"If-then" as a version of "Implies"

This simple defense of material implication helps clarify the debate between the orthodox logicians, who claim 'If p then q' and 'p |horseshoe| q' are not interderivable and the nonorthodox logicians, who claim that the two expressions are interderivable. The paper shows the orthodox logician must deny that ordinary language arguments of the form modus ponens and modus tollens are truth-functional in any consistent sense on three lines of the truth-table.

There is a profound, but frequently ignored relationship between logical consequence (formal implication) and material implication. The first repeats the patterns of the latter, but with a wider modal reach. It is argued that this kinship between formal and material implication simply means that they express the same kind of implication, but differ in scope. Formal implication is unrestricted material implication. This apparently innocuous observation has some significant corollaries: (1) conditionals are not connectives, but arguments; (2) the traditional examples of valid argumentative forms are metalogical principles that express the properties of logical consequence; (3) formal logic is not a useful guide to detect valid arguments in the real world; (4) it is incoherent to propose alternatives to the material implication while accepting the classical properties of formal implication; (5) some of the counter-examples to classical argumentative forms and known conditional puzzles are unsound.

SSRG-IJHSS, 2022

At first, George Boole introduced conditional propositions as the basis for the better development of propositional logic. A compound statement composed of two atomic statements (antecedent and consequent) by the connective 'if-then' is called propositional logic. For example: "If it is raining, then the roads are wet". Here the first statement of this conditional proposition, 'It is raining,' is called the antecedent, and the second, 'the roads are wet,' is called the consequent. The material conditional in which conditional is false only when the antecedent is true and the consequent is false, but it is always true in all other cases. So, a definition of a material implication isa conditional proposition is true when either the antecedent is false, or the consequent is true. The conditional is false when the antecedent is true and the consequent is false. Whereas, if there is no relation between the antecedent and the consequent, then the truth-functional definition of material implication leads to a paradox called the paradox of material implication. Basically, the twin paradox arises here. For example, the conditionals 'if the earth is a star, then Socrates is a philosopher and 'if the earth is a star, then Socrates is not a philosopher are both true because the antecedent is false. Secondly, the conditionals 'if Islamabad is the capital of Pakistan, then New Delhi is the capital of India' and 'if Islamabad is not the capital of Pakistan, then New Delhi is the capital of India' are both true because the consequent is true. C.I. Lewis clearly establishes these paradoxes in 1912 '(I) a false proposition implies any proposition; and (II) a true proposition is implied by any proposition'.

2023

Attention: There is a revised and updated version of this article on: https://www.academia.edu/114815496/Solving_the_Paradox_of_Material_Implication_2024 The paradox of material implication has been a mystery to philosophers and logicians since antiquity. This article brings the final solution to the problem. In the course of the conducted research, the true nature of the implication was identified, which turned out to be the competition, occurring in two varieties – strong and weak, which is the logical equivalent of the difference of sets in set theory. Therefore, postulates were put forward regarding changes in nomenclature, adding logical connectives and taking over the role of the existing implication by the biconditional. A proposal for a system of axioms of logic and changes to the formula of the rule of identity and the names of some other logic rules were given. Finally, the issue of agreeing the names of logical functions and the names of the corresponding Boolean functions in computer science and logic gates in electronics was discussed.

A many-valued logic in 8 truth-values based upon Classical logic termed ‘Universal Logic’, denoted U8, provides a correspondence to the ‘if-then’ implication meaning of natural language. Truth tables of implication and equivalence for U8 will be given, expanding the definition of validity. Accordingly, a new analysis of the ‘paradoxes of material implication’ will be undertaken. Material implication will be found to be identical with Universal logic when Boolean assignments are employed signifying that Classical is a subset of U8 logic. However, when 2 truth-values {true, false} are employed, denoted U2, implication resembles material equivalence and so validity is amended. As illustration of this approach five major ‘paradoxes of material implication’ will be analysed in terms of U2 validity. Remarkable results will be elucidated showing two ‘paradoxes’ to be affirmed while three were denied providing evidence that ‘Universal Logic’ offers an intuitive inductive logic.

The material account of indicative conditionals faces a legion of counterexamples that are the bread and butter in any entry about the subject. For this reason, the material account is widely unpopular among conditional experts. I will argue that this consensus was not built on solid foundations, since these counterexamples are contextual fallacies. They ignore a basic tenet of semantics according to which when evaluating arguments for validity we need to maintain the context constant, otherwise any argumentative form can be rendered invalid. If we maintain the context fixed, the counterexamples to the material account are disarmed. Throughout the paper I also consider the ramifications of this defence, make suggestions to prevent contextual fallacies, and anticipate some possible misunderstandings and objections.

Journal of Philosophical Logic, 2015

It is generally agreed that constructions of the form “if P, Q” are capable of conveying a number of different relations between antecedent and consequent, with pragmatics playing a central role in determining these relations. Controversy concerns what the conventional contribution of the if-clause is, how it constrains the pragmatic processes, and what those processes are. In this essay, I begin to argue that the conventional contribution of if-clauses to semantics is exhausted by the fact that these clauses introduce a proposition without presenting it as true so that the consequent can be understood in relation to it. Given our cognitive interests in such non-truth-presentational introductions, conditionals will make salient the wide but nevertheless disciplined variety of contents that we naturally attribute to them; no further substantial constraints of the sorts proposed by standard theories of conditionals are needed to explain the phenomena. If this is correct, it provides prima facie evidence for a radically contextualist account of conditionals according to which conditionals have no truth-evaluable or intuitively complete content absent some contextually provided, sufficiently salient relation between antecedent and consequent.

Synthese, 2020

The aim of this paper is to give a general solution to the paradoxes of the material conditional, including the paradoxes generated by embedded conditionals. The solution consists in a pragmatic reinterpretation of the formal languages of classical logic LK and relevant logic LR (which rejects the validity of weakening and splits the connectives into two) as presented in Paoli [2002],[2007]. In particular I argue that the material conditional in the classical logic LK captures the truth conditions of "if...then", but ignores certain pragmatic enrichments that are associated to it, while relevant logic LR can give a systematic diagnostic of the cases in which a conditional is pragmatically enriched and those cases in which it is not. This diagnostic shows the reason why the paradoxes seem unacceptable and why they are, nevertheless, truth-preserving. This reinterpretation will also cover the solution that Paoli [2005] gives to McGee's paradoxes of the Modus Ponens.

Topoi-an International Review of Philosophy, 1999