Modus Ponens

Tablas de verdad para aplicar al método de tablas de verdad (MTV). Reglas y leyes lógicas para aplicar al método demostrativo o derivativo (MD). Reglas para aplicar al MD en la lógica de predicados. Reglas de aplicación para el método de árboles lógicos.

Are modus ponens and its counterpart, modus tollens, universal laws of logic? A deconstruction of modus ponens in binary propositional logic finds that the propositional formula of binary logic has exceeded it's definition .

In a number of logic systems, including U4 , modus ponens is not a theorem. An examination of the truth table for modus ponens in U 4 will show that modus ponens as an inference rule can be preserved.

The efficiency of algorithms for probabilistic inference in Bayesian networks can be improved by exploiting independence of causal influence. The factorized rep- resentation of independence of causal influence offers a factorized decomposition of certain independence of causal influence models. We describe how lazy prop- agation - a junction tree based inference algorithm - easily can be extended to take advantage

The modus ponens (A → B, A ∴ B) is, along with modus tollens and the two logically not valid coun-terparts denying the antecedent (A → B, ¬A ∴ ¬B) and affirming the consequent, the argument form that was most often investigated in the psychol-ogy of human reasoning. The present contribution reports the results of three experiments on the proba-bilistic versions of modus ponens and denying the an-tecedent. In probability logic these arguments lead to conclusions with imprecise probabilities. In the modus ponens tasks the participants inferred probabilities that agreed much better with the coher-ent normative values than in the denying the an-tecedent tasks, a result that mirrors results found with the classical argument versions. For modus po-nens a surprisingly high number of lower and up-per probabilities agreed perfectly with the conjugacy property (upper probabilities equal one complements of the lower probabilities). When the probabilities of the premises are imprecise the parti...

In this paper, we propose a fuzzy-rule-based model for the representation of contextual preferences in a database querying framework. We discuss the augmentation of a query with preferences deduced from information regarding the current context of the user. To this end, we present an approach based on generalized modus ponens pattern.

In the present article I attribute to the common topic in the Rhetoric a two-fold suggestive function and a guarantee function. These three functions are possible because this type of topic, while often quite abstract, nevertheless contains thought-steering, substantial terms, and formulates a generally empirical or normative endoxon. Assuming that according to Aristotle an enthymeme has at least two premises, it would appear that a common topic is the abstract principle behind the often implicit major premise. This means that the topic may be regarded as the – generalizing – if-then statement in a modern argumentation scheme. Therefore it should be possible to see the enthymemes of Rhetoric 2.23 as a combination of a logical argument form (which can usually be reconstructed as modus ponens) and an argumentation scheme – even though we may not attribute this idea to Aristotle himself.

1. Consequential conditionals are defined as" if P then Q" statements, where P is an action, and Q a predicted outcome of this action, which is either desirable or undesirable to the agent. Experiment 1 shows that desirable (viz. undesirable) outcomes invite an inference to the truth (viz. falsity) of their antecedent. Experiment 2 shows that the more extreme the outcome is, the stronger the invited inference is. Experiment 3 shows that modus ponens from premises" If A then C, A" can be suppressed with the introduction of a consequential ...

This paper reports some novel approach on linguistic logic with our intention to realize CWW, Computing With Words, via a simple example which consists of only five words. As a by product, this simple example of the linguistic logical system may serve as a mathematical model, modeling the degree of truthfulness in daily usage. The five words set of a linguistic variable modeling the degree of truthfulness are; true, nearly true, undecided, nearly false and false. We subjectively choose trapezoidal fuzzy numbers as our linguistic truth values in order to model our linguistic logic system. Firstly, some natural operations and linguistic logic operators are defined to suit our objective of developing a closed linguistic variable set. Then the computation of linguistic truth values for this linguistic logical system is developed in order to facilitate us to perform the linguistic inferences. Properties of these natural operations can be derived accordingly. It is perhaps quite rewarding...

An important field of probability logic is the investigation of inference rules that propagate point probabilities or, more generally, interval probabilities from premises to conclusions. Conditional probability logic (CPL) interprets the common sense expressions of the form “if..., then...” by conditional probabilities and not by the probability of the material implication. An inference rule is probabilistically informative if the coherent probability interval of its conclusion is not necessarily equal to the unit interval $[0, 1] $. Not all logically valid ...

I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call “the Cut-off Point Problem” and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburg’s puzzle. Very briefly, I show that, in the Lottery Paradox, the premises “ticket n°1 will lose”, “ticket n°2 will lose”… “ticket n°1000 will lose” are equivalent to soritical premises of the form “~(the winning ticket is in {…, (tn)}) ⊃ ~(the winning ticket is in {…, tn, (tn + 1)})” (where “⊃” is the material conditional, “~” is the negation symbol, “tn” and “tn + 1” are “ticket n°n” and “ticket n°n + 1” respectively, and “{}” identify the elements of the lottery tickets’ set. The brackets in “(tn)” and “(tn + 1)” are meant to point out that in the antecedent of the conditional we do not always have a “tn” (and, as a result, a “tn + 1” in the consequent): consider the conditional “~(the winning ticket is in {}) ⊃ ~(the winning ticket is in {t1})”). As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises. In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions. One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox.

The purpose of this master thesis is to investigate the basic concepts of propositional logic for knowledge representation and formalization of reasoning in Artificial Intelligence. The different properties of logical propositions are discussed. The basic and derived logical connectives are used to establish the compound statements, and the truth tables are constructed to investigate the properties of logical connectives. Such propositions as tautology, satisfiability, contradiction, contingency, logical entailment and logical equivalence are analyzed. Three algebraic normal forms - negation normal form, disjunctive normal form and conjunctive normal form are studied. Horn clauses are implemented. Two forms of valid inferences as modus ponens and modus tollens are considered. Some examples are provided to better understand the main properties of propositional logic. Keywords: Propositional Logic, Logical Connectives, Normal Forms, Horn Clauses, Modus Ponens, Modus Tollens.

We consider a multiple-rule, generalized modus ponens inference scheme, with an interpretation based on compositional rule of inference (CRI) and a residuated implication. We show that such a system is equivalent, as far as CRI is concerned, to a system that satisfies the “basic requirement for fuzzy reasoning”, proposed by Turksen and Tian (Fuzzy Sets and Systems 58 (1993) 3–40). We establish an analogous conclusion for an alternative interpretation (due to Magrez and Smets) of generalized modus ponens, which we call consequent dilation rule (CDR). This method is computationally faster than CRI, but it produces less specific inference results.

Pavelka [10] had shown in 1979 that the only natural way of formalizing fuzzy logic for truth-value in the unit interval [0,1] is by using Lukasiewicz's implication operator a→b = min{1, 1−a+b} or some isomorphic form of it A considerable number of other papers around the same time had attempted to formulate alternative definitions for a→b by giving intuitive justifications for them. There continues to be some confusion, however, even today about the right notion of fuzzy logic. Much of this has its origin in the use of improper “and” (“or”) and the “not” operations and a misunderstanding of some of the key differences between “proofs” or inferencing in fuzzy logic and those in Lukasiewicz's logic. We point out the need for defining the strong conjunction operator “⊗” in connection with fuzzy Modus-ponens rule and why we do not need the fuzzy Syllogism rule. We also point out the shortcomings of many of the alternative definitions of a→b, which indicate further support for Pavelka's result. We hope that these discussions help to clarify the misconceptions about fuzzy logic.

In this article I analyse the way rationalism (and specifically ethical rationalism) has been presented both in the Greek world and in the Christian horizon. I highlight a specific feature of this rationalism, namely the identification between rational knowledge and knowledge ordered according to the causal logic of modus ponens. This consists in a relationship between causes and effects that knowledge has the task
of revealing and that the (political and legal) will has the task of implementing and respecting. I do not claim that this model can be applied to the Greek thought as a whole, or that Greek rationalism is a homogeneous and unitary paradigm of thought. More simply, I discuss the relationship between rational knowledge and understanding
of the causal order according to the modus ponens model. I point out how this feature came to dominate the very idea of rationality in Greek and Christian culture (the Middle Ages in particular) and the important consequences it has had for thinking about law and justice.

Vann McGee has presented a putative counterexample to modus ponens. I show that (a slightly modified version of) McGee’s election scenario has the same structure as a famous lottery scenario by Kyburg. More specifically, McGee’s election story can be taken to show that, if the Lockean Thesis holds, rational belief is not closed under classical logic, including classical-logic modus ponens. This conclusion defies the existing accounts of McGee’s puzzle.

Four intuitions are recurrent and influential in theories about conditionals: the Ramsey's test, the Adams' Thesis, the Equation, and the robustness requirement. For simplicity's sake, I call these intuitions 'the big four'. My aim is to show that: (1) the big four are interdependent; (2) they express our inferential dispositions to employ a conditional on a modus ponens; (3) the disposition to employ a conditional on a modus ponens doesn't have the epistemic significance that is usually attributed to them, since their acceptability or truth conditions are not necessarily associated with its employability on a modus ponens.

Conditional probability logic is about the propagation of imprecise probabilities from premises to conclusions. An inference rule is probabilis- tically informative if the coherent probability interval of its conclusion is not necessarily equal to the unit interval. Not all logically valid inference rules are probabilistically informative and vice versa. The relationship between logically valid and probabilistically informative inference rules is

In decision-making problems, several methods exist in the literature. This paper shows how abductive reasoning can be a useful approach to deal with incomplete information. Our contribution proposes a model based on fuzzy gradual rules and fuzzy observations and operating in two steps. In the first step, we construct abductive hypotheses by using an inference method based on mathematical properties of generalized modus ponens. These hypotheses are not uniquely defined. This is why in the second step, we define two types of criteria to construct the best ones according to the given observation. These criteria are, respectively, the similitude and the coherence, which we propose to study in detail in this paper.