Negation and Dichotomy

The present contribution might be regarded as a kind of defense of the common sense in logic. It is demonstrated that if the classical negation is interpreted as the minimal negation with n = 2 truth values, then deviant logics can be conceived as extension of the classical bivalent frame. Such classical apprehension of negation is possible in non- classical logics as well, if truth value is internalized and bivalence is replaced by bipartition

Depicting Negation in Diagrammatic Logic: Legacy and Prospects

Here are considered the conditions under which the method of diagrams is liable to include non-classical logics, among which the spatial representation of non-bivalent negation. This will be done with two intended purposes, namely: a review of the main concepts involved in the definition of logical negation; an explanation of the epistemological obstacles against the introduction of non-classical negations within diagrammatic logic

Sound Probabilistic Reasoning under Contradiction, Inconsistency and Incompleteness

I intend to show how probability theory can be regarded as logic-dependent, viewing probability as a branch of logic in a generalized way. A kind of meta-axiomatics permits us to define probability measures that are either classical, paraconsistent, intuitionistic, or simultaneously intuitionistic and paraconsistent, just by parameterizing on consequence relations. In particular, I intend to discuss theories of probability built upon the paraconsistent Logic of Formal Inconsistency Ci, and upon the paraconsistent and paracomplete Logic of Evidence and Truth LETj. I argue that Ci very naturally encodes an extension of the notion of probability able to express probabilistic reasoning under an excess of information (contradictions), while LETj encodes an extension of the notion of probability able to express probabilistic reasoning under lack of information (incompleteness), and is thus naturally connected to the notion of probability of evidence. I also discuss how interesting non-standard Bayesian updating can be defined in both cases. This is a joint project with J. Bueno-Soler and A. Rodrigues. and most results already appear in [1] and in [5]

Recovery operators, paraconsistency and duality

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices

Semantical Investigations on Non-classical Logics with Recovery Operators: Negation

We investigate mathematical structures that provide a natural semantics for families of (quantified) non-classical logics featuring special unary connectives, called recovery operators, that allow us to 'recover' the properties of classical logic in a controlled fashion. These structures are called topological Boolean algebras. They are Boolean algebras extended with additional unary operations, called operators, such that they satisfy particular conditions of a topological nature. In the present work we focus on the paradigmatic case of negation. We show how these algebras are well-suited to provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness, which feature recovery operators used to earmark propositions that behave 'classically' in interaction with non-classical negations. In contrast to traditional semantical investigations, carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras become encoded as algebras of sets via their Stone-type representation. We employ our higher-order meta-logic to define and interrelate several transformations on unary set operations (operators), which naturally give rise to a topological cube of opposition. Furthermore, our approach allows for a uniform characterization of propositional, first-order and higher-order quantification (also restricted to constant and varying domains). With this work we want to make a case for the utilization of automated theorem proving technology for doing computer-supported research in non-classical logics. All presented results have been formally verified (and in many cases obtained) using the Isabelle/HOL proof assistant

Structuring co-constructive logic for proofs and refutations

This paper considers a topos-theoretic structure for the interpretation of co-constructive logic for proofs and refutations following. It is notoriously tricky to define a proof-theoretic semantics for logics that adequately represent constructivity over proofs and refutations. By developing abstractions of elementary topoi, we consider an elementary topos as structure for proofs, and complement topos as structure for refutation. In doing so, it is possible to consider a dialogue structure between these topoi, and also control their relation such that classical logic (interpreted in a Boolean topos) is simulated where proofs and refutations are conclusive