scholarly journals The Laws of Natural Deduction in Inference by DNA Computer

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Łukasz Rogowski ◽  
Petr Sosík

We present a DNA-based implementation of reaction system with molecules encoding elements of the propositional logic, that is, propositions and formulas. The protocol can perform inference steps using, for example,modus ponensandmodus tollensrules and de Morgan’s laws. The set of the implemented operations allows for inference of formulas using the laws of natural deduction. The system can also detect whether a certain propositionacan be deduced from the basic facts and given rules. The whole protocol is fully autonomous; that is, after introducing the initial set of molecules, no human assistance is needed. Only one restriction enzyme is used throughout the inference process. Unlike some other similar implementations, our improved design allows representing simultaneously a factaand its negation ~a, including special reactions to detect the inconsistency, that is, a simultaneous occurrence of a fact and its negation. An analysis of correctness, completeness, and complexity is included.

2013 ◽  
Vol 11 (1) ◽  
pp. 2-7 ◽  
Author(s):  
V. Sgurev

Abstract In decision making logic it is often necessary to solve logical equations for which, due to the features of disjunction and conjunction, no admissible solutions exist. In this paper an approach is suggested, in which by the introduction of Imaginary Logical Variables (ILV), the classical propositional logic is extended to a complex one. This provides a possibility to solve a large class of logical equations.The real and imaginary variables each satisfy the axioms of Boolean algebra and of the lattice. It is shown that the Complex Logical Variables (CLV) observe the requirements of Boolean algebra and the lattice axioms. Suitable definitions are found for these variables for the operations of disjunction, conjunction, and negation. A series of results are obtained, including also the truth tables of the operations disjunction, conjunction, negation, implication, and equivalence for complex variables. Inference rules are deduced for them analogous to Modus Ponens and Modus Tollens in the classical propositional logic. Values of the complex variables are obtained, corresponding to TRUE (T) and FALSE (F) in the classic propositional logic. A conclusion may be made from the initial assumptions and the results achieved, that the imaginary logical variable i introduced hereby is “truer” than condition “T” of the classic propositional logic and i - “falser” than condition “F”, respectively. Possibilities for further investigations of this class of complex logical structures are pointed out


2019 ◽  
Author(s):  
Matheus Pereira Lobo

All nine axioms and a single inference rule of logic (Modus Ponens) within the Hilbert axiomatic system are presented using capital letters (ABC) in order to familiarize the beginner student in hers/his first contact with the topic.


Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1409
Author(s):  
Marija Boričić Joksimović

We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A→B, and B→C have probabilities a, b, c, r, and s, respectively, then for probability p of A→C, we have f(a,b,c,r,s)≤p≤g(a,b,c,r,s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner.


Author(s):  
ANA PRADERA ◽  
ENRIC TRILLAS ◽  
SUSANA CUBILLO

This paper investigates the use of functions other than t-norms to model the Modus Ponens rule in a fuzzy inference process. For that purpose, new definitions for fuzzy inference related concepts are suggested, that take into account the possibility of using a larger class of functions. In particular, the concept of "Modus Ponens generating function" is revisited, allowing to find out when and where (in which subset of the defined universe) an operator is able to generate the Modus Ponens scheme. In addition, given such an operator, the conditional relations that may be used along with it to model an inference process are found. These results are applied to some common operators, finding their Modus Ponens generation capacity as well as their corresponding residuated fuzzy conditionals. Finally, the relation between an operator's ability to describe the Modus Ponens rule and its conjunctive/disjunctive behaviour is also studied, by means of a series of sufficient and/or necessary conditions relating both concepts.


1990 ◽  
Vol 55 (1) ◽  
pp. 90-105 ◽  
Author(s):  
J. Roger Hindley ◽  
David Meredith

The condensed detachment rule, or ruleD, was first proposed by Carew Meredith in the 1950's for propositional logic based on implication. It is a combination of modus ponens with a “minimal” amount of substitution. We shall give a precise detailed statement of rule D. (Some attempts in the published literature to do this have been inaccurate.)The D-completeness question for a given set of logical axioms is whether every formula deducible from the axioms by modus ponens and substitution can be deduced instead by rule D alone. Under the well-known formulae-as-types correspondence between propositional logic and combinator-based type-theory, rule D turns out to correspond exactly to an algorithm for computing principal type-schemes in combinatory logic. Using this fact, we shall show that D is complete for intuitionistic and classical implicational logic. We shall also show that D is incomplete for two weaker systems, BCK- and BCI-logic.In describing the formulae-as-types correspondence it is common to say that combinators correspond to proofs in implicational logic. But if “proofs” means “proofs by the usual rules of modus ponens and substitution”, then this is not true. It only becomes true when we say “proofs by rule D”; we shall describe the precise correspondence in Corollary 6.7.1 below.This paper is written for readers in propositional logic and in combinatory logic. Since workers in one field may not feel totally happy in the other, we include short introductions to both fields.We wish to record thanks to Martin Bunder, Adrian Rezus and the referee for helpful comments and advice.


Author(s):  
Maria Dias ◽  
Antonio Roazzi

O objetivo deste estudo foi investigar o efeito do modo de faz-deconta, formas de silogismos e tipos de fatos, no desempenho de crianças inglesas de 5 anos de idade, escolarizadas, de nivel sócio-econômico (NSE) médio e de crianças brasileiras de 5 anos de idade, nao escolarizadas de NSE baixo. Os resultados mostraram que as crianças, independentemente se escolarizadas ou nao, produziram maior número de respostas corretas e maior número de justificativas teóricas na condição de brincadeira de faz-de-conta do que na condição verbal comum. Este efeito foi especialmente marcante para os fatos desconhecidos e para os fatos contrários às experiências das crianças. Este padrão foi mantido para ambas formas de silogismos, embora em geral o desempenho das crianças nos problemas envolvendo Modus Ponens tenha sido mais acurada do que nos problemas envolvendo Modus Tollens. As crianças não-escolarizadas obtiveram menores escores do que as escolarizadas. No entanto, esta diferença não foi tão forte como aquela relatada por Scribner (1977) onde as crianças não-escolarizadas obtiveram baixos escores adotando um viés empírico. Em nosso estudo, apesar das crianças não-escolarizadas terem freqüentemente recorrido a justificativas arbitrárias, puderam recorrer a uma atitude teórica quando a condição de faz-de-conta foi adotada. Abstract The main aim of this study was to investigate the effect of makebelieve mode, form of syllogisms and type of fact in 5-year-old schooled children from medium SES families in England and in non literate 5- year-old unschooled children from low SES families in Brazil. This study was a test of the claim that schooling is a pre requesite for deductive reasoning. The results showed that children, English or Brazilian, produced more correct responses and theoretical justifications in the makebelieve play condition than in the standard verbal mode. This was especially true for unknown facts and contrary facts. This pattern held for both form forms of syllogism, although in general children 's performance on Modus Ponens inferences was more accurate than on Modus Tollens inferences. Unschooled children s performance was poorer than schooled children s performance. However, this difference was not so strong as that found in the studies reported by Scribner (1977) where unschooled children performed at chance level, adopting the empirical bias. In this study, the unschooled children, although they very often appealed to arbitraty justifications, could adopt a theoretical attitude when the makebelieve mode was used. Résumé Ce travail de recherche a eu comme objectif principal l'investigation de Ì'effet du mode faire-semblant, des formes de syllogisme et des types défaits sur la réussite d'enfants anglais de cinq ans, scolarisés, niveau socio-économique moyen, et d'enfants brésiliens de même âge, nonscolarisés et de bas niveau socio-économique. Les résultats indiquent aue les enfants, scolarisés ou non, ont produit un plus grand nombre de réponses correctes, ainsi que de justificatives théoriques dans la condiction experimentale faire-semblant, par rapport à la condiction verbale commune. Cet effet a été plus net en ce qui concerne des faits nconnus et des faits contraires aux expériences des enfants. Ce résultat a été observé pour ¡ es deux formes de syllogisme étudiées, quoi que, en général, le nombre de réponses correctes a été plus important chez les enfants travaillant avec des problèmes em Modus Ponens, par raport à ceux travaillant sous modus Tollens. Les enfants non-scolarisés ont obtenu des scores plus bas que ceux des enfants scolarisés. Néanmoins, cette différence n 'a pas été aussi forte que celle mentionnée par Scribner (1977), qui a pu constater des scores bas chez des enfants non-scolarisés adoptant un biais empirique. Dans notre étude, bien que les enfants nonscolarisés aient fréquemment fait appel à des justifications arbitraires, ils ont, d'autre part, eu recours à une attitude théorique quand la condiction faire-semblant a été proposée. Resumen El objetivo principal de este estudio fue investigar el efecto del modo de fingir formas de silogismos y tipos de hechos en el rendimiento de niños ingleses de 5 anos de edad, escolarizadas, de NSE médio y de linos brasileños de 5 anos de edad, sin escolaridad de NSE bajo. Los resulados mostraron que ¡os niños, independientemente de si eran escolarizados o no, producían mayor número de respuestas correctas e mayor número de justificativas teóricas en la condición del juego de fingir que en la condición verbal común. Este efecto fue especialmente marcante para los hechos desconocidos y para los hechos contrários a Ias experiencias de los ninos. Este padrón fue conservado para ambas formas de silogismos, no obstante, en general el rendimiento de los ninos en los problemas envolviendo Modus Ponens haya sido mas preciso que en los problemas envolviendo Modus Tollens. Los ninos sin escolaridad obtuvieron resultados menores que los escolarizados. Sin embargo, esta diferencia no fue tan fuerte como aquella relatada por Scribner (1977) donde los ninos sin escolaridad obtuvieron bajos resultados adoptando una inclinación empírica. En nuestro estúdio, a pesar de los ninos sin escolaridad haber recurrido frecuentemente a justificativas arbitrarias, estos adoptaron una actitud teórica cuando fue usada Ia condición de fingir.


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