Search Results (6)

This study focuses on the Modus Tollens (MT) property induced by discrete uninorms. Specifically, we identify the set of necessary and sufficient criteria for a discrete implication function to comply with this logical property. This rule of inference is studied by using discrete residual implication functions derived from uninorms of two of the most important families of these discrete operators (${\mathcal{U}}_{min}$, idempotents), exploring which properties these operators must satisfy, as well as providing some characterizations of the Modus Tollens in this domain of definition. Our findings contribute to a deeper understanding of reasoning mechanisms in fuzzy logic, particularly in discrete settings. Full article

The overlap function, a particular kind of binary aggregate function, has been extensively utilized in decision-making, image manipulation, classification, and other fields. With regard to overlap function theory, many scholars have also obtained many achievements, such as pseudo-overlap function, quasi-overlap function, semi-overlap function, etc. The above generalized overlap functions contain commutativity and continuity, which makes them have some limitations in practical applications. In this essay, we give the definition of pseudo-quasi overlap functions by removing the commutativity and continuity of overlap functions, and analyze the relationship of pseudo-t-norms and pseudo-quasi overlap functions. Moreover, we present a structure method for pseudo-quasi overlap functions. Then, we extend additive generators to pseudo-quasi overlap functions, and we discuss additive generators of pseudo-quasi overlap functions. The results show that, compared with the additive generators generated by overlap functions, the additive generators generated by pseudo-quasi overlap functions have fewer restraint conditions. In addition, we also provide a method for creating quasi-overlap functions by utilizing pseudo-t-norms and pseudo automorphisms. Finally, we introduce the idea of left-continuous pseudo-quasi overlap functions, and we study fuzzy inference triple I methods of residual implication operators induced by left-continuous pseudo-quasi overlap functions. On the basis of the above, we give solutions of pseudo-quasi overlap function fuzzy inference triple I methods based on FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems. Full article

Two important basic inference models of fuzzy reasoning are Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT). In order to solve FMP and FMT problems, the full implication triple I algorithm, the reverse triple I algorithm and the Subsethood Inference Subsethood (SIS for short) algorithm are proposed, respectively. Furthermore, the existing reasoning algorithms are extended to intuitionistic fuzzy sets and interval-valued fuzzy sets according to different needs. The purpose of this paper is to study the relationship between intuitionistic fuzzy reasoning algorithms and interval-valued fuzzy reasoning algorithms. It is proven that there is a bijection between the solutions of intuitionistic fuzzy triple I algorithm and the interval-valued fuzzy triple I algorithm. Then, there is a bijection between the solutions of intuitionistic fuzzy reverse triple I algorithm and the interval-valued fuzzy reverse triple I algorithm. At the same time, it is shown that there is also a bijection between the solutions of intuitionistic fuzzy SIS algorithm and interval-valued fuzzy SIS algorithm. Full article

As a generalization of intuitionistic fuzzy sets, single-valued neutrosophic sets have certain advantages for solving indeterminate and inconsistent information. In this paper, we study the fuzzy inference full implication method based on a single-valued neutrosophic t-representable t-norm. Firstly, single-valued neutrosophic fuzzy inference triple I principles for fuzzy modus ponens and fuzzy modus tollens are shown. Then, single-valued neutrosophic $ℛ$-type triple I solutions for fuzzy modus ponens and fuzzy modus tollens are given. Finally, the robustness of the full implication of the triple I method based on a left-continuous single-valued neutrosophic t-representable t-norm is investigated. As a special case in the main results, the sensitivities of full implication triple I solutions, based on three special single-valued neutrosophic t-representable t-norms, are given. Full article

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, $A\to B$, and $B\to C$ have probabilities a, b, c, r, and s, respectively, then for probability p of $A\to C$, we have $f(a,b,c,r,s)\le p\le g(a,b,c,r,s)$, 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. Full article

Statistical Hypothesis Testing (SHT) is a class of inference methods whereby one makes use of empirical data to test a hypothesis and often emit a judgment about whether to reject it or not. In this paper, we focus on the logical aspect of this strategy, which is largely independent of the adopted school of thought, at least within the various frequentist approaches. We identify SHT as taking the form of an unsound argument from Modus Tollens in classical logic, and, in order to rescue SHT from this difficulty, we propose that it can instead be grounded in t-norm based fuzzy logics. We reformulate the frequentists’ SHT logic by making use of a fuzzy extension of Modus Tollens to develop a model of truth valuation for its premises. Importantly, we show that it is possible to preserve the soundness of Modus Tollens by exploring the various conventions involved with constructing fuzzy negations and fuzzy implications (namely, the S and R conventions). We find that under the S convention, it is possible to conduct the Modus Tollens inference argument using Zadeh’s compositional extension and any possible t-norm. Under the R convention we find that this is not necessarily the case, but that by mixing R-implication with S-negation we can salvage the product t-norm, for example. In conclusion, we have shown that fuzzy logic is a legitimate framework to discuss and address the difficulties plaguing frequentist interpretations of SHT. Full article