Heyting’s formalization of intuitionistic mathematics started many discussions about the meaning of the logical connectives in terms of proof and construction. We focus on the ideas and results related to the interpretation of implication and on formal systems that have different rules for implication. Some of these systems are not intended to contribute to the discussions mentioned above, but ...

A comparative study of the implication functions and t-norms behavior when they are used as implication operators in the fuzzy control inference process is carried out in this paper. In order to do it, the widely studied application of the Inverted Pendulum is considered. Moreover, a comparison methodology of the different fuzzy logic controllers designed is defined with the purpose of get more...

Before beginning the research that led to "Probabilistic logic" [11 ], I had participated with Richard Duda, Peter Hart, and Georgia Sutherland on the PROSPECTOR project [3]. There, we used Bayes' rule (with some assumptions about conditional independence) to deduce the probabilities of hypotheses about ore deposits given (sometimes uncertain) geologic evidence collected in the field [4]. At th...

This paper defends two theses about probabilistic reasoning. First, although modus ponens has a probabilistic analog, modus tollens does not – the fact that a hypothesis says that an observation is very improbable does not entail that the hypothesis is improbable. Second, the evidence relation is essentially comparative; with respect to hypotheses that confer probabilities on observation statem...

We discuss the problem of construction of inference procedures which can manipulate with uncertainties measured in ordinal scales and fulfil to the property of strict monotonic­ ity of conclusion. The class of A-valuations of plausibility is considered where operations based only on information about linear or­ dering of plausibility values are used. In this class the modus ponens generating fu...

A conjunction T ties an implication operator A if the identity A(a,A(b, z)) = A(T (a, b), z) holds [A.A. Abdel-Hamid, N.N. Morsi, Associatively tied implications, Fuzzy Sets and Systems 136 (2003) 291–311]. We study the class of tied adjointness algebras (which are five-connective algebras on two partially ordered sets), in which the implications are tied by triangular norms. This class contain...

In this paper intuitionistic set theory INC# ∞# in infinitary theoretical language is considered. External induction principle nonstandard arithmetic were derived. Non trivial application number considered.The Goldbach-Euler theorem obtained without anyreferences to Catalan conjecture.

This paper addresses fundamental issues on the nature of the concepts and structures of fuzzy logic, focusing, in particular, on the conceptual and functional differences that exist between probabilistic and possibilistic approaches. A semantic model provides the basic framework to define possibilistic structures and concepts by means of a function that quantifies proximity, closeness, or resem...

We give some simple examples of applying the well-known elementary probability theory inequalities and properties in field logical argumentation. A probabilistic version hypothetical syllogism inference rule is as follows: if propositions A, B, C, A→B, B→C have probabilities a, b, c, r, s, respectively, then for p A→C, we f(a,b,c,r,s)≤p≤g(a,b,c,r,s), functions f g given parameters. In this pape...

Two studies examined conditional reasoning with false premises. In Study 1, 12- and 16-year-old adolescents made "if-then" inferences after producing an alternative antecedent for the major premise. Older participants made more errors on the simple modus ponens inference than did younger ones. Reasoning with a false premise reduced this effect. Study 2 examined the relation between performance ...