Extensions of paraconsistent weak Kleene logic

2020 ◽  
Author(s):  
Francesco Paoli ◽  
Michele Pra Baldi
Keyword(s):  
General Result ◽  
Classical Logic ◽  
Order Ideal ◽  
Consequence Relation ◽  
Reduced Models ◽  
Logic Of Paradox

Abstract Paraconsistent weak Kleene ($\textrm{PWK}$) logic is the $3$-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α$\textrm{lg} (\textrm{PWK})$. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, $\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\textrm{PWK}_{\textrm{E}}\textrm{,}$ PWK logic plus explosion. This $6$-valued logic, unlike $\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models.

Subprevarieties versus extensions. Application to the logic of paradox

2000 ◽  
Vol 65 (2) ◽  
pp. 756-766 ◽  
Author(s):  
Alexej P. Pynko
Keyword(s):  
General Result ◽  
Classical Logic ◽  
Modus Ponens ◽  
Galois Connection ◽  
Normal Extension ◽  
Material Implication ◽  
Element Chain ◽  
Class Of Matrices ◽  
The One ◽  
Logic Of Paradox

AbstractIn the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule.

scholarly journals A non-transitive relevant implication corresponding to classical logic consequence

2019 ◽  
Vol 16 (2) ◽  
pp. 10
Author(s):  
Peter Verdée ◽  
Inge De Bal ◽  
Aleksandra Samonek
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Modus Ponens ◽  
Relevant Logic ◽  
Object Language ◽  
Cut Rule ◽  
Consequence Relation ◽  
Relevant Implication ◽  
Relevant Logics ◽  
Traditional Sense

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.

On the Decidability and Complexity of Autoepistemic Reasoning1

1992 ◽  
Vol 17 (1-2) ◽  
pp. 117-155
Author(s):  
Ilkka N.F. Niemelä
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Classical Logic ◽  
General Setting ◽  
Decision Problems ◽  
Consequence Relation ◽  
Autoepistemic Logic ◽  
Complete Problems ◽  
Decidability And Complexity ◽  
Polynomial Time Hierarchy

The decidability and computational complexity of autoepistemic reasoning is investigated in a general setting where the autoepistemic logic CLae built on top of a given classical logic CL is studied. Correct autoepistemic conclusions from a set of premises are defined in terms of expansions of the premises. Three classes of expansions are studied: Moore style stable expansions, enumeration based expansions, and L-hierarchic expansions. A simple finitary characterization for each type of expansions in CLae is developed. Using the characterizations conditions ensuring that a set of premises has at least one or exactly one stable expansion can be stated and an upper bound for the number of stable expansions of a set of premises can be given. With the aid of the finitary characterizations results on decidability and complexity of autoepistemic reasoning are obtained. E.g., it is shown that autoepistemic reasoning based on each of the three types of expansions is decidable if the monotonic consequence relation given by the underlying classical logic is decidable. In the propositional case decision problems related to the three classes of expansions are shown to be complete problems with respect to the second level of the polynomial time hierarchy. This implies that propositional autoepistemic reasoning is strictly harder than classical propositional reasoning unless the polynomial time hierarchy collapses.

Elementary intuitionistic theories

1973 ◽  
Vol 38 (1) ◽  
pp. 102-134 ◽  
Author(s):  
C. Smorynski
Keyword(s):  
Intuitionistic Logic ◽  
Model Theory ◽  
General Result ◽  
Classical Logic ◽  
Decision Problem ◽  
Quantifier Elimination ◽  
Kripke Model ◽  
Logic Model ◽  
Intuitionistic Theory ◽  
Classical Extension

The present paper concerns itself primarily with the decision problem for formal elementary intuitionistic theories and the method is primarily model-theoretic. The chief tool is the Kripke model for which the reader may find sufficient background in Fitting's book Intuitionistic logic model theory and forcing (North-Holland, Amsterdam, 1969). Our notation is basically that of Fitting, the differences being to favor more standard notations in various places.The author owes a great debt to many people and would particularly like to thank S. Feferman, D. Gabbay, W. Howard, G. Kreisel, G. Mints, and R. Statman for their valuable assistance.The method of elimination of quantifiers, which has long since proven its use in classical logic, has also been applied to intuitionistic theories (i) to demonstrate decidability ([9], [15], [17]), (ii) to prove the coincidence of an intuitionistic theory with its classical extension ([9], [17]), and (iii), as we will see below, to establish relations between an intuitionistic theory and its classical extension. The most general of these results is to be obtained from the method of Lifshits' quantifier elimination for the intuitionistic theory of decidable equality.Since the details of Lifshits' proof have not been published, and since the proof yields a more general result than that stated in his abstract [15], we include the proof and several corollaries.

PARACONSISTENT CIRCUMSCRIPTION: PRELIMINARY REPORT

1996 ◽  
Vol 10 (06) ◽  
pp. 679-686 ◽  
Author(s):  
ZUOQUAN LIN
Keyword(s):  
Classical Logic ◽  
Preliminary Report ◽  
Paraconsistent Logic ◽  
Nonmonotonic Logic ◽  
Logic Of Paradox

In this paper we describe the paraconsistent circumscription by the application of predicate circumscription in a paraconsistent logic, the logic of paradox LP. In addition to circumscribing the predicates, we also circumscribe the inconsistency. The paraconsistent circumscription can be well characterized by the minimal semantics which is both nonmonotonic and paraconsistent. It brings us advantages in two respects: nonmonotonic logic would be nontrivial while there was a contradiction, and paraconsistent logic would be equivalent to classical logic while there was no effect of a contradiction.

Hybrid Reduced Model of Rotor

2013 ◽  
Vol 60 (3) ◽  
pp. 319-333
Author(s):  
Rafał Hein ◽  
Cezary Orlikowski
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Hybrid Model ◽  
Rotor System ◽  
Reduced Model ◽  
Order Model ◽  
Gyroscopic Effect ◽  
Reduced Models ◽  
Rigid Finite Element Method ◽  
Low Order

Abstract In the paper, the authors describe the method of reduction of a model of rotor system. The proposed approach makes it possible to obtain a low order model including e.g. non-proportional damping or the gyroscopic effect. This method is illustrated using an example of a rotor system. First, a model of the system is built without gyroscopic and damping effects by using the rigid finite element method. Next, this model is reduced. Finally, two identical, low order, reduced models in two perpendicular planes are coupled together by means of gyroscopic and damping interaction to form one model of the system. Thus a hybrid model is obtained. The advantage of the presented method is that the number of gyroscopic and damping interactions does not affect the model range

New method for building ASP knowledge base from knowledge in classical logic

2010 ◽  
Vol 30 (11) ◽  
pp. 2932-2936
Author(s):  
Ling-zhong ZHAO ◽  
Xue-song WANG ◽  
Jun-yan QIAN ◽  
Guo-yong CAI
Keyword(s):  
Knowledge Base ◽  
Classical Logic ◽  
New Method

The Argument from Counterfactuals

2018 ◽  
Author(s):  
Alexander R. Pruss
Keyword(s):  
Natural Law ◽  
Classical Logic ◽  
Divine Command

It seems that counterfactuals and many other statements are subject to semantic underdetermination. Classical logic pushes one to an epistemicist account of this underdetermination, but epistemicism seems implausible. However epistemicism can be made plausible when conjoined with a divine institution account of meaning. This gives us some reason to accept that divine institution account, and hence some reason to think that God exists. This chapter evaluates the arguments for epistemicism and divine institution, including objections, and incorporates Plantinga’s consideration of counterfactuals when it comes to theism. In particular, an analogy is drawn with divine command and natural law theories in ethics.

Sign in / Sign up

Export Citation Format

Share Document