It is well known that reductio ad absurdum arguments raise a number of interesting philosophical questions. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? What kind of absurdity do we obtain? Moreover, in the mathematics education literature number of studies have shown that students find it difficult to truly comprehend the idea of reductio proofs, which indicates the cognitive complexity of these (...) constructions. In this paper, I start by discussing four philosophical issues pertaining to reductio arguments. I then briefly present a dialogical conceptualization of deductive arguments, according to which such arguments are best understood as a dialogue between two participants—Prover and Skeptic. Finally, I argue that many of the philosophical and cognitive difficulties surrounding reductio arguments are dispelled or at least further clarified once one adopts a dialogical perspective. (shrink)

Thomas Polger and Laurence Shapiro argue that Carl Gillett's much publicized dimensioned theory of realization is incoherent, being subject to a reductio. Their argument turns on the fact that Gillett's definition of realization makes property instances the exclusive relata of the realization relation, while his belief in multiple realization implies its denial, namely, that properties are the relata of the realization relation on occasions of multiple realization. Others like Sydney Shoemaker have also expressed their view of realization in terms (...) of property instances, yet they too have accepted the multiple realizability of properties. Thus I am interested in the more general issue raised by Polger and Shapiro's argument. Specifically, I show how to supplement a theory of realization with a category-inclusive auxiliary assumption, which avoids the stated reductio. I then offer a few reasons to justify the proposed category-inclusive view of realization, making some comparisons to supervenience and causation along the way. (shrink)

Epistemic infinitism, advanced in different forms by Peter Klein, Scott Aikin, and David Atkinson and Jeanne Peijnenburg, is the theory that justification of a proposition for a person requires the availability to that person of an infinite, non-repeating chain of propositions, each providing a justifying reason for its successor in the chain. The reductio argument is the argument to the effect that infinitism has the consequence that no one is justified in any proposition, because there will be an infinite (...) chain of reasons supporting any proposition (and similarly, a chain supporting its negation). Four ways of defending infinitism against the reductio argument are considered and found wanting: Peijnenburg and Atkinson’s use of probabilistic chains of reasons; Klein’s concept of emergent justification; Aikin’s insistence that there be non-propositional input in the justification of any proposition; and Klein’s use of the distinction between reasons that are and are not available to a person. I contend that, in the absence of some further defence, the reductio argument makes infinitism untenable. (shrink)

We introduce a novel expansion of the four-valued Belnap–Dunn logic by a unary operator representing reductio ad contradictionem and study its algebraic semantics. This expansion thus contains both the direct, non-inferential negation of the Belnap–Dunn logic and an inferential negation akin to the negation of Johansson’s minimal logic. We formulate a sequent calculus for this logic and introduce the variety of reductio algebras as an algebraic semantics for this calculus. We then investigate some basic algebraic properties of this (...) variety, in particular we show that it is locally finite and has EDPC. We identify the subdirectly irreducible algebras in this variety and describe the lattice of varieties of reductio algebras. In particular, we prove that this lattice contains an interval isomorphic to the lattice of classes of finite non-empty graphs with loops closed under surjective graph homomorphisms. (shrink)

This chapter focuses on one of the common fallacies in Western philosophy called 'Reductio ad Hitlerum (RAH)'. RAH is a species of the reductio ad hominem genre of logically fallacious reasoning. It is clear that ad hominem arguments, such as RAH, may be understood as “fallacies of relevance”. The most notorious example of the RAH in philosophy is the association of Martin Heidegger with Hitler and the Nazi Party. The RAH makes it seem as though philosophically critiquing the (...) Cartesian worldview or providing proofs of valid inference in a formal logic course would be tantamount to being desirous of perpetrating the Holocaust. Thus, the logical question regarding the truth‐value of Heidegger's philosophical claims is separate from the moral question of whether actions perpetrated by “Nazis”, or in the name of “Nazism”, were evil. (shrink)

Howell Pacific Philosophical Quarterly, 89, 348–358 and Mizrahi and Morrow Ratio, 28, 1–13 offer a group of reductio arguments to rebut the CP thesis, i.e., the hypothesis that ideal conceivability entails possibility. Each of them has a conceivability premise: it is ideally conceivable that CP is false. According to the same CP, one can infer that CP’s being false is possible, from which it follows that CP is false. In other words, CP is shown to be self-defeating and therefore (...) false if any of these reductio arguments is proved sound. In this paper, I aim to disprove these arguments. I first show that neither Howell nor Mizrahi & Morrow provide adequate justification for their conceivability premises. I go on to argue that these reductio arguments face a dilemma: either their conceivability premises cannot be adequately justified, or they are redundant. Finally, I argue that, although this dilemma resists the reductio arguments against CP, on the other hand, it also reveals that CP lacks epistemic fruitfulness: it is of no practical use in the realm of apriority as well as necessity. (shrink)

Cartesianism was always a subject to Wittgenstein's criticism. In his case against it, he employed a general strategy that I have called "Reductio ad vacuum". There is something right in Cartesianism but without a hidden confusing premise, the truth of Cartesianism is empty. According to the early Wittgenstein, Cartesianism was right be-cause Solipsism is true: the Self is the center of the world. But without confusing this metaphysical Self with the psychological one, Solipsism becomes empty and no different from (...) Realism. (shrink)

An argument is presented which shows that coherence theories of justification are committed to a conception of epistemic support which conflicts with an axiom of probability theory.

Despite the fact that the reductio ad absurdum argument is a valid deductive form, while the slippery slope argument is most often presented as a fallacious form of inductive argument, the two argument types bear some striking similarities. Investigation of these similarities reveals some more universal difficulties in the teaching of informal logic, and, in particular the difference between strong informal arguments and fallacious ones.

The present article contains part of results from my lecture delivered at II Flemish-Polish workshop on Ontological Foundation of Paraconsistency.

Is there any number of people you should save from paralysis rather than saving one person from death? Is there any number of people you should save from a migraine rather than saving one person from death? Many people answer “yes” and “no,” respectively. The aim of partially aggregative moral views is to capture and justify combinations of intuitions like these. In this article, I develop a risk-based reductio argument that shows that there can be no adequate partially aggregative (...) view. I then argue that the only plausible response to this reductio is to accept a fully aggregative view. (shrink)

A central part of Kripke's influential interpretation of Wittgenstein's sceptical argument about meaning is the rejection of dispositional analyses of what it is for a word to mean what it does. In this paper I show that Kripke's arguments prove too much: if they were right, they would preclude not only the idea that dispositional properties can make statements about the meanings of words true, but also the idea that dispositional properties can make true statements about paradigmatic dispositional properties such (...) as a cup's fragility or a person's bravery. However, since dispositional properties can make such statements true, Kripke-Wittgenstein's arguments against dispositionalism about meaning are mistaken. (shrink)

In “A Reliabilist Solution to the Problem of Promiscuous Bootstrapping”, Hilary Kornblith (2009) proposes a reliabilist solution to the bootstrapping problem. I’m going to argue that Kornblith’s proposal, far from solving the bootstrapping problem, in fact makes the problem much harder for the reliabilist to solve. Indeed, I’m going to argue that Kornblith’s considerations give us a way to develop a quick reductio of a certain kind of reliabilism. Let’s start with a crude statement of the problem. The bootstrapper, (...) call them S, looks at a device D1 that happens to be reliable, though at this stage S doesn’t know this. We assume that S is a reliable reader of devices. S then draws the following conclusions. (shrink)

At the center of the following essay is an analysis of At the Mind's Limits by Jean Améry––philosopher and survivor of Auschwitz. The essay tries to define and refine, via comparison and contrast with works by Hannah Arendt and René Descartes, the unique conception of morality that arises from Améry's text. “Victim morality,” as it will be called here, is a non-normative morality which is patient and victim-based rather than agent or actor-based. It is grounded in a heightened exposure and (...) sensitivity to reality and suffering, which results from a reduction of a human being to “a bundle of reactions.” This process cannot be voluntarily undertaken but is violently undergone. The claim is that morality not only survives the loss of humanity but is only fully exposed where humanity falters. My torturers were not interested in degrees of pain. They were interested only in demonstrating to me what it meant to live in a body, as a body, a body that can entertain notions of justice only as long as it is whole and well …. They did not come to force the story out of me of what I had said to the barbarians and what the barbarians had said to me …. They came to my cell to show me the meaning of humanity, and in the space of an hour they showed me a great deal. –J. M. Coetzee, Waiting for the Barbarians. (shrink)

In this paper, I present an argument to show that the doctrine of divine temporality (the view that God is in time, but everlastingly eternal) is incoherent. The doctrine of divine temporality entails that God has traversed an actually infinite series of moments in order to reach the present. But I show that an actually infinite series of moments cannot be traversed. Hence, God could not have traversed his infinite past to reach the present. Therefore, the doctrine of divine temporality (...) is necessarily false. I defend this argument against various responses and objections. (shrink)

Theists believe that God is eternal, but they differ as to just what God's eternality means . The traditional, historic view of most Christian philosophers is that eternality means that God is timeless. He is ‘outside’ of time and not subject to any kind of temporal change. Indeed, God is the creator of time. Lets call this view divine timelessness.

As Federal Indian Law has evolved, many questions have been posed regarding tribal jurisdiction. This paper examines the jurisdiction tribes have over member Indians, non-member Indians, and non-member, non-Indians. It addresses the ethical challenge faced by tribal attorneys who represent non-member Indian clients in a manner that ultimately undermines tribal sovereignty.

Pierre Bayle is perhaps most well-known for arguing in his Dictionary (1697) that the problem of evil cannot be solved by reason alone. This skepticism about theodicy is usually credited to a religious crisis suffered by Bayle in 1685 following the unjust imprisonment and death of his brother, the death of his father, and the revocation of the Edict of Nantes. But in this paper I argue that Bayle was skeptical about theodicy a decade earlier than these events, from at (...) least the time of his Sedan philosophy course (1675–77). I then argue that both the Various Thoughts on the Comet (1683) and Philosophical Commentary on Luke 4:23 (1686–88), which are usually read as treatments of superstition and toleration respectively, are works that also closely engage the problem of evil and demonstratethe skepticism of Bayle toward theodicy. (shrink)

Pierre Bayle is perhaps most well-known for arguing in his Dictionary (1697) that the problem of evil cannot be solved by reason alone. This skepticism about theodicy is usually credited to a religious crisis suffered by Bayle in 1685 following the unjust imprisonment and death of his brother, the death of his father, and the revocation of the Edict of Nantes. But in this paper I argue that Bayle was skeptical about theodicy a decade earlier than these events, from at (...) least the time of his Sedan philosophy course (1675–77). I then argue that both the Various Thoughts on the Comet (1683) and Philosophical Commentary on Luke 4:23 (1686–88), which are usually read as treatments of superstition and toleration respectively, are works that also closely engage the problem of evil and demonstratethe skepticism of Bayle toward theodicy. (shrink)

In this paper, we argue that formal ontological relations avoid Bradley's reductio of relations, including his famous relation regress.

Don Marquis’ future-like-ours account argues that abortion is seriously immoral because itdeprives the embryo or fetus of a valuable future much like our own. Marquis was mindful ofcontraception being reductio ad absurdum of his reasoning, and argued that prior tofertilisation, there is not an identifiable subject of harm. Contra Marquis, Tomer Chaffercontends that the ovum is a plausible subject of harm, and therefore contraception deprives theovum of a future-like-ours. In response, I argue that being an identifiable subject of harm (...) is notsufficient for Marquis’ argument to succeed. In addition, we must also share our identity withan ovum. I show that on the account of personal identity utilised by Marquis’ defenders, weare not identical with an ovum. As a result, Chaffer’s reductio fails. (shrink)

Ibn Sīnā proposed an analysis of arguments by reductio ad absurdum. His analysis contains, perhaps for the first time, a workable method for handling the making and discharging of assumptions in a formal proof. We translate the relevant text of Ibn Sīnā and put his analysis into the context of his general approach to logic.

By means of a reductio argument, Wittgenstein’s Tractatus calls into question the very idea that we can represent logical form. My paper addresses three interrelated questions: first, what conception of logical form is at issue in this argument? Second, whose conception of logic is this argument intended to undermine? And third, what could count as an adequate response to it? I show that the argument construes logical form as the universal, underlying correlation of any representation and the reality it (...) represents. I further show that the argument seeks to undermine core commitments of Frege’s and Russell’s. But the reductio, as I read it, is not intended to establish the falsity of any of their specific assumptions. Rather, its aim is to make manifest the indeterminacies that underlie the language in which these assumptions are framed, and establish the need for a transformation of that language. So understood, Wittgenstein’s argument exemplifies his idea that philosophy is not a theory, but an activity of elucidation. The interpretation I propose bears on one of the central debates in the literature, namely how we should understand Wittgenstein’s contention that his elucidations succeed despite being nonsensical. (shrink)

In recent years two‐dimensional semantics has become one of the most serious alternatives to Millianism for the proper interpretation of modal discourse. It has origins in the works of a diverse group of philosophers, and it has proven popular as an interpretation of both language and thought. It has probably received most of its attention, however, because of its use by David Chalmers in his arguments against materialism. It is this more metaphysical application of two‐dimensionalism that is the concern in (...) this paper. For though there is probably something salvageable from two‐dimensionalism as a way to explain the content of thought, as a metaphysical tool it should be abandoned. In this paper I aim to establish this point by reductio: if ‘metaphysical’ two‐dimensionalism is assumed, it can be shown to be false. (shrink)

Recently, a number of cases have been proposed which seem to show that – contrary to widely held opinion – a subject can inferentially come to know some proposition p from an inference which relies on a false belief q which is essential. The standard response to these cases is to insist that there is really an additional true belief in the vicinity, making the false belief inessential. I present a new kind of case suggesting that a subject can inferentially (...) come to know a proposition from an essential false belief where no truth in the vicinity seems to be present. (shrink)

ABSTRACT William Hurlbut, a Stanford University bioethicist and member of the President's Council on Bioethics, recently proposed a solution to the current impasse over human embryonic stem cell research in the United States. He suggested that researchers could use genetic engineering and somatic cell nuclear transfer (i.e. cloning) to develop human ‘pseudo‐embryos’ that have no potential to develop fully into human persons. According to Hurlbut, even thinkers who typically ascribe high moral status to human embryos could approve of destroying these (...) ‘pseudo‐embryos’ for the sake of harvesting human embryonic stem cells. This essay argues, first, that an argument based on the ‘paradox of the heap’ (an argument that many ‘pro‐life’ thinkers employ in order to defend the notion that human embryos have high moral value from the moment of conception) challenges the ethical legitimacy of Hurlbut's proposal. Second, the paper argues that this conflict may illustrate a reductio ad absurdum for this ‘pro‐life’ argument itself rather than being a problem for Hurlbut's proposal. As a result, the paper challenges the ‘pro‐life’ strategy of arguing that one should respond to uncertainty about the moral status of developing embryos by being morally ‘cautious’ and granting all human embryos full moral status from the moment of conception. It appears that one is faced with a complex series of choices (about where to draw the moral line between entities that are human persons and entities that are not), and a strict moral ‘cautiousness’ about this series of choices may ultimately lead to absurdity. (shrink)

Principles of procreative beneficence (PPBs) hold that parents have good reasons to select the child with the best life prospects. Sparrow (2010) claims that PPBs imply that we should select only female children, unlesswe attach normative significance to “normal” human capacities. We argue that this claim fails on both empirical and logical grounds. Empirically, Sparrow’s argument for greater female wellbeing rests on a selective reading of the evidence and the incorrect assumption that an advantage for females would persist even when (...) a serious gender imbalance is obtained. Logically, PPBs cite only pro tanto reasons and allow that the good of an individual child could be outweighed by other morally relevant considerations, such as those which take collectively suboptimal outcomes into account. There is thus no need to attach value to the “normal.”. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Bonaventure's Reductio of the Nine Choirs of Angels:How Bonaventure Compressed Two Monumental Traditions into Nine Words and Nine Short PhrasesRandall B. Smith"There is probably no better illustration in medieval thought of how the genius of the symbolic imagination also involves deep speculative insight." So wrote Bernard McGinn of Bonaventure's Itinerarium mentis in deum in The Flowering of Mysticism: Men and Woman in the New Mysticism, 1200–1350.1 There is (...) no question that the Itinerarium is a book suffused with symbolic imagery; so much so, in fact, that it can sometimes seem to lapse into a confusing jumble.Certainly there are many sections of the Itinerarium that have baffled readers over the years, often because Bonaventure's use of imagery is so unique and creative and the truths he is attempting to enunciate are so far beyond our normal ways of thinking about God. But there is a short passage in chapter 4 of the Itinerarium that seems to defy interpretation precisely because of what one might describe as an almost austere terseness and brevity. The point Bonaventure wishes to make is one he communicates in little more than a list of nine words and nine short, parallel phrases.The passage I have in mind is this one in Itinerarium 4.4.When these things have been accomplished, and our spirit has been brought into conformity with the heavenly Jerusalem, it is ordered [End Page 583] hierarchically so that it can ascend upward. For no one enters into that city unless that city has first descended into the person's heart by means of grace, as John sees in his Apocalypse. It descends into the heart when our spirit has been made hierarchical by the reformation of the image, by means of the theological virtues, the enjoyment of the spiritual senses, and the ecstasy of rapture; for it has been purged, illumined and brought to perfection.In this way our spirit is adorned with nine orderly levels when within it the following are found in an appropriate order: announcing, dictating, leading, ordering, strengthening, commanding, receiving, revealing, and anointing. These correspond to the nine choirs of angels. The first three of the foregoing levels in the human mind relate to nature; the next three relate to work; and the final three relate to grace. When it has attained these, the soul, by entering into itself, enters into the heavenly Jerusalem where, as it considers the choirs of angels, it sees in them the God who dwells in them and who works in all their operations.Therefore, Bernard says to Pope Eugene that "God loves in the Seraphim as charity; knows in the Cherubim as truth; sits in the Thrones as justice; reigns in the Dominations as majesty; rules in the Principalities as a guiding principle; protects in the Powers as salvation; is at work in the Virtues as strength; reveals in the Archangels as light; assists in the Angels as kindness." From all this, God is seen as all in all when we contemplate God in our minds where God dwells through the gifts of the most generous charity.2Notice in the passage I have quoted above that Bonaventure says our spirit must be "brought into conformity with the heavenly Jerusalem," and that this requires it be "ordered hierarchically so that it can ascend upward." This serves as a transition into a rather odd pair of lists—one with nine words and the other with nine short, parallel phrases—that Bonaventure associates with the nine angelic choirs. These two lists of the nine choirs of angels do not, however, correspond exactly with one [End Page 584] another, for reasons I will explain below. And although these two lists of the nine ranks of angels in the angelic hierarchy take up just a few lines in Bonaventure's text, there is a lot of background information we will need to fill in to provide the necessary context to appreciate them. Indeed, as we will see, in this short passage, Bonaventure was able using the methods of the thirteenth-century sermo modernus style to reduce two monumental commentary traditions regarding the nine choirs of angels—one that... (shrink)

The uses and interpretation of reductio ad absurdum argumentation in mathematical proof and discovery are examined, illustrated with elementary and progressively sophisticated examples, and explained. Against Arthur Schopenhauer’s objections, reductio reasoning is defended as a method of uncovering new mathematical truths, and not merely of confirming independently grasped mathematical intuitions. The application of reductio argument is contrasted with purely mechanical brute algorithmic inferences as an art requiring skill and intelligent intervention in the choice of hypotheses and attribution (...) of contradictions deduced to a particular assumption in a contradiction’s derivation base within a reductio proof structure. (shrink)

Joe Horton argues that partial aggregation yields unacceptable verdicts in cases with risk and multiple decisions. I begin by showing that Horton’s challenge does not depend on risk, since exactly similar arguments apply to riskless cases. The underlying conflict Horton exposes is between partial aggregation and certain principles of diachronic choice. I then provide two arguments against these diachronic principles: they conflict with intuitions about parity, prerogatives, and cyclical preferences, and they rely on an odd assumption about diachronic choice. Finally, (...) I offer an explanation, on behalf of partial aggregation, for why these diachronic principles fail. (shrink)

Plato’s Parmenides and Lysis have a surprising amount in common from a methodological standpoint. Both systematically employ a method that I call ‘exploring both sides’, a philosophical method for encouraging further inquiry and comprehensively understanding the truth. Both have also been held in suspicion by interpreters for containing what looks uncomfortably similar to sophistic methodology. I argue that the methodological connections across these and other dialogues relieve those suspicions and push back against a standard developmentalist story about Plato’s method. This (...) allows for a better understanding of why exploring both sides is explicitly recommended in the Parmenides and its role within Plato’s broader methodological repertoire. (shrink)

Used informally, the Reductio ad Absurdum (RAA) consists in reasoning appealing to the logically implied, absurd consequences of a hypothetical proposition, in order to refute it. This kind of reasoning resembles the Argument from Consequences, which appeals to causally induced consequences. These types of argument are sometimes confused, since it is not worked out how these different kinds of consequences should be distinguished. In this article it is argued that the logical consequences in RAA-argumentation can take different appearances and (...) that it therefore must be concluded that RAA cannot be characterised by a specific content, but must instead be characterised as an argument form. Furthermore, clues are provided to distinguish RAA reasoning from the Argument from Consequences. (shrink)

Harman [8] would concede that (1)–(3) are inconsistent, and (as a result) that something is wrong with premises (1)–(3). But, he would reject the relevantists’ diagnosis that (1) must be rejected. I take it he’d say it’s (2) that is to blame here. (2) is a bridge principle [12] linking entailment and inference. (2) is correct only for consistent B’s. [Even if B is consistent, the correct response may rather be to reject some Bi’s in B.].