A decidable paraconsistent relevant logic: Gentzen system and Routley-Meyer semantics

2016 ◽  
Vol 62 (3) ◽  
pp. 177-189 ◽  
Author(s):  
Norihiro Kamide
Keyword(s):  
Relevant Logic ◽  
Gentzen System

Simple Gentzenizations for the normal formulae of contraction-less logics

1996 ◽  
Vol 61 (4) ◽  
pp. 1321-1346
Author(s):  
Ross T. Brady
Keyword(s):  
Relevant Logic ◽  
Conservative Extension ◽  
Structural Rules ◽  
Relevant Logics ◽  
Innovative Method ◽  
Gentzen System ◽  
Branching Rules ◽  
Good Range

In [1], we established Gentzenizations for a good range of relevant logics with distribution, but, in the process, we added inversion rules, which involved extra structural connectives, and also added the sentential constant t. Instead of eliminating them, we used conservative extension results to relate them back to the original logics. In [4], we eliminated the inversion rules and t and established a much simpler Gentzenization for the weak sentential relevant logic DW, and also for its quantificational extension DWQ, but a restriction to normal formulae (defined below) was required to enable these results to be proved. This method was quite general and hope was expressed about extending it to other relevant logics.In this paper, we develop an innovative method, which makes essential use of this restriction to normality, to establish two simple Gentzenizations for the normal formulae of the slightly weaker logic B, and then extend the method to other sentential contraction-less logics. To obtain the first of these Gentzenizations, for the logics B and DW, we remove the two branching rules (F&) and (T∨), together with the structural connective ‘,’, to simplify the elimination of the inversion rules and t. We then eliminate the rules (T&) and (F∨), thus reducing the Gentzen system to one containing only ˜ and → and their four associated rules, and reduce the remaining types of structures to four simple finite types. Subsequently, we re-introduce (T&) and (F∨), and also (F&) and (T∨), to obtain the second Gentzenization, which contains ‘,’ but no structural rules.

scholarly journals Why Intuitionistic Relevant Logic Cannot Be a Core Logic

2017 ◽  
Vol 58 (2) ◽  
pp. 241-248
Author(s):  
Joseph Vidal-Rosset
Keyword(s):  
Relevant Logic

scholarly journals Reasoning-predicting Model based on Strong Relevant Logic in Road Traffic Forecasting

2012 ◽  
Vol 33 ◽  
pp. 1105-1110 ◽  
Author(s):  
Dancheng Li ◽  
Zhiliang Liu ◽  
Cheng Liu ◽  
Binsheng Liu ◽  
Wei Zhang
Keyword(s):  
Road Traffic ◽  
Relevant Logic ◽  
Predicting Model ◽  
Traffic Forecasting ◽  
Model Based

BILATERAL RELEVANT LOGIC

2014 ◽  
Vol 7 (2) ◽  
pp. 250-272
Author(s):  
NISSIM FRANCEZ
Keyword(s):  
Relevant Logic

Whither relevant arithmetic?

1992 ◽  
Vol 57 (3) ◽  
pp. 824-831 ◽  
Author(s):  
Harvey Friedman ◽  
Robert K. Meyer
Keyword(s):  
Relevant Logic ◽  
Peano Arithmetic ◽  
Central Question ◽  
Proof Methods ◽  
Classical Proof ◽  
Natural Way

AbstractBased on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R#, which does formalize arithmetic on relevant principles, but also admits γ in a natural way?

scholarly journals Boolean negation and non-conservativity II: The variable-sharing property

2020 ◽  
Author(s):  
Tore Fjetland Øgaard
Keyword(s):  
Classical Logic ◽  
Relevant Logic ◽  
Relevant Logics ◽  
Acute Form ◽  
Vast Range ◽  
Boolean Extension

Abstract Many relevant logics are conservatively extended by Boolean negation. Not all, however. This paper shows an acute form of non-conservativeness, namely that the Boolean-free fragment of the Boolean extension of a relevant logic need not always satisfy the variable-sharing property. In fact, it is shown that such an extension can in fact yield classical logic. For a vast range of relevant logic, however, it is shown that the variable-sharing property, restricted to the Boolean-free fragment, still holds for the Boolean extended logic.

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