The paradox of material implication has remained unresolved since antiquity because it was believed that the nature of implication was entailment. The article shows that this nature is opposition and therefore the name "implication" should be replaced with the name "competition". A solution to the paradox is provided along with appropriate changes in nomenclature, the addition of connectives and the postulate that the biconditional take over the role of the previous implication. In addition, changes to (...) the nomenclature of logic gates reflecting the competition in electronics are proposed. (shrink)

Here I present a new objection to the material or "hook" analysis of indicative conditionals - the thesis that an indicative conditional 'If A then C' has the truth-conditions of the so-called material conditional - based on Liar-like reasoning. This objection seems invulnerable to any Grice-Lewis-Jackson-inspired pragmatic rejoinder.

This paper is concerned with the paradox of decrease. Its aim is to defend the answer to this puzzle that was propounded by its originator, namely, the Stoic philosopher Chrysippus. The main trouble with this answer to the paradox is that it has the seemingly problematic implication that a material thing could perish due merely to extrinsic change. It follows that in order to defend Chrysippus’ answer to the paradox, one has to explain how it could be that (...) Theon is destroyed by the amputation without changing intrinsically. In this paper, I shall answer this challenge by appealing to the broadly Aristotelian idea that at least some of the proper parts of a material substance are ontologically dependent on that substance. I will also appeal to this idea in order to offer a new solution to the structurally similar paradox of increase. In this way, we will end up with a unified solution to two structurally similar paradoxes. (shrink)

Demonstrates that the "paradoxes of material implication" are only apparent, sticking entirely within the confines of classical logic.

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle (...) ex falso quodlibet, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. (shrink)

In this paper Grice's requirements for assertability are imposed on the disjunction of Classical Logic. Defining material implication in terms of negation and disjunction supplemented by assertability conditions, results in the disappearance of the most important paradoxes of material implication. The resulting consequence relation displays a very strong resemblance to Schurz's conclusion-relevant consequence relation.

It is shown that the paradoxes of confirmation are closely linked to the paradoxes of material implication and that they can be avoided by formulating natural laws in terms of a genuine if-Connective rather than the material conditional. However, Natural laws so expressed are not confirmed by simple conjunctions. The question then is whether the common assumption that simple conjunctions do confirm universal generalizations is correct. The answer given is that it is not. In particular, (...) A confirming proposition of the form 'this is a black raven' is not equivalent to 'this is a raven and this is black'. (shrink)

Any religion has an ethical component. Thus, the examination of ethical problems is very important for religious studies. Consequentialism is an ethical doctrine according to which a fact is good only if it has good consequences. In order to avoid infinite regression, there is the need for a moral foundation in conformity with the criterion of goodness. The consequentialists proposed various criteria for goodness, such as pleasure, happiness or utility. Any fact will be judged as good only if its consequences (...) belong to the moral basis. The correctness of ethical judgments depends on the analysis of consequence relations. This study takes into account material and strict implication as representations of consequence. Both of them generate ethical paradoxes. If consequence is represented by material implication, according to the Manichean ethics, a fact is either good or bad; there are no neutral ethical facts. If consequence is analyzed by strict implication, the necessary facts will not have a moral value. The present study is to suggest an original solution to these paradoxes. (shrink)

In this paper, we aim to devise a logic that can deal with both the paradoxes that motivate dialetheism and the paradoxes related to the irrelevance of material implication. We propose the semantics and the sequent calculus of a relevant logic inspired by difference-making accounts of causation and arguably true to Graham Priest’s Logic of Paradox \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document}: a relevant logic that validates those and only those \documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document}-consequences that are not irrelevant. The new logic’s relevant implication is then added to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LP}$$\end{document} to obtain a logic with the logical vocabulary required to cope with the mentioned paradoxes. (shrink)

The aim of this paper is to clarify the truth-functional interpretation of the logical connective of the material implication. The importance of such clarification lies in the fact that it allows avoiding the supposed paradoxes introduced by C. I. Lewis (1918). I argue that an adequate understanding of the history and purposes of logic is enough to dissolve them away. The defense is based on an exposition of propositional compositionalism. To compare, I also present Stalnaker’s (1968) alternative (...) that seeks to explain the implicit concept of conditionality of conditionals assertions. In order to reinforce my aim I expound the twelve different meanings of ‘implication’ stated by Corcoran (1993) in order to hold that a proper understanding of these meanings allows us to avoid ambiguities and confusions. Finally, I expect to show that despite the possible alternatives, compositionalism is still the best explanation for the truth-functionality account of logical connectives. (shrink)

This paper introduces the logics of super-strict implications, where a super-strict implication is a strengthening of C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the modal cube. it is shown (...) that all logics of super-strict implications are connexive logics in that they validate Aristotle's Theses and (weak) Boethius's Theses. A proof-theoretic characterisation of logics of super-strict implications is given by means of G3-style labelled calculi, and it is proved that the structural rules of inference are admissible in these calculi. It is also shown that validity in the S5-based logic of super-strict implications is equivalent to validity in G. Priest's negation-as-cancellation-based logic. Hence, we also give a cut-free calculus for Priest's logic. (shrink)

Paradoxes have played an important role both in philosophy and in mathematics and paradox resolution is an important topic in both fields. Paradox resolution is deeply important because if such resolution cannot be achieved, we are threatened with the charge of debilitating irrationality. This is supposed to be the case for the following reason. Paradoxes consist of jointly contradictory sets of statements that are individually plausible or believable. These facts about paradoxes then give rise to a deeply (...) troubling epistemic problem. Specifically, if one believes all of the constitutive propositions that make up a paradox, then one is apparently committed to belief in every proposition. This is the result of the principle of classical logical known as ex contradictione (sequitur) quodlibetthat anything and everything follows from a contradiction, and the plausible idea that belief is closed under logical or material implication (i.e. the epistemic closure principle). But, it is manifestly and profoundly irrational to believe every proposition and so the presence of even one contradiction in one’s doxa appears to result in what seems to be total irrationality. This problem is the problem of paradox-induced explosion. In this paper it will be argued that in many cases this problem can plausibly be avoided in a purely epistemic manner, without having either to resort to non-classical logics for belief (e.g. paraconsistent logics) or to the denial of the standard closure principle for beliefs. The manner in which this result can be achieved depends on drawing an important distinction between the propositional attitude of belief and the weaker attitude of acceptance such that paradox constituting propositions are accepted but not believed. Paradox-induced explosion is then avoided by noting that while belief may well be closed under material implication or even under logical implication, these sorts of weaker commitments are not subject to closure principles of those sorts. So, this possibility provides us with a less radical way to deal with the existence of paradoxes and it preserves the idea that intelligent agents can actually entertain paradoxes. (shrink)

It can be said that Formal Logic begun by studying an idealization of the statements ’if p, then q’, something coming from long ago in both Greek and Scholastic Philosophy. Nevertheless, only in the XX Century it arrived at a stage of formalization once in 1910 Russell introduced and identified the ’material conditional’ with the expresion ”not p or q”. In 1918, and from paradoxical conditionals like ”If the Moon is a cheese, it is a Lyon’s face”, Lewis critiziced (...) the material conditional and introduced the so-called ’strict conditional’ as the modal necessity of the material one. In 1934, Huntington proved that both material and strict implications are coincidental in the setting of Boolean algebras and, hence, that they only can be actually different in algebraic structures weaker than Boolean algebras. Boolean algebras have too much laws for supporting the difference of both conditionals. This paper is nothing else than an algebraic trial to find simple binary Boolean operations able to express Lewis’ strict implication, and contains a new and simpler proof than that of Huntington, made by identifying ’possible’ with ’non self-contradictory’. By the way, it is shown that this proof is only valid in Boolean algebras, but neither in proper De Morgan algebras, nor in proper Ortholattices, with which it still remains open where the two conditional relations are actually different. (shrink)

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. $${p\Rightarrow q}$$ p ⇒ q is thus defined as $${\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}$$ ¬ ◊ (...) ( p ∧ ¬ q ) ∧ ◊ p ∧ ◊ ¬ q. 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. (shrink)

When the student of logic is first initiated into the problem of the relation of implication as expressed by the hypothetical proposition, he enters into a portion of logic which seems to him an impenetrable jungle. Elementary statements of the meaning of implication, as he is acquainted with them, are so meagre that they have failed to give any indication of the problems and paradoxes involved. Adequate discussions, on the other hand, are so abstruse that the initiate (...) is lost in their incomprehensibility, for these are issues discussed by the technical logicians largely among themselves. When one considers the important bearings implicit in the interpretation of implication, such a state of affairs is greatly to be deplored. Through a simple but adequate presentation of the meaning of a hypothetical proposition one may be led not only to an understanding of a fundamental paradox of logic but to the whole question of modal logic and of the philosophy of logic. It is my purpose in what follows to present as simply as possible the various interpretations of the meaning of implication and to show their relations to modal logic. Having done this I believe it will be possible to state the case in favor of material implication in a new light. (shrink)

In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox. In (...) order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule. (shrink)

Abstract Pavel Florensky solves Lewis Carroll’s ‘Barbershop’ paradox to support his reasoning in a previous chapter. Our discussion includes a) the problem (which we also refer to as the p paradox), b) Carroll’s solution, c) Bertrand Russell’s solution, d) Florensky’s solution and then e) a material example proffered by Florensky. Both Russell and Florensky disagree with Carroll’s solution, yet, (ostensibly) unbeknownst to themselves they offer the same solution, which is ‘p implies not-q’. Given Florensky’s material example, the solution (...) seems to tell us something about the logic of belief. We ask whether Florensky’s example has reverse implications for Russell’s solution. Content Type Journal Article Pages 1-10 DOI 10.1007/s11406-011-9333-6 Authors Michael Rhodes, Philosophy and Religious Studies (NPD), Chicago, IL, USA Journal Philosophia Online ISSN 1574-9274 Print ISSN 0048-3893. (shrink)

Many of the psychological studies carried out within virtual environments are motivated by the idea that virtual research findings are generalizable to the non-virtual world. This idea is vulnerable to the paradox of fiction, which questions whether it is possible to express genuine emotion toward a character (or event) known to be fictitious. As many of these virtual studies are designed to elicit, broadly speaking, emotional responses through interactions with fictional characters (avatars) or objects/places, the issue raised by the paradox (...) seems particularly apt. This paper assesses the extent to which the paradox of fiction constitutes a legitimate challenge to psychological research within virtual environments, and argues that any alleged conflict is in fact a product of an overly simplistic view of emotions which a more complete understanding resolves. Moreover, through a more detailed analysis of why the paradox cannot be sustained, one finds justification for the claim that emotions elicited through interactions with virtual (fictitious) objects/events are valid. However, their generalizability to the non-virtual world must still be treated with caution. (shrink)

It is widely assumed that a geometric model of boundaries, which prescribes a tripartite topological characterisation of the boundaries for material objects – fully open, fully closed, or partially open/closed – can be unproblematically extended from regions to material objects. Drawing on a disanalogy between regions and material objects – that only the latter move – I demonstrate the incoherence of open material objects through two arguments relating to the ability for such objects to freely move. (...) The first is a dilemma considering separately open material objects occupying their location directly or indirectly. It is argued that movement in the former case would involve a miraculous topological transformation; whilst in the latter case, would involve miraculous reorganising movements in the object's proper parts. The second argument reignites a problem regarding the moment of change specifically for the movement of such objects. (shrink)

This paper extends the defense of a simple theory of indicative conditionals previously proposed by the author, in which the truth conditions are material, and Grice-style assertability conditions are given to explain the paradoxes of material implication. The paper discusses various apparent counter-examples to the material account in which conditionals are not asserted, and so the original theory cannot be applied; it is argued that, nevertheless, the material theory can be defended.

It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by (...) classical philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the co-incidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato’s Theory of Forms and Aristotle’s metaphysics and logic. Plato’s and Aristotle’s systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a ‘principle of contradiction’ is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing “space” as the seat of stability, and “time” as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion ‘separable system’, related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of ‘non-locality’ as related to it, indicating the need to re-think the notions system, entity, as well as the implications of the operation ‘measurement’, which is seen here as an application of classical logic (including its ontological consequences) on the material world. (shrink)

This paper falls into two parts. In the first part, I argue that consideration of general indicative conditionals, e.g., sentences like If a donkey brays it is beaten, provides a powerful argument that a pure material implication analysis of indicative if p, q is correct. In the second part I argue, opposing writers like Jackson, that a Gricean style theory of pragmatics can explain the manifest assertability conditions of if p, q in terms of its conventional content – (...) assumed to be merely (p⊃q) – and the conversational implicature contents which utterance of if p, q may gain in certain contexts. I also defend the pragmatic approach against a recent objection by Edgington that appeal to pragmatics cannot explain what we are inclined to say about the believability conditions, as opposed to the assertability conditions, of indicative if p, q. (shrink)

It has been maintained that we are quite able to express (1*)–(4*) without the introduction of a dyadic deontic operator, provided only that we supply our standard deontic logic with a stronger conditional than material implication. The lesson learned from Chisholm's paradox has been the eminently convincing, indeed obvious, one: that what we ought to do is not determined by what is the case in some perfect world, but by what is the case in the best world we (...) can ‘get to’ from this world. What we ought to do depends upon how we are circumstanced. (shrink)

This paper resolves a paradox concerning colour constancy. On the one hand, our intuitive, pre-theoretical concept holds that colour constancy involves invariance in the perceived colours of surfaces under changes in illumination. On the other, there is a robust scientific consensus that colour constancy can persist in cerebral achromatopsia, a profound impairment in the ability to perceive colours. The first stage of the solution advocates pluralism about our colour constancy capacities. The second details the close relationship between colour constancy and (...) contrast. The third argues that achromatopsics retain a basic type of colour constancy associated with invariants in contrast processing. The fourth suggests that one person-level, conscious upshot of such processing is the visual awareness of chromatic contrasts ‘at’ the edges of surfaces, implicating the ‘colour for form’ perceptual function. This primitive type of constancy sheds new light on our most basic perceptual capacities, which mark the lower borders of representational mind. (shrink)

Many contemporary logicians acknowledge a plurality of logical theories and accept that theory choice is in part motivated by logical evidence. However, just as there is no agreement on logical theories, there is also no consensus on what constitutes logical evidence. In this paper, I outline Jean Piaget’s psychological theory of reasoning and show how he used it to diagnose and solve one of the paradoxes of material implication. I assess Piaget’s use of psychology as a source (...) of evidence for logical theory in light of reservations raised by psychologism, and I highlight some ramifications for exceptionalism and anti-exceptionalism about logic by considering his use of psychology as logical evidence in the framework of genetic epistemology, Piaget’s research programme. I conclude that Piaget’s psychological theory of reasoning not only plausibly serves as a source of evidence for logical theory but also makes a strong case for anti-exceptionalism about logic. (shrink)

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions (Section 1), we argue that a stronger account of conditional can be obtained in two steps: firstly, by reminding its historical roots inside modal logic and set-theory (Section 2); secondly, by revising the meaning of logical values, thereby getting rid of the paradoxes of material implication whilst showing the bivalent roots of conditional as a speech-act (...) based on affirmations and rejections (Section 3). Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed through a broader definition of logical consequence that encompasses both a normal relation of truth propagation and a weaker relation of falsity non-propagation from premises to conclusion (Section 3). (shrink)

Most instantiations of the inference ‘y; so if x, y’ seem intuitively odd, a phenomenon known as one of the paradoxes of the material conditional. A common explanation of the oddity, endorsed by Mental Model theory, is based on the intuition that the conclusion of the inference throws away semantic information. We build on this explanation to identify two joint conditions under which the inference becomes acceptable: (a) the truth of x has bearings on the relevance of asserting (...) y; and (b) the speaker can reasonably be expected not to be in a position to assume that x is false. We show that this dual pragmatic criterion makes accurate predictions, and contrast it with the criterion defined by the mental model theory of conditionals, which we show to be inadequate. (shrink)

As the title of this paper indicates, I’m going to discuss what we ought to do in situations where our actions affect future generations. More specifically, I shall focus on the moral problems raised by cases where our actions affect who’s going to live, their number and their well being. I’ll start, however, with population axiology. Most discussion in population ethics has concentrated on how to evaluate populations in regard to their goodness, that is, how to order populations by the (...) relations “is better than” and “is as good as”. This field has been riddled with “paradoxes” which purport to show that our considered beliefs are inconsistent in cases where the number of people and their welfare varies. Derek Parfit’s Mere Addition Paradox is a case in point. The main question of my paper concerns the implication of such axiological paradoxes for normative theories. Do the axiological paradoxes translate into paradoxes for normative theories or will they, as some believe, disappear if we switch to a normative framework? (shrink)

This paper replies to Politzer’s (2007) criticisms of the mental model theory of conditionals. It argues that the theory provides a correct account of negation of conditionals, that it does not provide a truth-functional account of their meaning, though it predicts that certain interpretations of conditionals yield acceptable versions of the ‘paradoxes’ of material implication, and that it postulates three main strategies for estimating the probabilities of conditionals.

The relation between indicative conditionals in natural language and material implication wasn’t a major topic in the Lvov-Warsaw school. However, a major defense of the claim that the truth conditions of these two are the same has been developed by Ajdukiewicz. The first major goal of this paper is to present, assess, and improve his strategy. It turns out that it is quite similar to the approach developed by Grice, so our second goal is to compare these two (...) and to argue that the accuracy of Ajdukiewicz’s explanation is less dependent on controversial properties of a systematic but convoluted general theory of cooperative communicative behavior. In Lvov-Warsaw school the relation between material implication and indicative conditionals was also discussed by Goł ąb :27–29, 1949) and Słupecki :32–35, 1949), so the third part of our paper is devoted to their discussion and relating it to Ajdukiewicz’s views. (shrink)

In the face of the puzzles of material constitution, some philosophers have been moved to posit a distinction between an object's matter and its form. A familiar difficulty for contemporary hylomorphism is to say which properties are eligible as forms: for example, it seems that it would be intolerably arbitrary to say that being statue shaped is embodied by some material object, but that other complex shape properties aren't. Anti-arbitrariness concerns lead quickly to a plenitudinous ontology. The usual (...) complaint is that the super-abundance of material objects is too extraordinary to accept, but I want to raise a different worry: I argue that the most natural way of developing this picture is already inconsistent. I show that a simple version of plenitudinous hylomorphism is subject to a Russellian argument, but argue that we cannot treat the problem straightforwardly as an instance of Russell's Paradox of Sets. (shrink)