Content, Context, and Logic

2021 ◽  
pp. 143-152
Author(s):  
Una Stojnić

Non-propositionalism has additionally been fueled by the fact that modals and conditionals seem to give raise to failure of classical patterns of inference, for instance, modus ponens and modus tollens. Since non-propositionalist accounts typically invalidate some of these patterns of inference, the apparent counterexamples have been taken as further data in support of such treatments. This chapter argues that this is a mistaken reaction to the apparent counterexamples. The seeming violations of classical patterns of inference yet again result from a faulty assumption of the extra-linguistic account of context-sensitivity resolution, and a failure to account for the effects of discourse conventions on the interpretation of modal discourse. The chapter’s account of modality, which treats them as prominence-sensitive expressions whose interpretation is determined by discourse conventions, predicts and explains the apparent counterexamples, while nevertheless preserving classical logic.

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.


2021 ◽  
pp. 153-168
Author(s):  
Una Stojnić

This chapter develops a formal model of context-sensitivity of modal discourse. Much like demonstrative pronouns, modals are prominence-sensitive, selecting the most prominent candidate interpretation. The prominence ranking of candidate interpretations is recorded in the conversational record, and is maintained through the effects of discourse conventions represented in the logical form of a discourse. In this way arguments are individuated as structured discourses that underwrite a particular propositional pattern. It is shown that such account provably preserves classical logic. Further, this chapter argues that its model offers a more satisfactory account of the individuation of argument patterns in natural language discourse then the competing alternatives. Any adequate account, it is here argued, will have to take into account not just the contribution of individual sentences, but also of discourse conventions. Indeed, the contribution of discourse conventions is crucial for determining the contribution of individual sentences in the first place.


Author(s):  
G.M. Bierman

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.


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.


2020 ◽  
Vol 8 ◽  
pp. 73-89
Author(s):  
Sonil Kwak ◽  
Unha Kim ◽  
Kumju Kim ◽  
Ilmyong Son ◽  
Chonghan Ri

This paper shows a basic and original fuzzy reasoning method that can draw a novel study direction of the approximate inference in fuzzy systems with uncertainty. Firstly we propose a criterion function for checking of the reductive property about fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT). Secondly unlike fuzzy reasoning methods based on the similarity measure, we propose a principle of new fuzzy reasoning method based on distance measure and then present two theorems for FMP and FMT. Thirdly through the several computational experiments, we show that proposed method is simple and effective, and in accordance with human thinking. Finally we pointed out conclusion that proposed method does satisfy the convergence of the fuzzy control and has not information loss.


1970 ◽  
Vol 32 (1) ◽  
Author(s):  
Enric Trillas ◽  
Itziar García-Honrado
Keyword(s):  

El cálculo proposicional se basa proposiciones que son tautologías, antilogías y de tipo contingente, construidas con las operaciones de disyunción, conjunción, y negación, así como con el llamado condicional material para trasladar los enunciados condicionales. Con todo ello se obtiene una estructura de álgebra de Boole, que permite equiparar el cálculo proposicional al cálculo algebraico booleano. En parte, la simplificación conceptual que se obtiene resulta de ser el orden parcial del álgebra equivalente a la afirmación del condicional material. Además, con tal metodología cabe ver qué partes del cálculo proposicional clásico pueden trasladarse, aun con limitaciones, a estructuras más débiles y, a la vez, más generales, como son los ortoretículos no booleanos o las álgebras de De Morgan.El cálculo proposicional trata con la deducción. Por ello, el artículo lo desarrolla a partir de un operador de consecuencias en el sentido de Tarski. Ampliado el modelo deductivo, es fácil ver otras consecuencias que las usualmente consideradas en los textos elementales; en particular, se analizan cuatro esquemas de razonamiento deductivo típicos y, entre ellos, se presta especial atención a los usuales Modus Ponens y Modus Tollens. En el caso de las estructuras algebraicas de conjuntos ‘borrosos’, ninguna es de álgebra de Boole ni siquiera de orto-retículo y sólo cabe trabajar, en algunos casos, con álgebras de De Morgan-Kleene. Por lo tanto el cálculo proposicional ‘fuzzy’ debe construirse de acuerdo con la estructura algebraica que corresponda en cada caso y una vez elegida como representación del condicional una función que verifique el Modus Ponens (‘condicional borroso’).


Author(s):  
Suzanne M. Egan ◽  
Ruth M. J. Byrne

We examine how people understand and reason from counterfactual threats, for example, “if you had hit your sister, I would have grounded you” and counterfactual promises, for example, “if you had tidied your room, I would have given you ice-cream.” The first experiment shows that people consider counterfactual threats, but not counterfactual promises, to have the illocutionary force of an inducement. They also make the immediate inference that the action mentioned in the “if” part of the counterfactual threat and promise did not occur. The second experiment shows that people make more negative inferences (modus tollens and denial of the antecedent) than affirmative inferences (modus ponens and affirmation of the consequent) from counterfactual threats and promises, unlike indicative threats and promises. We discuss the implications of the results for theories of the mental representations and cognitive processes that underlie conditional inducements.


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