe custom as merely a minor source of law, formal law is actually only one source of the legal customs that govern us. Many laws grow out of custom, and one measure of a law's success is by its creation of an enduring legal custom. Yet custom and customary law have long been neglected topics in unsettled jurisprudential debate. Smaller concerns, such as whether customs can be legitimized by practice or by stipulation, stipulated by (...) an authority or by general consent, or dictated by law or vice versa, lead to broader questions of law and custom as alternative or mutually exclusive modes of social regulation, and whether rational reflection in general ought to replace sub-rational prejudice. Can legal rules function without customary usage, and does custom even matter in society? The Philosophy of Customary Law brings greater theoretical clarity to the often murky topic of custom by showing that custom must be analyzed into two more logically basic concepts: convention and habit. James Bernard Murphy explores the nature and significance of custom and customary law, and how conventions relate to habits in the four classic theories of Aristotle, Francisco Suarez, Jeremy Bentham, and James C. Carter. He establishes that customs are conventional habits and habitual conventions, and allows us to better grasp the many roles that custom plays in a legal system by offering a new foundation of understanding for these concepts. (shrink)

Due to the failure of the classical principles of bivalence and verifunctionality, the logic of experimental propositions relative to quantum systemscannot be interpreted in Boolean algebras. However, we cannot say neither that this logic is captured by orthomodular lattices, as claimed by many authors along the line of Birkhoff‘s and von Neumann‘s standard approach. For the alleged violation of distributivity is based on the possibility of combining statements relative to complementary contexts, which does not refer to any (...) experience and, consequently, has no meaning. Indeed, quantum logic should be interpreted in partial, transitive Boolean algebras whose compatibility relation limits the application of the connectives within each of its Boolean sub-algebras, which refer to partial, classical descriptions. Moreover, this approach of quantum logic makes it possible to deal with composite systems, which was not possible to do within the standard approach, and then to deal with the fundamental notion of quantum entanglement. The latter notion can be represented by a series of axioms of the object language that restrict the set of experimental statements bearing on a composite system, while its close link to the notion of complementarity can be expressed in the metalanguage. (shrink)

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. (...) We will claim that a logic is to be identified with an infinite sequence of consequence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting consequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 361 ing should gravitate, it is no wonder that many say: " There are no clear answers." Finally, I wonder if casuistry can even deal with the most significant ethical issue facing medicine in the immediate future: The construction of a system in the United States which will provide adequate health care for all citizens. Director, Center for Health Care Ethics Saint Louis University Medical Center KEVIN (...) O'ROURKE, O.P. Christian Uniqueness Reconsidered: The Myth of a Pluralistic Theology of Religions. Edited By GAVIN D'CosTA. Maryknoll, New York: Orbis Books, 1990. Pp. xii + 218. $16.95 (paper). There are two ways of reading ·this remarkably stimulating collection of essays. At one level it is a vigorous rebuttal of an earlier book in the "Faith Meets Faith Series" entitled The Myth of Christian Uniqueness: Toward a Pluralistic Theology of Religions, edited by John Hick and Paul Knitter; D'Costa's sub-title, "The Myth of a Pluralistic Theology of Religions," is a conscious polemical riposte. On the second level it purports to propose an alternative theology of religions which would on the one hand retain the claim of Christian uniqueness but on the other be not exclusivistic but genuinely pluralistic and, therefore, fruitful for interreligious dialogue. In my judgment, the book achieves its first objective well, furnishing an impressive array of counter-arguments to the pluralistic thesis. Indeed, like a swarm of tacklers ganging up on the hapless quarterback, so many contributors attack the same points of the pluralistic proposal that readers must have the impression of witnessing an overkill. On the other hand, the book's positive construction of a theology of religions suffers from vagueness and even internal contradictions. It is a classical case of people banding together because they know what they are against (in this case, the proposal to regard all religions as equally valid ways of salvation, with none allowed to claim superiority and exclusiveness) but not yet able to determine what they are for (except to retain the claim of Christian uniqueness). The volume contains 14 essays divided into three groups. The first three discuss the implications of the Christian doctrine of the Trinity for interreligious dialogue (Rowan Williams, Gavin D'Costa, and Christoph Schwobel); the next five explore the relevance of christology in the context of religious pluralism (M. M. Thomas, Francis X. Clooney, John B. Cobb, Jr., Wolfhart Pannenberg, and Monika Hell- 362 BOOK REVIEWS wig) ; the last six examine the epistemological and hermeneutical issues of religious pluralism (J. A. DiNoia, Lesslie Newbigin, Jurgen Moltmann, Paul J. Griffith, John Milbank, and Kenneth Surin). The editor has done an excellent job of summarizing the main points of each essay; it is therefore unnecessary to replicate his effort. My intention in this review is not to examine each essay individually; space would not permit such an undertaking. Rather I shall list the major criticisms made by the contributors against the pluralistic thesis and then examine their rather diverse positive proposals. Before doing so, however, it would be useful to describe briefly the essential thesis of The Myth of Christian Uniqueness, which is under attack. Its contributors argue that due to the rise of historical consciousness, the nature of God as Absolute Mystery, and the obligation to promote peace and justice, the Christian claim to uniqueness and superiority as a way to salvation should be abandoned. Instead of the exclusivist and inclusivist theologies of religions, they propose the pluralist position that Christianity is one among the many religions, equally valid and mutually complementary. What do the contributors to Christian Uniqueness Reconsidered find wrong with this pluralist thesis? A complete list of their objections cannot be furnished here, but these are some of the more important ones. First, the pluralists are as imperialistic as the exclusivists and inclusivists they denounce because they impose their (Western) notions of religion, dialogue, social justice, and so on, on other religionists (D'Costa, Newbigin, Griffiths). Hence the pluralist thesis is logically incoherent. Secondly, the pluralists wrongly presume that there is such a thing as a common core of religious experience which functions as a genus of the different species of religions (Newbigin, Milbank, Cobb). Thirdly, pluralists neglect the... (shrink)

This text, serving as an introduction to Aristotelian, symbolic, inductive, and practical logic, presents the techniques of these approaches to logic, some of the philosophical problems of logic, and some of the attempts to solve philosophical problems by means of various logical techniques. It discusses the problem of universals and null classes; briefly introduces the theory of types; and presents Lukasiewicz's formalization of Aristotelian syllogistic logic as an example of a formal system. Classical logic (...) and propositional logic are carefully and fully presented. The text, however, is weak in its presentation of quantificational logic. After a correct, but brief and very inadequate introduction to quantificational logic, the readers are referred to Quine, Methods of Logic, or to Copi, Symbolic Logic, for a more adequate development. The chapter on inductive logic discusses the uses and dangers of generalizations and introduces the probability calculus; the chapter on informal logic discusses criteria of rationality and presents the usual informal fallacies. There are problems and exercises after many of the sections, and answers in the back of the book.—R. L. S. (shrink)

The paraconsistent system CPQ-ZFC/F is defined. It is shown using strong non-finitary methods that the theorems of CPQ-ZFC/F are exactly the theorems of classical ZFC minus foundation. The proof presented in the paper uses the assumption that a strongly inaccessible cardinal exists. It is then shown using strictly finitary methods that CPQ-ZFC/F is non-trivial. CPQ-ZFC/F thus provides a formulation of set theory that has the same deductive power as the corresponding classical system but is more reliable in that (...) non-triviality is provable by strictly finitary methods. This result does not contradict Gödel's incompleteness theorem because the proof of the deductive equivalence of the paraconsistent and classical systemss use non-finitary methods. (shrink)