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)

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)

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)

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)

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.

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.

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

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.

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)

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)

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.

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)

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)

El Menón es un buen ejemplo del uso de la ironía socrática como forma de refutación. Específicamente, la forma lógica de reducción al absurdo constituye un dispositivo muy relacionado con la ironía socrática. En este artículo, se examinan analíticamente los cuatro elementos de la ironía que se presentan en el diálogo: 1. el conocimiento falso como una posición inicial asumida por los interlocutores de Sócrates; 2. la igno­ rancia socrática; 3. la reducción al absurdo del falso saber inicial; y 4. (...) el reconocimiento del error y la ignorancia. (shrink)

Don Marquis' future‐like‐ours account is regarded as the best secular anti‐abortion position because he frames abortion as a wrongful killing via deprivation of a valuable future. Marquis objects to the reductio ad absurdum of contraception as being immoral because it is too difficult to identify an individual that is deprived of a future. To demonstrate why Marquis’ treatment of the contraception reductio is flawed by his own future‐like‐ours line of reasoning, I offer an argument for why there is (...) indeed a candidate for harm—the ovum—for it can be viewed as providing the functional foundation for a new life through (1) mitochondrial DNA inheritance, (2) paternal histone restructuring during fertilization, and (3) ability to initiate parthenogenesis. As evidenced by these distinct and natural features of ova, candidate (2) “some ovum or other” should be morally prioritized as the direct candidate for harm in the contraception reductio. By assessing the philosophical inconsistencies in Marquis’ future‐like‐ours argument, this paper provides strong metaphysical grounds for rejecting the best secular anti‐abortion position. (shrink)

Considerando que o cogito possa ser tomado, nas Meditações, como uma conclusão de uma demonstração, pode-se avançar a tese de que essa demonstração está consoante ao método analítico, que Descartes reconhece empreender nesse texto. Esse método teria entre as suas funções nas Meditações aquela de apresentar – sob a forma de uma rede de implicações ontológicas – o raciocínio que conduz à certeza da existência. Como cumpre no referido texto determinar a certeza da existência sem tomar como base nenhuma certeza (...) preestabelecida, o método analítico financiaria, segundo a nossa interpretação, uma reconstrução do argumento do cogito sob uma base indireta, mais precisamente através de uma reductio ad absurdum, cujo objetivo consiste em mostrar a contradição inelutável que surge na tentativa de um indivíduo de provar a sua não existência. (shrink)

Considerando que o cogito possa ser tomado, nas Meditações, como uma conclusão de uma demonstração, pode-se avançar a tese de que essa demonstração está consoante ao método analítico, que Descartes reconhece empreender nesse texto. Esse método teria entre as suas funções nas Meditações aquela de apresentar – sob a forma de uma rede de implicações ontológicas – o raciocínio que conduz à certeza da existência. Como cumpre no referido texto determinar a certeza da existência sem tomar como base nenhuma certeza (...) preestabelecida, o método analítico financiaria, segundo a nossa interpretação, uma reconstrução do argumento do cogito sob uma base indireta, mais precisamente através de uma reductio ad absurdum, cujo objetivo consiste em mostrar a contradição inelutável que surge na tentativa de um indivíduo de provar a sua não existência. (shrink)

The ‘Private Language’ sections of the Philosophical Investigations §§ 243–315 serve to undermine the idea that our ordinary felt sensations, e.g., of heat, or cold, or pain, together with our experienced impressions of colour or of sound, are ‘private’ or ‘inner’ objects, where an object mirrors in the mental realm what we associate with that of the physical. This paper explores Wittgenstein's method in these sections, together with the work of several of his commentators who agree with his ‘therapeutic’ approach (...) to the denial of a possible private language. More significantly, however, it discovers a paradox in the fact that a number of philosophers who claim that his ‘Argument Against Private Language’ is unsuccessful point in reaching this conclusion to the very features concerning first-person sensation ascription that Wittgenstein employs to reveal that our ordinary sensations are not ‘private’ to us in that sense identified in § 243, which talks of ‘a language used to refer to what only its speaker can know’, viz, ‘his immediate private sensations’. This helps to explain why there is still no general agreement on the significance of §§ 243–315. (shrink)

Norman Finkelstein, a prominent political scientist specializing in the Palestine-Israel conundrum, on which he has authored five highly praised books, was denied tenure at DePaul University by the President, Rev. Dennis H. Holtschneider, on June 8, 2007. After examining the particulars of the case, it strikes me as so obviously wrong to deny him tenure that the tenure procedure at DePaul constitutes a reductio ad absurdum of a university system which allows such a thing to happen.

Kimi zaman akademik hayatımızı doğrudan etkileyen kararların kimler tarafından nerede ve nasıl alındığını bilemeyebiliriz. Kimi zaman da bilsek bile bu akademik toplumun şeffaf olduğu anlamına gelmeyebilir. Doktora sonrası yurda dönmeden önce, tezimi okuyan hocalarımdan “Orada seni anlayan birilerini bulabilecek misin?” gibi bir soru gelmişti. Sorunun doğru yanıtının ne olduğundan emin değildim; hala da değilim.

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)