Nonuniform Substitutions Can’t Get you From Classical Logic to any Traditional Relevant or Quasi-Relevant Logic

In this work we argue for relevant logics as a basis for paraconsistent epistemic logics. In order to do so, a paraconsistent nonmonotonic multi-agent epistemic logic, MDR (for Modal Defeasible Relevant), is briefly introduced. In MDR each agent has two kinds of belief: an absolute belief that P, represented by AiP, and a defeasible belief that P, represented by DiP. Therefore, an agent can reason with his own absolute and defeasible beliefs about the world and also reason about his beliefs about other agents' beliefs both absolute and defeasible. A theorem is presented showing some patterns of reasoning in MDR. Avron's relevant logic RMI→ s compared proof theoretically with Anderson and Belnap's relevant logics R and RM and also with Da Costa's paraconsistent logic CI, with respect to some desired properties of absolute and defeasible beliefs. Then we can understand why RMI was chosen to be the underlying logic for the monotonie part (the absolute beliefs) of MDR, and had to be modified and integrated with a nonmonotonic logic in order to be the underlying logic for the nonmonotonic part (defeasible beliefs).

There is a strong case to be made for thinking that an obscure logic, KR, is better than classical logic and better than any relevant logic. The argument for KR over relevant logics is that KR counts disjunctive syllogism valid, and this is the biggest complaint about relevant logics. The argument for KR over classical logic depends on the normativity of logic and the paradoxes of implication. The paradoxes of implication are taken by relevant logicians to justify relevant logic, but considerations on the normativity of logic show that only some of the paradoxes of implication are genuine. KR avoids all the genuine paradoxes of implication, unlike classical logic. Overall, KR avoids the genuine paradoxes of implication and avoids the major objection to relevant logics. This combination of features provides strong reason to give KR a place in the conversation about the right logic(s).

The formalization of human deductive reasoning is a main issue in artificial intelligence. Although classical logic is one of the most useful ways for the formalization, the material implication of classical logic has some fallacies. Relevant logic has been studied for the removal of material implication fallacies in classical logic for the formalization of human deductive reasoning. Relevant logic \(ER\) is free from fallacies of material implication and has more provability than typical relevant logic \(R\). Moreover, ER has a decision procedure, even though almost all relevant logics are undecidable. This is one of the reasons why \(ER\) is an appropriate logic for the formalization of human deductive reasoning. This paper implements automated theorem prover of Relevant logic \(ER\) and experiments on this theorem prover to check the processing time of implemented automated theorem prover.

Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing.

Modal Logic’s theorems and rules are valid in possible worlds but Relevant Logic’s theorems and rules are valid, respectively, in logical worlds and situations. Robert Meyer in 1974 removed this asymmetry between the theorems and the rules of Relevant Logic by establishing a logical system, whose theorems and rules are valid in all situations. Introducing a new kind of truth and falsity operators, the authors in this article sketch a logical system defined on the basis of Relevant Logic. Such a system can not only preserve symmetry, but also remove some inconsistency between Modern Classical Logic and Relevant Logic, because the latter, like the former, puts possible worlds as a criterion for validity.

Fitch-style natural deduction was first introduced into relevant logic by Anderson in [1960], for the sentential logic E of entailment and its quantincational extension EQ. This was extended by Anderson and Belnap to the sentential relevant logics R and T and some of their fragments in [ENT1], and further extended to a wide range of sentential and quantified relevant logics by Brady in [1984]. This was done by putting conditions on the elimination rules, →E, ~E, ⋁E and ∃E, pertaining to the set of dependent hypotheses for formulae used in the application of the rule. Each of these rules were subjected to the same condition, this condition varying from logic to logic. These conditions, together with the set of natural deduction rules, precisely determine the particular relevant logic. Generally, this is a simpler representation of a relevant logic than the standard Routley-Meyer semantics, with its existential modelling conditions stated in terms of an otherwise arbitrary 3-place relation R, which is defined over a possibly infinite set of worlds. Readers are urged to refer to Brady [1984], if unfamiliar with the above natural deduction systems, but we will introduce in §2 a modified version in full detail.Natural deduction for classical logic was invented by Jaskowski and Gentzen, but it was Prawitz in [1965] who normalized natural deduction, streamlining its rules so as to allow a subformula property to be proved. (This key property ensures that each formula in the proof of a theorem is indeed a subformula of that theorem.)

To design and develop anticipatory reasoning-reacting systems with three-dimensional moving objects, it is indispensable to decide the fundamental logic basis to underlie anticipatory reasoning about three-dimensional moving objects. This paper investigates the fundamental logic basis for anticipatory reasoning-reacting systems with three-dimensional moving objects. The paper presents some basic requirements for the fundamental logic basis and shows that three-dimensional spatio-temporal relevant logic is more suitable than classical mathematical logic and its various classical conservative extensions and various relevant logics to be the fundamental logic basis by two case studies.

This paper continues the work of Priest and Sylvan in Simplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such as B+ and B. We show that the simplified semantics can also be used for a large number of extensions of the positive base logic B+, and then add the dualising'*' operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.

This paper proposes a new relevant logic B+⊓⊔, which is obtained by adding two binary connectives, intensional conjunction ⊓ and intensional disjunction ⊔, to Meyer–Routley minimal positive relevant logic B+, where ⊓ and ⊔ are weaker than fusion ˚ and fission +, respectively. We give Kripke-style semantics for B+⊓⊔, with →, ⊓ and ⊔ modelled by ternary relations. We prove the soundness and completeness of the proposed semantics. A number of axiomatic extensions of B+⊓⊔, including negation-extensions, are also considered, together with the corresponding semantic conditions required for soundness and completeness to be maintained.

In this paper, I set out a semantics for identity in relevant logic that is based on an analogy between the biconditional and identity. This analogy supports the semantics that Priest has set out for identity in basic relevant logic and it motivates a version of the Routley–Meyer semantics in which identities can be viewed as constraints on the ternary relation that is used to treat implication.

Relevance Logics

Routley-Meyer type ternary relational semantics are defined for relevant logics including Routley and Meyer’s basic logic B plus the reductio rule \( \vdash A\rightarrow \lnot A\Rightarrow \vdash \lnot A\) and the disjunctive syllogism. Standard relevant logics such as E and R (plus γ) and Ackermann’s logics of ‘strenge Implikation’ Π and Π′ are among the logics considered.

Clark Glymour has argued that hypothetico-deductivism, which many take to be an important method of scientific confirmation, is hopeless because it cannot be reconstructed in classical logic. Such reconstructions, as Glymour points out, fail to uphold the condition of relevance between theory and evidence. I argue that the source of the irrelevant confirmations licensed by these reconstructions lies not with hypothetico-deductivism itself, but with the classical logic in which it is typically reconstructed. I present a new reconstruction of hypothetico-deductivism in relevance logic that does maintain the condition of relevance between theory and evidence. Hence, if hypothetico-deductivism is an important rationale in science, we have good reason to believe that the logic underlying scientific discourse is captured better by relevance logic than by its classical counterpart.

Pretabular logics are those that lack finite characteristic matrices, although all of their normal proper extensions do have some finite characteristic matrix. Although for Anderson and Belnap’s relevance logic R, there exists an uncountable set of pretabular extensions (Swirydowicz in J Symb Log 73(4):1249–1270, 2008), for the classical relevance logic $$ \hbox {KR} = \hbox {R} + \{(A\,\, \& \sim A)\rightarrow B\}$$ there has been known so far a pretabular extension: $${\mathcal L}$$ (Galminas and Mersch in Stud Log 100:1211–1221, 2012). In Section 1 of this paper, we introduce some history of pretabularity and some relevance logics and their algebras. In Section 2, we introduce a new pretabular logic, which we shall name $${\mathcal M}$$ , and which is a neighbor of $${\mathcal L}$$ , in that it is an extension of KR. Also in this section, an algebraic semantics, ‘ $${\mathcal M}$$ -algebras’, will be introduced and the characterization of $${\mathcal M}$$ to the set of finite $${\mathcal M}$$ -algebras will be shown. In Section 3, the pretabularity of $${\mathcal M}$$ will be proved.

Demonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this concise undergraduate textbook covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. The book balances accessibility, breadth, and rigor, and is designed so that its materials will fit into a single semester. Its distinctive presentation of traditional logic material will enhance readers' capabilities and mathematical maturity. The proof theory portion presents classical propositional logic and first-order logic using a computer-oriented (resolution) formal system. Linear resolution and its connection to the programming language Prolog are also treated. The computability component offers a machine model and mathematical model for computation, proves the equivalence of the two approaches, and includes famous decision problems unsolvable by an algorithm. The section on nonclassical logic discusses the shortcomings of classical logic in its treatment of implication and an alternate approach that improves upon it: Anderson and Belnap's relevance logic. Applications are included in each section. The material on a four-valued semantics for relevance logic is presented in textbook form for the first time. Aimed at upper-level undergraduates of moderate analytical background, Three Views of Logic will be useful in a variety of classroom settings. Gives an exceptionally broad view of logic Treats traditional logic in a modern format Presents relevance logic with applications Provides an ideal text for a variety of one-semester upper-level undergraduate courses