A non-transitive relevant implication corresponding to classical logic consequence

In this paper we first develop a logic independent account of relevant implication. We propose a stipulative denition of what it means for a multiset of premises to relevantly L-imply a multiset of conclusions, where L is a Tarskian consequence relation: the premises relevantly imply the conclusions iff there is an abstraction of the pair <premises, conclusions> such that the abstracted premises L-imply the abstracted conclusions and none of the abstracted premises or the abstracted conclusions can be omitted while still maintaining valid L-consequence.          Subsequently we apply this denition to the classical logic (CL) consequence relation to obtain NTR-consequence, i.e. the relevant CL-consequence relation in our sense, and develop a sequent calculus that is sound and complete with regard to relevant CL-consequence. We present a sound and complete sequent calculus for NTR. In a next step we add rules for an object language relevant implication to thesequent calculus. The object language implication reflects exactly the NTR-consequence relation. One can see the resulting logic NTR-> as a relevant logic in the traditional sense of the word.       By means of a translation to the relevant logic R, we show that the presented logic NTR is very close to relevance logics in the Anderson-Belnap-Dunn-Routley-Meyer tradition. However, unlike usual relevant logics, NTR is decidable for the full language, Disjunctive Syllogism (A and ~AvB relevantly imply B) and Adjunction (A and B relevantly imply A&B) are valid, and neither Modus Ponens nor the Cut rule are admissible

Adaptive Logics using the Minimal Abnormality strategy are Π11\Pi^1_1-complex

In this article complexity results for adaptive logics using the minimal abnormality strategy are presented. It is proven here that the consequence set of some recursive premise sets is Pi(1)(1)-complete. So, the complexity results in ( Horsten and Welch, Synthese 158: 41- 60, 2007) are mistaken for adaptive logics using the minimal abnormality strategy

A proof procedure for adaptive logics

In this article, I present a procedure that generates proofs for finally derivable adaptive logic consequences. A proof procedure for the inconsistency adaptive logic CLuNr is already presented in [7]. In this article a procedure for CLuNm is presented and the results for both logics are generalized to all adaptive logics, on the presupposition that there exists a proof procedure for the lower limit logic. The generated proofs are so called goal-directed proofs, i.e. proofs that (i) start with the formula (the goal) of which one wants to know whether it is a consequence of a certain premise set and (ii) only consist of lines that may potentially be useful for proving or disproving the goal. The goal-directed proofs form good explications of actual problem-solving reasoning processes.In this article, I present a procedure that generates proofs for finally derivable adaptive logic consequences. A proof procedure for the inconsistency adaptive logic CLuNr is already presented in [7]. In this article a procedure for CLuNm is presented and the results for both logics are generalized to all adaptive logics, on the presupposition that there exists a proof procedure for the lower limit logic. The generated proofs are so called goal-directed proofs, i.e. proofs that (i) start with the formula (the goal) of which one wants to know whether it is a consequence of a certain premise set and (ii) only consist of lines that may potentially be useful for proving or disproving the goal. The goal-directed proofs form good explications of actual problem-solving reasoning processes

Adaptive logics using the minimal abnormality strategy are Pi11Pi^1_1 -complex

In this article complexity results for adaptive logics using the minimal abnormality strategy are presented. It is proven here that the consequence set of some recursive premise sets is Π11 -complete. So, the complexity results in (Horsten and Welch, Synthese 158:41–60, 2007) are mistaken for adaptive logics using the minimal abnormality strategy

Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations

Obtaining infinitely many degrees of inconsistency by adding a strictly paraconsistent negation to classical logic

This paper is devoted to a consequence relation combining the negation of Classical Logic (CL) and a paraconsistent negation based on Graham Priest’s Logic of Paradox (LP). We give a number of natural desiderata for a logic L that combines both negations. They are motivated by a particular property-theoretic perspective on paraconsistency and are all about warranting that the combining logic has the same characteristics as the combined logics, without giving up on the radically paraconsistent nature of the paraconsistent negation. We devise the logic CLP by means of an axiomatization and three equivalent semantical characterizations (a non-deterministic semantics, an infinite-valued set-theoretic semantics and an infinite-valued semantics with integer numbers as values). By showing that this logic is maximally paraconsistent, we prove that CLP is the only logic satisfying all postulated desiderata. Finally we show how the logic’s infinite-valued semantics permits defining different types of entailment relations

Truthmakers and Relevance for FDE, LP, K3, and CL

In this paper, we first develop truthmaker semantics for four relevance logics defined as the X-relevant cores (as introduced in [34]) of the well-known propositional logics CL (classical logic), LP (the logic of paradox), K3 (strong Kleene logic) and FDE (first degree entailment). The semantics is similar to Kit Fine’s truthmaker semantics for classical logic, but we define the notion of exact verification similarly to Fine’s inexact notion of loose verification. Dropping Fine’s principle of the Downward Closure of the set of consistent states makes our verification notion nevertheless exact. In order to prove soundness and completeness of the logics w.r.t. the new semantics, we make a detour via a sequent calculus that is adequate both for the four relevance logics and for the corresponding semantics’ exact consequence notion, i.e. each exact verifier of the premises exactly verifies the conclusion. The sequent calculus is interesting in its own right. Finally, we argue that the four presented truthmaker semantics are also interesting semantics for the original (irrelevant) consequence relations FDE, LP, K3, and CL. The most interesting difference with Fine’s approach (seen as a semantics for CL) is the way in which tautologies are handled: next to their usual verifiers, they are also made true by the empty state. We provide philosophical arguments for the plausibility of such an account