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A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence according to which α 1 , … , α n entails β just in case □ α 1 , … , □ α n entails □ β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status.

Recently, it has been proposed to understand a logic as containing not only a validity canon for inferences but also a validity canon for metainferences of any finite level. Then, it has been shown that it is possible to construct infinite hierarchies of ‘increasingly classical’ logics—that is, logics that are classical at the level of inferences and of increasingly higher metainferences—all of which admit a transparent truth predicate. In this paper, we extend this line of investigation by taking a somehow different route. We explore logics that are different from classical logic at the level of inferences, but recover some important aspects of classical logic at every metainferential level. We dub such systems meta-classical non-classical logics. We argue that the systems presented deserve to be regarded as logics in their own right and, moreover, are potentially useful for the non-classical logician.

Priest and others have presented their \"most telling\" argument for paraconsistent logic: that only paraconsistent logics allow non-trivial inconsistent theories. This is a very prevalent argument; occurring as it does in the work of many relevant and more generally paraconsistent logicians. However this argument can be shown to be unsuccessful. There is a crucial ambiguity in the notion of non-triviality. Disambiguated the most telling reason for paraconsistent logics is either question-begging or mistaken. This highlights an important confusion about the role of logic in our development of our theories of the world. Does logic chart good reasoning or our commitments? We also consider another abductive argument for paraconsistent logics which also is shown to fail.

In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular 'justifications' are demonstrably without epistemic value (sec. 1). Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus nonclassical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, can be translated into classical logic by introducing additional intensional operators into the language (sec. 2-5). Based on this result it is argued that classical logic is representationally optimal. In sec. 6 it is investigated whether non-classical logics can be likewise representationally optimal. The answer is predominantly negative but partially positive. Nevertheless the situation is not symmetric, because classical logic has important ceteris paribus advantages as a unifying metalogic.

Propongo que el pensamiento crítico y las lógicas comparten tres supuestos: asumen intuiciones sobre la realidad, permiten al agente “creer” y realizan actividades racionales en torno a creencias. Muestro dos actividades complementarias cuando de elucidar y conocer se trata: las prácticas formal y conceptual-pragmática del agente; quien intuye, cree, formaliza y satisface criterios metodológicos mediante ciertas virtudes en su discurso y práctica racional: Orden en su pensamiento, rigor lógico, claridad y brevedad conceptual, relevancia epistémica en sus afirmaciones, congruencia pensamiento-acción. Comparo novedosamente las tareas del lógico y concluyendo que, si el agente no cuenta con habilidades de pensamiento desarrolladas, no logra la construcción de una visión de la realidad, representada en modelos y teorías; algo necesario en ciencias. 

In this article, we will present a number of technical results concerning Classical Logic, S T 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 S T 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 S T , 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.

Natural deduction with alternatives extends Gentzen–Prawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion natural deduction proof system in which the connective rules are unmodi_ed from the usual Prawitz introduction and elimination rules — the extension is purely structural. This framework is general: it can be used for (1) classical logic, (2) relevant logic without distribution, (3) affine logic, and (4) linear logic, keeping the connective rules fixed, and varying purely structural rules. The key result of this paper is that the two principles that introduce kinds of irrelevance to natural deduction proofs: (a) the rule of explosion (from a contradiction, anything follows); and (b) the structural rule of vacuous discharge; are shown to be two sides of a single coin, in the same way that they correspond to the structural rule of weakening in the sequent calculus. The paper also includes a discussion of assumption classes, and how they can play a role in treating additive connectives in substructural natural deduction.

This paper investigates the univocity (or uniqueness) of connectives in intuitionistic and classical sentential logic. Specifically, unlike Gentzen systems, Hilbert systems for (various fragments of) intuitionistic and classical logic do not always determine univocal (or unique) conditional connectives. This paper explains when univocal conditional connectives are achieved in Hilbert systems for intuitionistic and classical sentential logic (and when they are not). In the final section, we discuss the (non-)univocity of the Sheffer stroke in Hilbert vs. Gentzen systems for classical sentential logic.

Classical logic is based on an underlying view of the world, according to which there are elementary facts and compound facts, which are logical combinations of these elementary facts. Sentences are true if they correspond to, in last instance, the elementary facts in the world. This world view has no place for rules, which exist as individuals in the world, and which create relations between the most elementary facts. As a result, classical logic is not suitable to deal with rules, and is therefore unsuitable to deal with legal reasoning. A logic that is more suitable should take into account that law is a part of social reality, in particular a part that consists of constructivist facts, and that rules play a central role in law. This article gives a superficial description of how social reality exists and of the place of law and legal rules in it. It uses this description to argue that traditional techniques to reason with and about legal rules provide a better logic for law than classical logic. These techniques can be accommodated in a logic that treats rules as logical individuals.

In their recent article “A Hierarchy of Classical and Paraconsistent Logics”, Eduardo Barrio, Federico Pailos and Damien Szmuc (BPS hereafter) present novel and striking results about meta-inferential validity in various three valued logics. In the process, they have thrown open the door to a hitherto unrecognized domain of non-classical logics with surprising intrinsic properties, as well as subtle and interesting relations to various familiar logics, including classical logic. One such result is that, for each natural number n , there is a logic which agrees with classical logic on tautologies, inferences, meta-inferences, meta-meta-inferences, meta-meta-...( n - 3 times)-meta-inferences, but that disagrees with classical logic on n + 1-meta-inferences. They suggest that this shows that classical logic can only be characterized by defining its valid inferences at all orders. In this article, I invoke some simple symmetric generalizations of BPS’s results to show that the problem is worse than they suggest, since in fact there are logics that agree with classical logic on inferential validity to all orders but still intuitively differ from it. I then discuss the relevance of these results for truth theory and the classification problem.