The strong relevance logics

The tautology p - q - p is not a theorem of the various relevance logics (see Anderson and Belnap [1]) because q is not considered to be relevant in the derivation of final p. We can take this lack of relevance to mean simply that p-q-p could have been proved without q and its -, i.e., p-p. By the same criterion we could say that in ((p-p) -q) -q p-p is not relevant. In general we will say that any theorem A of an implicational logic is strongly relevant if there is no subpart B ! which can be removed from A, leaving the rest still a theorem of the same logic. Such a subpart B - is said to be superfluous

Choosing Your Nonmonotonic Logic: A Shopper’s Guide

The paper presents an exhaustive menu of nonmonotonic logics. The options are individuated in terms of the principles they reject. I locate, e.g., cumulative logics and relevance logics on this menu. I highlight some frequently neglected options, and I argue that these neglected options are particularly attractive for inferentialists

Modal and Relevance Logics for Qualitative Spatial Reasoning

Qualitative Spatial Reasoning (QSR) is an alternative technique to represent spatial relations without using numbers. Regions and their relationships are used as qualitative terms. Mostly peer qualitative spatial reasonings has two aspect: (a) the first aspect is based on inclusion and it focuses on the ”part-of” relationship. This aspect is mathematically covered by mereology. (b) the second aspect focuses on topological nature, i.e., whether they are in ”contact” without having a common part. Mereotopology is a mathematical theory that covers these two aspects. The theoretical aspect of this thesis is to use classical propositional logic with non-classical relevance logic to obtain a logic capable of reasoning about Boolean algebras i.e., the mereological aspect of QSR. Then, we extended the logic further by adding modal logic operators in order to reason about topological contact i.e., the topological aspect of QSR. Thus, we name this logic Modal Relevance Logic (MRL). We have provided a natural deduction system for this logic by defining inference rules for the operators and constants used in our (MRL) logic and shown that our system is correct. Furthermore, we have used the functional programming language and interactive theorem prover Coq to implement the definitions and natural deduction rules in order to provide an interactive system for reasoning in the logic

Structural completeness in relevance logics

It is proved that the relevance logic R (without sentential constants) has no structurally complete consistent axiomatic extension, except for classical propositional logic. In fact, no other such extension is even passively structurally complete.http://link.springer.com/journal/112252017-06-30hb201

Singly generated quasivarieties and residuated structures

A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably generated algebra in K can serve as A. A consequence of this demand, called "passive structural completeness" (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K-simple algebra. It is a minimal quasivariety if, moreover, it is PSC but fails to unify some finite set of equations. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC iff its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics

Relevance logics, paradoxes of consistency and the K rule II. A non-constructive negation

The logic B+ is Routley and Meyer’s basic positive logic. We define the logics BK+ and BK'+ by adding to B+ the K rule and to BK+ the characteristic S4 axiom, respectively. These logics are endowed with a relatively strong non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency

"A Smack of Irrelevance" in Inconsistent Mathematics?

Recently, some proponents and practitioners of inconsistent mathe- matics have argued that the subject requires a conditional with ir- relevant features, i.e. where antecedent and consequent in a valid conditional do not behave as expected in relevance logics —by shar- ing propositional variables, for example. Here we argue that more fine-grained notions of content and content-sharing are needed to ex- amine the language of (inconsistent) arithmetic and set theory, and that the conditionals needed in inconsistent mathematics are not as irrelevant as it is suggested in the current literature