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.

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.

scholarly journals Deformation of finite morphisms and smoothing of ropes

2008 ◽  
Vol 144 (3) ◽  
pp. 673-688 ◽  
Author(s):  
Francisco Javier Gallego ◽  
Miguel González ◽  
Bangere P. Purnaprajna
Keyword(s):  
General Theory ◽  
Projective Space ◽  
General Result ◽  
The Other ◽  
Independent Interest ◽  
Higher Dimension ◽  
Smooth Curves ◽  
Other Hand ◽  
Finite Covers ◽  
The One

AbstractIn this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1:1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.

Linear logic

2018 ◽  
Author(s):  
G.M. Bierman
Keyword(s):  
Water Molecule ◽  
Classical Logic ◽  
Linear Logic ◽  
Modus Ponens ◽  
Double Negation ◽  
Current State ◽  
Logical Connectives ◽  
Excluded Middle ◽  
Intuitionistic Logics ◽  
Oxygen Atoms

Linear logic was introduced by Jean-Yves Girard in 1987. Like classical logic it satisfies the law of the excluded middle and the principle of double negation, but, unlike classical logic, it has non-degenerate models. Models of logics are often given only at the level of provability, in that they provide denotations of formulas. However, we are also interested in models which provide denotations of deductions, or proofs. Given such a model two proofs are said to be equivalent if their denotations are equal. A model is said to be ‘degenerate’ if there are no formulas for which there exist at least two non-equivalent proofs. It is easy to see that models of classical logic are essentially degenerate because any formula is either true or false and so all proofs of a formula are considered equivalent. The intuitionist approach to this problem involves altering the meaning of the logical connectives but linear logic attacks the very connectives themselves, replacing them with more refined ones. Despite this there are simple translations between classical and linear logic. One can see the need for such a refinement in another way. Both classical and intuitionistic logics could be said to deal with static truths; both validate the rule of modus ponens: if A→B and A, then B; but both also validate the rule if A→B and A, then A∧B. In mathematics this is correct since a proposition, once verified, remains true – it persists. Many situations do not reflect such persistence but rather have an additional notion of causality. An implication A→B should reflect that a state B is accessible from a state A and, moreover, that state A is no longer available once the transition has been made. An example of this phenomenon is in chemistry where an implication A→B represents a reaction of components A to yield B. Thus if two hydrogen and one oxygen atoms bond to form a water molecule, they are consumed in the process and are no longer part of the current state. Linear logic provides logical connectives to describe such refined interpretations.

Extensions of the Lewis system S5

1951 ◽  
Vol 16 (2) ◽  
pp. 112-120 ◽  
Author(s):  
Schiller Joe Scroggs
Keyword(s):  
Broad Class ◽  
Characteristic Matrix ◽  
Normal Extension ◽  
Strict Implication ◽  
Finite Characteristic ◽  
Material Implication ◽  
Sentential Calculus ◽  
The Third ◽  
Simple Class ◽  
Definition Of

Dugundji has proved that none of the Lewis systems of modal logic, S1 through S5, has a finite characteristic matrix. The question arises whether there exist proper extensions of S5 which have no finite characteristic matrix. By an extension of a sentential calculus S, we usually refer to any system S′ such that every formula provable in S is provable in S′. An extension S′ of S is called proper if it is not identical with S. The answer to the question is trivially affirmative in case we make no additional restrictions on the class of extensions. Thus the extension of S5 obtained by adding to the provable formulas the additional formula p has no finite characteristic matrix (indeed, it has no characteristic matrix at all), but this extension is not closed under substitution—the formula q is not provable in it. McKinsey and Tarski have defined normal extensions of S4* by imposing three conditions. Normal extensions must be closed under substitution, must preserve the rule of detachment under material implication, and must also preserve the rule that if α is provable then ~◊~α is provable. McKinsey and Tarski also gave an example of an extension of S4 which satisfies the first two of these conditions but not the third. One of the results of this paper is that every extension of S5 which satisfies the first two of these conditions also satisfies the third, and hence the above definition of normal extension is redundant for S5. We shall therefore limit the extensions discussed in this paper to those which are closed under substitution and which preserve the rule of detachment under material implication. These extensions we shall call quasi-normal. The class of quasi-normal extensions of S5 is a very broad class and actually includes all extensions which are likely to prove interesting. It is easily shown that quasi-normal extensions of S5 preserve the rules of replacement, adjunction, and detachment under strict implication. It is the purpose of this paper to prove that every proper quasi-normal extension of S5 has a finite characteristic matrix and that every quasi-normal extension of S5 is a normal extension of S5 and to describe a simple class of characteristic matrices for S5.

Strong completeness of fragments of the propositional calculus

1951 ◽  
Vol 16 (3) ◽  
pp. 204-204 ◽  
Author(s):  
Alan Rose
Keyword(s):  
Propositional Calculus ◽  
Modus Ponens ◽  
Primitive Function ◽  
Strong Completeness ◽  
Substitution Rule ◽  
Material Implication ◽  
Rules Of Procedure

There has recently been developed a method of formalising any fragment of the propositional calculus, subject only to the condition that material implication is a primitive function of the fragmentary system considered. Tarski has stated, without proof, that when implication is the only primitive function a formulation which is weakly complete (i.e., has as theorems all expressible tautologies) is also strongly complete (i.e., provides for the deduction of any expressible formula from any which is not a tautology). The methods used by Henkin suggest the following proof of theTheorem. If in a fragment of the propositional calculus material implication can be defined in terms of the primitive functions, then any weakly complete formalisation of the fragmentary system which has for rules of procedure the substitution rule and modus ponens is also strongly complete.

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.

An integer-valued expression of Dedekind sums

2017 ◽  
Vol 13 (05) ◽  
pp. 1253-1259
Author(s):  
Simon Macourt
Keyword(s):  
General Result ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
The One

We prove a conjecture of Myerson and Phillips on when an expression involving Dedekind sums is an integer. We also provide a more general result and use this to extend the work of Myerson and Phillips studying whether the points of the graph of the one-variable Dedekind sum that fall on the line are dense.

Classical logic, conditionals and “nonmonotonic” reasoning

2009 ◽  
Vol 32 (1) ◽  
pp. 85-85
Author(s):  
Nicholas Allott ◽  
Hiroyuki Uchida
Keyword(s):  
Classical Logic ◽  
Nonmonotonic Reasoning ◽  
Modus Ponens

AbstractReasoning with conditionals is often thought to be non-monotonic, but there is no incompatibility with classical logic, and no need to formalise inference itself as probabilistic. When the addition of a new premise leads to abandonment of a previously compelling conclusion reached by modus ponens, for example, this is generally because it is hard to think of a model in which the conditional and the new premise are true.

One-loop divergences

2021 ◽  
pp. 363-387
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
General Result ◽  
Asymptotic Freedom ◽  
Yukawa Model ◽  
Yang Mills ◽  
Starting Point ◽  
The One ◽  
Mills Field ◽  
Matter Fields

This chapter focuses on one-loop calculations and related issues such as practical renormalization and the derivation of beta functions. The general result for the one-loop divergences from chapter 13 is applied to a sequence of practical calculations. The starting point is the derivation of vacuum divergences of free matter fields. The beta functions in the vacuum sector are calculated. Asymptotic freedom is discussed. In addition, examples of one-loop divergences in interacting theories are elaborated, including the Yang-Mills field coupled to fermions and scalars, and the Yukawa model.

Enthymematic classical recapture 1

2018 ◽  
Vol 28 (5) ◽  
pp. 817-831
Author(s):  
Henrique Antunes
Keyword(s):  
Classical Logic ◽  
Paraconsistent Logics ◽  
Logical Theory ◽  
Oxford University ◽  
The One ◽  
Oxford University Press

AbstractPriest (2006, Ch.8, 2nd edn. Oxford University Press), argues that classical reasoning can be made compatible with his preferred (paraconsistent) logical theory by proposing a methodological maxim authorizing the use of classical logic in consistent situations. Although Priest has abandoned this proposal in favour of the one in G. Priest (1991, Stud. Log., 50, 321–331), I shall argue that due to the fact that the derivability adjustment theorem holds for several logics of formal (in)consistency (cf. W. A. Carnielli and M. E. Coniglio, 2016, Springer), these paraconsistent logics are particularly well suited to accommodate classical reasoning by means of a version of that maxim, yielding thus an enthymematic account of classical recapture.

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