Modus Ponens Research Articles

Approximate reasoning based on the preference implication

In fuzzy logic, most of the implication operators are based on generalizations of the classical, material implication. That is, these implications are defined as the disjunction of the negated value of the first argument and the value of the second argument, while the underlying disjunction operators are associative triangular conorms. In our study, we concentrate on how a class of implication operators, called the preference implication operators, can be used in approximate reasoning. Using this implication operator family, we present a novel, Modus Ponens-like approximate reasoning method, in which we have two premises: (1) a statement and (2) a preference implication with an antecedent of this statement. Here, we show how the continuous logical value of the consequent of the preference implication can be derived from the continuous logical values of the premises. We point out that this novel approximate reasoning method is strongly connected with the so-called aggregative operator, which is a representable uninorm. Next, we also present a threshold value-based generalization of the Modus Ponens syllogism and demonstrate that the Modus Tollens syllogism can be generalized in the same way. Lastly, we provide an illustrative example.

Rough sets, modal logic and approximate reasoning

This paper introduces an approximate reasoning method based on rough sets and modal logic. Various Approximate Modus Ponens rules are investigated and defined in Modal Logic systems interpreted in the rough set language. Although this is primarily theoretical work, we expect natural applications of the technique in real-life scenarios. An attempt in this direction is made in a real case analysis to logically model some issues of legal interest.

Deep and shallow conditionals – and three alleged counterexamples

ABSTRACT We analyze three interesting arguments from the literature, where ascribing a probability of 1 to a certain right-nested conditional A→(B→C) leads to strong theses concerning conditionals: they serve as counterexamples to important general claims. The first is the classic and much discussed McGee’s counterexample to Modus Ponens from McGee [“A Counterexample to Modus Ponens.” The Journal of Philosophy 82 (9): 462–471]. The second example was given by Santorio [“Trivializing Informational Consequence.” Philosophy and Phenomenological Research 104:297–320] and is intended to undermine the probabilistic version of Modus Ponens. The third example was presented in Cantwell [“Revisiting McGee’s Probabilistic Analysis of Conditionals.” Journal of Philosophical Logic 51 (5): 973–1017] in order to reject the general version of the Ramsey Test (i.e. PCCP) and Modus Ponens. These examples are based on a specific interpretation of right-nested conditional which is not the only possible one. We provide examples demonstrating that alternative interpretations are feasible. As a result, the arguments presented in the three cited works possess limited efficacy, as they are confined solely to the particular interpretation of the conditional.

Global Realism with Bipolar Strings: From Bell Test to Real-World Causal-Logical Quantum Gravity and Brain-Universe Similarity for Entangled Machine Thinking and Imagination

Following Einstein’s prediction that “Physics constitutes a logical system of thought” and “Nature is the realization of the simplest conceivable mathematical ideas”, this topical review outlines a formal extension of local realism limited by the speed of light to global realism with bipolar strings (GRBS) that unifies the principle of locality with quantum nonlocality. The related literature is critically reviewed to justify GRBS which is shown as a necessary and inevitable consequence of the Bell test and an equilibrium-based axiomatization of physics and quantum information science for brain–universe similarity and human-level intelligence. With definable causality in regularity and mind–light–matter unity for quantum superposition/entanglement, bipolar universal modus ponens (BUMP) in GRBS makes quantum emergence and submergence of spacetime logically ubiquitous in both the physical and mental worlds—an unexpected but long-sought simplification of quantum gravity with complete background independence. It is shown that GRBS forms a basis for quantum intelligence (QI)—a spacetime transcendent, quantum–digital compatible, analytical quantum computing paradigm where bipolar strings lead to bipolar entropy as a nonlinear bipolar dynamic and set–theoretic unification of order and disorder as well as linearity and nonlinearity for energy/information conservation, regeneration, and degeneration toward quantum cognition and quantum biology (QCQB) as well as information-conservational blackhole keypad compression and big bang data recovery. Subsequently, GRBS is justified as a real-world quantum gravity (RWQG) theory—a bipolar relativistic causal–logical reconceptualization and unification of string theory, loop quantum gravity, and M-theory—the three roads to quantum gravity. Based on GRBS, the following is posited: (1) life is a living bipolar superstring regulated by bipolar entropy; (2) thinking with consciousness and memory growth as a prerequisite for human-level intelligence is fundamentally mind–light–matter unitary QI logically equivalent to quantum emergence (entanglement) and submergence (collapse) of spacetime. These two posits lead to a positive answer to the question “If AI machine cannot think, can QI machine think?”. Causal–logical brain modeling (CLBM) for entangled machine thinking and imagination (EMTI) is proposed and graphically illustrated. The testability and falsifiability of GRBS are discussed.

Sequent calculi and an efficient theorem prover for conditional logics with selection function semantics

Abstract In this paper we present our final solution to the problem of designing an efficient theorem prover for Conditional Logics with the selection function semantics. Conditional Logics recently have received a renewed attention and have found several applications in knowledge representation and artificial intelligence. In order to provide an efficient theorem prover for Conditional Logics, we introduce labelled sequent calculi for the logics characterized by well-established axioms systems including the axiom of strong centering CS, the axiom of conditional identity ID, the axiom of conditional modus ponens MP, as well as the conditional third excluded middle CEM, rejected by Lewis but endorsed by Stalnaker, as well as for the whole cube of extensions. The proposed calculi revise and improve the calculi SeqS introduced in Olivetti et al. (2007, ACM Trans. Comput. Logics, 8). We also present an implementation of these calculi in SWI Prolog, including a graphical interface in Python as well as standard heuristics and refinements that allow us to obtain an efficient theorem prover for the logics under consideration. Moreover, we present some statistics about the performances of the theorem prover, which are promising and significantly better than those of its predecessor CondLean, an implementation of the calculi SeqS.

A note on Mcgee’s counterexample to Modus Ponens

In this article I will review McGee's famous counterexample to Modus Ponens and I will argue that it is not a real counterexample. I will claim that the problem lies in an infelicitous assertion of the second premise. As result of this diagnosis, I will suggest that when dealing with indicative conditionals a pragmatic theory is needed.

Research on intuitionistic fuzzy implications. Part 4

Continuing the research from Parts 1, 2 [2, 3] where intuitionistic fuzzy implications, determined as implications with “good” properties, were investigated, here we correct the list of the implications that satisfy the Modus Ponens from Part 3, [4], and further select among them those implications that satisfy the Modus Tollens, as well. We discuss some applications of these implications and show the relationship between every two of them.

On the quantum circuit implementation of modus ponens

The process of inference reflects the structure of propositions with assigned truth values, either true or false. Modus ponens is a fundamental form of inference that involves affirming the antecedent to affirm the consequent. Inspired by the quantum computer, the superposition of true and false is used for the parallel processing. In this work, we propose a quantum version of modus ponens. Additionally, we introduce two generations of quantum modus ponens: quantum modus ponens inference chain and multidimensional quantum modus ponens. Finally, a simple implementation of quantum modus ponens on the OriginQ quantum computing cloud platform is demonstrated.

Solvability of systems of partial fuzzy relational equations revisited – a short note

Fuzzy inference systems have been widely investigated from different perspectives, including their logical correctness. It is not surprising that the logical correctness led mostly to the questions on the preservation of modus ponens. Indeed, whenever such a system processes an input equivalent to one of the rule antecedents, it is natural to expect the modus ponens to be preserved and the inferred output to be identical to the respective rule consequent. This leads to the related systems of fuzzy relational equations where the antecedent and consequent fuzzy sets are known values, the inference is represented either by the direct product related to the compositional rule of inference or by the Bandler-Kohout subproduct, and the fuzzy relation that represents the fuzzy rule base is the unknown element in the equations. The most important question is whether such systems are solvable, i.e., whether there even exists a fuzzy relation that models the given fuzzy rule base in such a way that the modus ponens is preserved.Partiality allows dealing with partially defined truth values which allows to deal with situations, where we cannot define the truth value for a given predicate. The partial logics have been recently extended to partial fuzzy logics and the partial fuzzy set theory has been developed. Partial fuzzy sets then may have undefined membership degrees for some values from the given universe.This background leads naturally to the problem of solvability of partial fuzzy relational equations, which are equations with partial fuzzy sets in the role of antecedents and consequents and with partial fuzzy relation as the model of the given fuzzy rule base. Such a setting led to a recent publication that uncovers the solvability and even the shape of the solutions. In this contribution, we revisit the problem and consider a specific case that allows partiality only in the input. In other words, antecedents, as well as consequents expressing the knowledge, are fully defined. Thus, the model of the fuzzy rules is one of the standard and fully defined relations. We investigate what happens if the input differs from one of the antecedents only by a few undefined values. This mimics the situation when a description of an observed object misses a few values in its feature vector. We show that under specific conditions, we still may preserve the modified modus ponens, i.e., that the inferred output is identical with the fully defined consequent of the respective rule.

Fuzzy Inference Quintuple Implication Method Based on Single Valued Neutrosophic t-representable t-norm

In this paper, we study the single valued neutrosophic fuzzy inference quintuple implication method. Firstly, single valued neutrosophic fuzzy inference quintuple implication principles for fuzzy modus ponens and fuzzy modus tollens are given. Then, single-valued neutrosophic R-type quintuple implication solutions for fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) are given. Finally, the robustness of the quintuple implication method based on a left-continuous single-valued neutrosophic t-representable t-norm is investigated. As a special case of the main results, the sensitivity of quintuple implication solutions based on special single-valued neutrosophic residual implications is given.

Fuzzy reasoning method based on picture similarity measure and its application in medical diagnosis

In previous studies, based on picture fuzzy sets, full implication algorithm and quintuple implication principle have been studied. The problem is, full implication algorithm did not consider the relationship between the input and antecedent of rule, while quintuple implication principle only considered the inclusion relationship between the rule’s antecedent and input. Both algorithms did not consider the similarity of rule’s antecedent and input. This leads to unreasonable calculation results in some cases. Therefore, we study fuzzy reasoning methods based on picture similarity measure for generalized modus ponens and generalized modus tollens in this paper. First of all, based on picture biresiduation, a new picture similarity measure between picture fuzzy sets is constructed. Then, representations of solutions of fuzzy reasoning methods based on picture similarity measure are given. Moreover, reducibility and robustness of fuzzy reasoning methods based on picture similarity measure are studied. Finally, two issues related to medical diagnosis are addressed by the proposed methods, demonstrate the rationality and effectiveness of our suggested methods.

Sign-inferences in Greek and Buddhist Logic

The Yogācāra school of logic developed a theory of sign-inferences that has many features of the Stoic and Epicurean logical teachings with small inclusions of Aristotelian ideas. In the Nyāyabindu of Dharmakīrti, we can find the following schemes of formal reasoning: modus Barbara (Figure I) and modus Camenes (Figure IV) of the Aristotelian syllogistic, and all the inference rules of the Stoic logic: modus ponens, modus tollens, modus ponendo tollens, modus tollendo ponens I, modus tollendo ponens II. The three premises of Yogācāra inference (if p, then q; there is an example that p and q occur together; p; then q) corresponds to the Epicurean doctrine of three premises with a necessary exemplification. Thus, the paper is a comparative study of Greek and Buddhist theory of sign-inferences, showing a deep similarity between them.

A New Distance-Type Fuzzy Inference Method Based on Characteristic Parameters

Reasoning is a cognitive activity that leverages knowledge to generate solutions to problems. Knowledge representations in the brain require both symbolic and graphical information since visual information is figurative and conveys a large amount of information. Consequently, graphical knowledge representation is often employed in reasoning. Distance-type fuzzy inference utilizes the distance information between the antecedent and the set of facts as the basis for inference. Compared to Mamdani inference, the distance-type fuzzy inference method not only satisfies the convexity and asymptotic properties of the inference results but also adheres to the separation rule (modus ponens), a fundamental principle in inference. This paper discusses extensions of distance-type fuzzy inference methods to handle spatial figures. In this paper, we first explain the distance-type fuzzy inference method. Then, we discuss the concept representation in the feature space and independent parameters that can completely express the characteristics of a figure in space, which are defined as “characteristic parameters”. Furthermore, we describe the correspondence between figures and vectors in the feature space, propose a new distance-type fuzzy inference method based on characteristic parameters and describe its characteristics. Finally, an example is used to demonstrate the inference results of this new distance-type fuzzy inference method.

Fuzzy Inference Full Implication Method Based on Single Valued Neutrosophic t-representable t-norm: Purposes, Strategies, and a Proof-of-Principle Study


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

Research on intuitionistic fuzzy implications. Part 3

Continuing the research from [1, 2], here we give the lists of the intuitionistic fuzzy implications, introduced in [2], that satisfy at least one of two forms of Modus Ponens (MP) – a standard and a new one, called “mixed”, forms. We show the relationship between every two of these implications that satisfy the mixed form of MP.

What linguists emphasize about conditions and what logicians emphasize and what each should take from the other

Abstract In the first part of the study, we proposed that logicians adopt from linguists their detailed distinction of conditions on a scale of varying degrees of satisfiability from factually true via potentially satisfiable to absolutely irreal. On this basis, we have proposed a suitable distinction for the logico-semantic investigation of conditionals of the following kinds: Factual, hypothetical (agnostic), counterfactual, and counterpossible. We consider the syntactic-semantic distinction between potentially and absolutely irreal conditions to be crucial. Therefore, we think that the conditional mode in the form of antepreterite is an appropriate syntactic signal for identifying the absolute irreality of the condition and, especially in scientific and artistic style, we consider it functional. On the other hand, the distinction between sufficient and necessary conditions could be inspiring for linguists. In classical logic, these are the basic kinds of conditions and are associated with the rules of deductive reasoning modus ponens and modus tollens, respectively. These rules are central to the application of logic. It turns out that these tools are not sufficiently established in Slovak linguistics, as there is no systematic distinction between sufficient and necessary conditions. The situation in czech dictionaries is similar. Finally, we have put forward a hypothesis why the conjunction unless is considered in Slovak linguistics as a double – in our opinion wrongly – of the conjunction only: There is a “forgotten” ellipsis behind it, which is not brought to the reader’s attention.

The Implicative Conditional

This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent. p⇒q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\\Rightarrow q}$$\\end{document} is thus defined as ¬◊(p∧¬q)∧◊p∧◊¬q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\lnot } \\Diamond {(p \\wedge \\lnot q) \\wedge } \\Diamond {p \\wedge } \\Diamond {\\lnot q}$$\\end{document}. We explore the logical properties of this conditional in a reflexive normal Kripke semantics, provide an axiomatic system and prove it to be sound and complete for our semantics. The implicative conditional validates transitivity and contraposition, which we take to be integral parts of reasoning and communication. But it only validates restricted versions of strengthening the antecedent, right weakening, simplification, and rational monotonicity. Apparent counterexamples to some of these properties are explained as due to contextual factors. Finally, the implicative conditional avoids the paradoxes of material and strict implication, and validates some connexive principles such as Aristotle’s theses and weak Boethius’ thesis, as well as some highly entrenched principles of conditionals, such as conjunction of consequents, disjunction of antecedents, modus ponens, cautious monotonicity and cut.

A probabilistic analysis of selected notions of iterated conditioning under coherence

It is well known that basic conditionals satisfy some desirable basic logical and probabilistic properties, such as the compound probability theorem. However checking the validity of these becomes trickier when we switch to compound and iterated conditionals. Herein we consider de Finetti's notion of conditional both in terms of a three-valued object and as a conditional random quantity in the betting framework. We begin by recalling the notions of conjunction and disjunction among conditionals in selected trivalent logics. Then we analyze the notions of iterated conditioning in the frameworks of the specific three-valued logics introduced by Cooper-Calabrese, by de Finetti, and by Farrel. By computing some probability propagation rules we show that the compound probability theorem and other important properties are not always preserved by these formulations. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem as well as some other desirable properties. We also check the validity of two generalized versions of Bayes' Rule for iterated conditionals. We study the p-validity of generalized versions of Modus Ponens and two-premise centering for iterated conditionals. Finally, we observe that all the basic properties are satisfied within the framework of iterated conditioning followed in recent papers by Gilio and Sanfilippo in the setting of conditional random quantities.

MP and MT properties of fuzzy inference with aggregation function

As the two basic fuzzy inference models, fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) have the important application in artificial intelligence. In order to solve FMP and FMT problems, Zadeh proposed a compositional rule of inference (CRI) method. This paper aims mainly to investigate the validity of A-compositional rule of inference (ACRI) method, as a generalized CRI method based on aggregation functions, from a logical view and an interpolative view, respectively. Specifically, the modus ponens (MP) and modus tollens (MT) properties of ACRI method are discussed in detail. It is shown that the aggregation functions to implement FMP and FMT problems provide more generality than the t-norms, uninorms and overlap functions as well-known the laws of T-conditionality, U-conditionality and O-conditionality, respectively. Moreover, two examples are also given to illustrate our theoretical results. Especially, Example 6.2 shows that the output B′ in FMP (FMT) problem is close to B (DC) with our proposed inference method when the fuzzy input and the antecedent of fuzzy rule are near (the fuzzy input near with the negation of the succedent in fuzzy rule).

Accuracy and Response Time for Modus Ponens Syllogisms Vary by Controversial Topic and Categorical Emotion.

Researchers have documented differential effects of emotion on cognitive processes, debating whether emotion may increase or decrease the response time and accuracy of logical thinking. The current study proposed that differences may be due to variability occurring across topic and categorical emotions, such that assessment of several basic emotional responses in the context of performing logical reasoning tasks may provide an initial indication of these differences. Utilizing modus ponens syllogisms composed of controversial statements, the current study evoked a variety of emotional responses and tested the accuracy of participants' basic logical thinking. Results indicated that logical skills were largely preserved despite the topic and emotion, nonetheless, accuracy varied across syllogism type (controversial vs. control), with increased accuracy on controversial syllogisms. Syllogisms rated as evoking no emotion were answered more accurately than those that evoked any emotion, with disgust and anger associated with less accuracy than no emotion and gladness associated with increased accuracy. Response times also differed across syllogism type, emotion, and emotion intensity.