A logic for evidence and truth *

Artificial Intelligence, 1992

Roos, N., A logic for reasoning with inconsistent knowledge, Artificial Intelligence 57 (1992) 69-103.

Synthese, 2017

The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence (BLE), and extends it to the Logic of Evidence and Truth (LETJ). The latter is a logic of formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. LETJ is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. Adequate semantics and a decision method are presented for both BLE and LETJ , as well as some technical results that fit the intended interpretation. * The first author acknowledges support from FAPESP (Fundação de Amparoà Pesquisa do Estado de São Paulo, thematic project LogCons) and from a CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) research grant. The second author acknowledges support from FAPEMIG (Fundação de Amparoà Pesquisa do Estado de Minas Gerais, research project 21308). 1 Although the dialetheist view has some antecedents in the history of philosophy, especially in Hegel and, according to some interpreters, also Heraclitus, the claim that there are 'A holds' means 'there is evidence that A is true'; 'A does not hold' means 'there is no evidence that A is true'; '¬A holds' means 'there is evidence that A is false';

arXiv (Cornell University), 2022

Accounting for the epistemic contribution of deduction has been a pervasive problem for logicians interested in deduction, such as, among others, Jakko Hintikka. The problem arises because the conclusion validly deduced from a set of premises is said to be "contained" in that set; because of this containment relation, the conclusion would be known from the moment the premises are known. Assuming this, it is problematic to explain how we can gain knowledge by deducing a logical consequence implied in a set of known premises. To address this problem, we offer an alternative account of the epistemic contribution of deduction as the process required to deduce a conclusion or a theorem, understanding such process not only in terms of the number of steps in the derivation but, more importantly, the reason for or justification for every step. That is, we do not know a proposition unless we have a justification or proof to hold that proposition. With this goal in mind, we develop a justification logic system which exhibits the epistemic contribution of a deductive derivation as the resulting justified formula.

Journal of Logic and Computation, 1998

We study a framework, RLF, for defining natural deduction presentations of linear and other relevant logics. RLF consists in a language together, in a manner similar to that of LF, with a representation mechanism. The language of RLF, the AA«-calculus, is a system of first-order linear ...

We often think of a logic as a three-part system composed of an uninterpreted language, a semantic system, and a deductive system. Keeping the language fixed, different logics can be produced by changing the semantic system or the deductive system or both. Choosing a logic is not just a matter of convenience. Although two logics that differ only in deductive system can be equivalent in the sense of always deriving the same conclusions from the same premises, it is often true that each system corresponds to a specific approach to logic. Natural deduction, specifically, corresponds to approaching logic as a mathematical model of correct deductive reasoning. In natural deduction, the flow of reasoning is not just a linear sequencing in which each formula in each step is derived just from its previous lines. Rather, natural deduction involves, in addition to steps derived just from their previous lines, also steps derived from subderivations that may contain further subderivations, each subderivation starting from a supposition. In natural deduction, the actual deductive process is more explicitly analyzed than in other deductive systems—with the same language and semantic system. This paper also investigates some consequences of understanding natural-deduction systems in an epistemological framework.

In this note we briefly discuss the relationship between the deduction theorem and rules of inference. Ultimately, we point out that although it is a mistake to say that the deduction theorem fails for modal logics, we can generalize the deduction theorem. This generalization can be used to encode, or as we say 'reflect' different kinds of rules. What we show, however, is that there is a limit on which logics can reflect certain kinds of rules. This limit can be shown to hold for a large class of logics which make up the majority of the logics studied.

Axioms, 2019

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

2021

We study a conservative extension of classical propositional logic distinguishing between four modes of statement: a proposition may be affirmed or denied, and it may be strong or classical. Proofs of strong propositions must be constructive in some sense, whereas proofs of classical propositions proceed by contradiction. The system, in natural deduction style, is shown to be sound and complete with respect to a Kripke semantics. We develop the system from the perspective of the propositions-as-types correspondence by deriving a term assignment system with confluent reduction. The proof of strong normalization relies on a translation to System F with Mendler-style recursion.

Trends in Logic, 2014

The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.

Data & Knowledge Engineering, 1995

Classical logic has many appealing features for knowledge representation and reasoning. But unfortunately it is awed when reasoning about inconsistent information, since anything follows from a classical inconsistency. This problem is addressed by introducing the notions of \argument" and of \acceptability" of an argument. These notions are used to introduce the concept of \argumentative structures". Each de nition of acceptability selects a subset of the set of arguments, and an argumentative structure is a subset of the power set of arguments. In this paper, we consider, in detail, a particular argumentative structure, where each argument is de ned as a classical inference together with the applied premisses. For such arguments, a variety of de nitions of acceptability are provided, the properties of these de nitions are explored, and their inter-relationship described. The de nitions of acceptability induce a family of logics called argumentative logics which we explore. The relevance of this work is considered and put in a wider perspective.