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)

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)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we (...) go through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (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.

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. I provide a counter-example to epistemic closure for reductio proofs. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

In his well-known paper on Euclid’s geometry, Ken Manders sketches an argument against conceiving the diagrams of the Elements in ‘semantic’ terms, that is, against treating them as representations—resting his case on Euclid’s striking use of ‘impossible’ diagrams in some proofs by contradiction. This paper spells out, clarifies and assesses Manders’s argument, showing that it only succeeds against a particular semantic view of diagrams and can be evaded by adopting others, but arguing that Manders nevertheless makes a compelling case that (...) semantic analyses ought to be relegated to a secondary role for the study of mathematical practices. (shrink)

This paper targets a series of potential issues for the discussion of, and modal resolution to, the alethic paradoxes advanced by Scharp (2013). I proffer four novel extensions of the theory, and detail six issues that the theory faces. I provide a counter-example to epistemic closure for reductio proofs.

Using a quantified propositional logic involving the operators it is known that and it is possible to know that, we formalize various interesting philosophical claims involved in the realism debate. We set out inferential rules for the epistemic modalities, ranging from ones that are obviously analytic, to ones that are epistemologically more substantive or even controversial. Then we investigate various aporias for the realism debate. These are constructively inconsistent triads of claims from our list: a claim expressing some sort of (...) common ground in the debate; a characteristically anti-realist thesis about truth and knowability; and a characteristically realist thesis about determinacy of truth value. Various patterns of reductio proof for these inconsistent triads are generated, so as to display their variety. The reductio proofs use only the inferential rules set out earlier. The philosophical utility of each aporia for the anti-realist is then assessed. This involves consideration of the acceptability of the premiss expressing common ground; the strength and plausibility of the anti-realist premiss; the strength of the realist premiss that is the ultimate target of the reductio; and the analytic status of the inferential rules applied within the proof. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. I provide a counter-example to epistemic closure for reductio proofs. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Frege famously claimed that logic is the science of truth: “To discover truths is the task of all science; it falls to logic to discern the laws of truth” (Frege, 1956, p. 289). But just like the other foundational concept of set, truth at that time was intimately associated with paradox; in the case of truth, the Liar paradox. The set-theoretical paradoxes had their teeth drawn by being recognised as reductio proofs of assumptions that had seemed too obvious to (...) warrant stating explicitly, but were now seen to be substantive, and more importantly inconsistent. Tarski includes the Liar paradox in his classic discussion of the concept of truth (Tarski, 1956), and developed it, in the form of his famous theorem on the undefinability of truth, as a reductio of the assumption that a language could be semantically closed, in the sense of being able to contain its own truth-predicate.Frege famously claimed that logic is the science of truth: “To discover truths is the task of all science; it falls to logic to discern the laws of truth” (Frege, 1956, p. 289). But just like the other foundational concept of set, truth at that time was intimately associated with paradox; in the case of truth, the Liar paradox. The set-theoretical paradoxes had their teeth drawn by being recognised as reductio proofs of assumptions that had seemed too obvious to warrant stating explicitly, but were now seen to be substantive, and more importantly inconsistent. Tarski includes the Liar paradox in his classic discussion of the concept of truth (Tarski, 1956), and developed it, in the form of his famous theorem on the undefinability of truth, as a reductio of the assumption that a language could be semantically closed, in the sense of being able to contain its own truth-predicate. (shrink)

Frege famously claimed that logic is the science of truth: “To discover truths is the task of all science; it falls to logic to discern the laws of truth”. But just like the other foundational concept of set, truth at that time was intimately associated with paradox; in the case of truth, the Liar paradox. The set-theoretical paradoxes had their teeth drawn by being recognised as reductio proofs of assumptions that had seemed too obvious to warrant stating explicitly, but (...) were now seen to be substantive, and more importantly inconsistent. Tarski includes the Liar paradox in his classic discussion of the concept of truth, and developed it, in the form of his famous theorem on the undefinability of truth, as a reductio of the assumption that a language could be semantically closed, in the sense of being able to contain its own truth-predicate.Frege famously claimed that logic is the science of truth: “To discover truths is the task of all science; it falls to logic to discern the laws of truth”. But just like the other foundational concept of set, truth at that time was intimately associated with paradox; in the case of truth, the Liar paradox. The set-theoretical paradoxes had their teeth drawn by being recognised as reductio proofs of assumptions that had seemed too obvious to warrant stating explicitly, but were now seen to be substantive, and more importantly inconsistent. Tarski includes the Liar paradox in his classic discussion of the concept of truth, and developed it, in the form of his famous theorem on the undefinability of truth, as a reductio of the assumption that a language could be semantically closed, in the sense of being able to contain its own truth-predicate. (shrink)

In my dissertation, I argue that contemporary interpretive work on Kant's philosophy of geometry has failed to understand properly the diagrammatic aspects of Euclidean reasoning. Attention to these aspects is amply repaid, not only because it provides substantial insight into the role of intuition in Kant's philosophy of mathematics, but also because it brings out both the force and the limitations of Kant's philosophical account of geometry. ;Kant characterizes the predecessors with which he was engaged as agreeing that mathematical judgments (...) are analytic a priori. This is an inadequate account of geometrical judgments for Kant, because it cannot be reconciled with the evident success of the Euclidean practice in producing a priori knowledge of the nature of space. The ubiquitous presence of diagrams in Euclidean proofs suggests that, at least in Kant's time, any plausible account of the establishment of geometrical judgments must provide for the role of quasi-perceptual representations of individuals. Kant's claim that mathematical judgments are synthetic a priori reflects, in its positive aspect, his recognition of the importance of these 'intuitive' representations to Euclidean reasoning. Furthermore, Kant's doctrine that geometrical knowledge is established by the construction of concepts in intuition can be usefully understood as an attempt to accommodate the role of diagrams in the establishment of synthetic a priori geometrical judgments. ;Both Kant's rejection of the analytic a priori account of mathematics and his positive account of geometry depend on the plausibility of his appeals to Euclidean practice. Attention to the role of diagrams in Euclidean proof supports Kant's contention that intuitive representations are an essential component of Euclidean reasoning; however, the role of diagrams in both reductio proofs and in proofs by cases brings out tensions in Kant's positive account. The use of diagrams to represent impossible geometrical situations in reductio proofs indicates that Kant's notion of 'construction in intuition' can have a broader application than the construction of instances of geometrical concepts . Similarly, proofs by cases undermine Kant's attempts to explain the generality of judgments established through the consideration of individuals by appeal to the rules for the construction of those individuals. ;Contemporary interpretive debates about Kant's philosophy of geometry have centered on the philosophical role played by his appeal to intuition. This appeal cannot be primarily understood as an attempt to establish that the fundamental assumptions of Euclidean geometry are necessary truths about space. Nor can the role for intuition be regarded as merely licensing moves in geometrical proofs like an inference rule in modern logic. Instead, by understanding Kant's appeal to intuition as, at least in part, an attempt to accommodate the diagrammatic features of Euclidean reasoning, a much richer role for intuition is revealed. Intuition is required to play a logical role by expressing spatial relationships and establishing the existence of geometrical entities. In addition, it plays a quasi-perceptual role by supporting the truth of certain claims appealed to during the course of Euclidean proof. (shrink)

My purpose in this paper is to argue for two separate, but related theses. The first is that contemporary analytic philosophy is incoherent. This is so, I argue, because its methods contain as an essential constituent a conception of intuition that cannot be rendered consistent with a key tenet of analytic philosophy unless we allow a Bayesian-subjectivist epistemology. I argue for this within a discussion of two theories of intuition: a classical account as proposed by Descartes and a modern reliabilist (...) account as proposed by Kornblith, maintaining that reliabilist accounts require a commitment to Bayesian subjectivism about probability. However, and this is the second thesis, Bayesian subjectivism is itself logically incoherent given two simple assumptions: (1) that some empirical propositions are known and (2) any proposition that is known is assigned a degree of subjective credence of 1. I establish this using a formal reductio proof. I argue for the first thesis in section 2 and for the second in section 3. The final section contains a summary and conclusion. (shrink)

Bilateralists hold that the meanings of the connectives are determined by rules of inference for their use in deductive reasoning with asserted and denied formulas. This paper presents two bilateral connectives comparable to Prior's tonk, for which, unlike for tonk, there are reduction steps for the removal of maximal formulas arising from introducing and eliminating formulas with those connectives as main operators. Adding either of them to bilateral classical logic results in an incoherent system. One way around this problem is (...) to count formulas as maximal that are the conclusion of reductio and major premise of an elimination rule and to require their removability from deductions. The main part of the paper consists in a proof of a normalisation theorem for bilateral logic. The closing sections address philosophical concerns whether the proof provides a satisfactory solution to the problem at hand and confronts bilateralists with the dilemma that a bilateral notion of stability sits uneasily with the core bilateral thesis. (shrink)

Free Semantics is based on normalized natural deduction for the weak relevant logic DW and its near neighbours. This is motivated by the fact that in the determination of validity in truth-functional semantics, natural deduction is normally used. Due to normalization, the logic is decidable and hence the semantics can also be used to construct counter-models for invalid formulae. The logic DW is motivated as an entailment logic just weaker than the logic MC of meaning containment. DW is the logic (...) focussed upon, but the results extend to MC. The semantics is called 'free semantics' since it is disjunctively and existentially free in that no disjunctive or existential witnesses are produced, unlike in truth-functional semantics. Such 'witnesses' are only assumed in generality and are not necessarily actual. The paper sets up the free semantics in a truth-functional style and gives a natural deduction interpetation of the meta-logical connectives. We then set out a familiar tableau-style system, but based on natural deduction proof rather than truth-functional semantics. A proof of soundness and completeness is given for a reductio system, which is a transform of the tableau system. The reductio system has positive and negative rules in place of the elimination and introduction rules of Brady's normalized natural deduction system for DW. The elimination-introduction turning points become closures of threads of proof, which are at the points of contradiction for the reductio system. (shrink)

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the (...) given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations. (shrink)

In this paper we revisit Pavel Tichý’s novel distinction between one-dimensional and two-dimensional conception of inference, which he presented in his book Foundations of Frege’s Logic (1988), and later in On Inference (1999), which was prepared from his manuscript by his co-author Jindra Tichý. We shall focus our inquiry not only on the motivation behind the introduction of this non-classical concept of inference, but also on further inspection of selected Tichý’s arguments, which we see as the most compelling or simply (...) most effective in providing support for his two-dimensional account of inference. Main attention will be given to exposing the failure of one-dimensional theory of inference in its explanation of indirect (reductio ad absurdum) proofs. Lastly, we discuss shortly the link between two-dimensional inference and deduction apparatus of Tichý’s Transparent Intensional Logic. (shrink)

This paper presents the main aspects of the views of the Tibetan logician Phya pa Chos kyi seng ge (1109–1169) on argumentation “by consequence” (thal ʼgyur, Skt. prasaṅga) based on his exposition of the topic in the fifth chapter of his Tshad ma yid kyi mun sel and on a parallel excursus in his commentary on Dharmakīrti’s Pramānaviniścaya. It aims at circumscribing primarily the nature and function of consequences (thal ʼgyur/thal ba) for this author—in particular the distinction between “proving consequences” (...) and “refuting consequences”—and the form prescribed for their enunciation in the context of debate. In addition to pointing out differences with the systems adopted by his predecessors, contemporaries and successors, the paper also discusses some of the similarities and differences between Phya pa’s understanding of argumentation by consequence and the notion of reductio ad absurdum in Western logic. (shrink)

Corcoran, John. 1974. Aristotelian Syllogisms: Valid arguments or true generalized conditionals?, Mind 83, 278–81. MR0532928 (58 #27178) This tightly-written and self-contained four-page paper must be studied and not just skimmed. It meticulously analyses quotations from Aristotle and Lukasiewicz to establish that Aristotle was using indirect deductions—as required by the natural-deduction interpretation—and not indirect proofs—as required by the axiomatic interpretation. Lukasiewicz was explicit and clear about the subtle fact that Aristotle’s practice could not be construed as correctly performed indirect proof. (...) Lukasiewicz evidence is presented fully; it is irrefutable. But, instead of considering the possibility that Aristotle’s discourses were not intended to express indirect proofs of universalized conditions presupposing axiomatic premises, Lukasiewicz came to the amazing conclusion that Aristotle did not understand indirect proof. This paper builds on the admirable Lukasiewicz scholarship to establish a conclusion diametrically opposed to the one Lukasiewicz asserted. This paper points out that if Aristotle had not been establishing an underlying logic but he was instead axiomatizing a theory of terms, Aristotelians could not claim for Aristotle the title Founder of Logic. People who take Euclid, Peano, and Zermelo to have first axiomatized geometry, arithmetic, and set theory, respectively, do not think that this qualifies them for the titles Founder of Geometry, Founder of Arithmetic, and Founder of Set Theory, respectively. Such people do think that this would qualify the three for the titles Founder of Axiomatic Geometry, Founder of Axiomatic Arithmetic, and Founder of Axiomatic Set Theory, respectively: titles that carry no honors in logic. Being the Founder of Axiomatic Term Theory would likewise carry no honors in the field of logic. CORRECTION: The paper inadvertently implied that Euclid was the first to axiomatize geometry. In order to understand Aristotle’s Analytics as a treatise on demonstration it helps to realize that axiomatized geometry was studied in Plato’s Academy when Aristotle was a student. Euclid was the author of the only ancient axiomatized geometry now available. (shrink)

In the literature on supervaluationism, a central source of concern has been the acceptability, or otherwise, of its alleged logical revisionism. I attack the presupposition of this debate: arguing that when properly construed, there is no sense in which supervaluational consequence is revisionary. I provide new considerations supporting the claim that the supervaluational consequence should be characterized in a ‘global’ way. But pace Williamson (1994) and Keefe (2000), I argue that supervaluationism does not give rise to counterexamples to familiar inference-patterns (...) such as reductio and conditional proof. (shrink)

There is an argument which seems to show that if all truths are knowable then all truths are known. It may be viewed as a "reductio ad absurdum" of certain forms of antirealism. However, The claim has been made elsewhere that the argument fails against antirealists who employ constructivist rather than classical logic. The paper defends and amplifies this claim against criticisms by crispin wright and others. Relations between knowability and time are discussed. Suggestions are also made about the (...) proper form of the constructivist semantics for the conditional, Involving a non-Platonistic version of the distinction between types and tokens of proofs. (shrink)

The Aristotelian syllogistic cannot account for the validity of certain inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether (...) class='Hi'>reductio ad absurdum is needed. Thus our main goal is to derive results on the existence (or nonexistence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments. (shrink)

Our question is “Are there Cultural Universals?” I propose a reductio ad absurdum proof for an affirmative answer as follows. Suppose there were no cultural universals. Then inter-cultural communication would be impossible. But there is inter-cultural communication. Therefore, there are cultural universals. Let me now try to unpack this epitome of a proof. I start with the premiss that there is inter-cultural communication. This is too visible in the present-day world to be disputed; what may need arguing (...) is what it implies. But not everything regarding its implications is open to debate. For example, it is tautologically obvious that for any two persons to communicate at all they must share some common medium of communication. In turn this implies that at some level they must share a conceptual scheme, however minimal its dimensions. Any such scheme of concepts is a universal for, at least, the given participants in the communication. The question now is “Is there any scheme of concepts which can be shared by all the cultures of humankind?”. (shrink)

We study how to postpone the application of the reductio ad absurdum rule (RAA) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of RAA, which induces a negative translation from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

This paper studies the relationship between Argumentation Logic, a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic. In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of sentences in AL and Natural Deduction proofs of the complement of these sentences. The proof of this equivalence uses a restricted (...) form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies. (shrink)

According to the “paradox of knowability”, the moderate thesis that all truths are knowable – ... – implies the seemingly preposterous claim that all truths are actually known – ... –, i.e. that we are omniscient. If Fitch’s argument were successful, it would amount to a knockdown rebuttal of anti-realism by reductio. In the paper I defend the nowadays rather neglected strategy of intuitionistic revisionism. Employing only intuitionistically acceptable rules of inference, the conclusion of the argument is, firstly, not (...) ..., but .... Secondly, even if there were an intuitionistically acceptable proof of ..., i.e. an argument based on a different set of premises, the conclusion would have to be interpreted in accordance with Heyting semantics, and read in this way, the apparently preposterous conclusion would be true on conceptual grounds and acceptable even from a realist point of view. Fitch’s argument, understood as an immanent critique of verificationism, fails because in a debate dealing with the justification of deduction there can be no interpreted formal language on which realists and anti-realists could agree. Thus, the underlying problem is that a satisfactory solution to the “problem of shared content” is not available. I conclude with some remarks on the proposals by J. Salerno and N. Tennant to reconstruct certain arguments in the debate on anti-realism by establishing aporias. (shrink)

Brogaard and Salerno (2005, Nous, 39, 123–139) have argued that antirealism resting on a counterfactual analysis of truth is flawed because it commits a conditional fallacy by entailing the absurdity that there is necessarily an epistemic agent. Brogaard and Salerno's argument relies on a formal proof built upon the criticism of two parallel proofs given by Plantinga (1982, "Proceedings and Addresses of the American Philosophical Association", 56, 47–70) and Rea (2000, "Nous," 34, 291–301). If this argument were conclusive, antirealism (...) resting on a counterfactual analysis of truth should probably be abandoned. I argue however that the antirealist is not committed to a controversial reading of counterfactuals presupposed in Brogaard and Salerno's proof, and that the antirealist can in principle adopt an alternative reading that makes this proof invalid. My conclusion is that no reductio of antirealism resting on a counterfactual analysis of truth has yet been provided. (shrink)

This thesis is a critique of Jaakko Hintikka’s reconstruction of Kantian intuition in logical and mathematical reasoning. I argue that Hintikka’s reconstruction of Kantian intuition in particular and his reconstruction of Kant's philosophy of mathematics in general fails to be successful in two ways: First, the logical formula which contains an instantiated term (henceforth, instantial term) that is introduced by the rule of existential instantiation in the ecthesis part of a proof of an argument is not even a proper (...) singular proposition whose relation to its object is supposed to be immediate. It is not a proper singular proposition because its truth conditions are general, i.e., it makes a general statement about a class of individuals of the sort instantiated- a statement whose analysis is based on quantifiers. Second, I show that certain proofs in mathematics- those in the form of reductio ad absurdum- are not captured by Hintikka’s reconstruction of Kant’s philosophy of mathematics either. (shrink)

This book consists of 220 logic problems on which students can practice their beginner's logic skills. At least one solution is provided for each exercise. The point is to provide a vehicle for practice that will not make additional demands on the instructor's time. In addition, Logic Problems, unlike most other "secondary" texts, does not require the additional purchase of a primary text. It includes sentential and predicate arguments, and employs truth tables, formal proofs, conditional proofs and reductio. Contents: (...) The Apparatus; Sentential Problems; Predicate Problems; Sentential Solutions; Predicate Solutions. (shrink)

I argue for an approach to Philosophical Fragments that allows it to be philosophical and fragmentary, and that pays particular attention to the fragments, or crumbs, that seem least important. One such overlooked crumb is the theory of merely human education in the book—education that does not enlist God as the teacher, where humans simply try to teach and learn from each other. I argue that Philosophical Fragments defends this theory of education with several reductio ad absurdum proofs that (...) are especially useful because they clarify why merely human education so often fails. Finally I apply the theory of merely human education in Philosophical Fragments to Kierkegaard’s authorship as a whole and argue that it gives us a paradigm for understanding all of Kierkegaard’s texts that is more fundamental than the distinction between direct and indirect communication. (shrink)

Can the so-ca\led first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gбdel built his proof on the ground of self-reference: а statement which claims its unprovabllity. So, he demonstrated that undecidaЬle propositions exist in any enough rich axiomatics (i.e. such one which contains Peano arithmetic in some sense). What about the decidabllity of the very first incompleteness theorem? We can display that it fulfills its conditions. (...) That's why it can Ье applied to itself, proving that it is an undecidaЬle statement. It seems to Ье а too strange kind of proposition: its validity implies its undecidabllity. If the validity of а statement implies its untruth, then it is either untruth (reductio ad absurdum) or an antinomy (if also its negation implies its validity). А theory that contains а contradiction implies any statement. Appearing of а proposition, whose validity implies its undecidabllity, is due to the statement that claims its unprovability. Obviously, it is а proposition of self-referential type. Ву Gбdel's words, it is correlative with Richard's or liar paradox, or even with any other semantic or mathematical one. What is the cost, if а proposition of that special kind is used in а proof? ln our opinion, the price is analogous to «applying» of а contradictory in а theory: any statement turns out to Ье undecidaЬ!e. Ifthe first incompleteness theorem is an undecidaЬ!e theorem, then it is impossiЬle to prove that the very completeness of Peano arithmetic is also an tmdecidaЬle statement (the second incompleteness theorem). Hilbert's program for ап arithmetical self-foundation of matheшatics is partly rehabllitated: only partly, because it is not decidaЬ!e and true, but undecidaЬle; that's wby both it and its negation шау Ье accepted as true, however not siшultaneously true. The first incompleteness theoreш gains the statute of axiom of а very special, semi-philosophical kind: it divides mathematics as whole into two parts: either Godel шathematics or Нilbert matheшatics. Нilbert's program of self-foundation ofmatheшatic is valid only as to the latter. (shrink)

ABSTRACT I suggest how a broadly Kantian critique of classical logic might spring from reflections on constructibility conditions. According to Kant, mathematics is concerned with objects that are given through ‘arbitrary synthesis,’ in the form of ‘constructions of concepts’ in the medium of ‘pure intuition.’ Logic, by contrast, is narrowly constrained – it has no objects of its own and is fixed by the very forms of thought. That is why there is not much room for developments within logic, as (...) compared to the progress in mathematics. Kant’s view of logic remains critical, though – through considerations that are effectively on the scope and limits of classical logic and which play a part in his transcendental idealism. The most important ones are to be found in his critique of the use of reductio ad absurdum proofs in metaphysics and his solutions to the ‘antinomies of pure reason.’ Arguably, these considerations carry over to mathematics as well – by way of ‘analogues’ to the antinomies – in particular in resolving infinity paradoxes. (shrink)

We advance a theory of the representational role of Euclidean diagrams according to which they are samples of co-exact features. We contrast our theory with two other conceptions, the instantial conception and Macbeth’s iconic view, with respect to how well they accommodate three fundamental constraints on theories of the Euclidean diagrammatic practice— that Euclidean diagrams are used in proofs whose results are wholly general, that Euclidean diagrams indicate the co-exact features that the geometer is allowed to infer from them and (...) that Euclidean diagrams play the same role in both direct proofs and indirect proofs by reductio—and argue that our view is the one best suited to account for them. We conclude by illustrating the virtues of our conception of Euclidean diagrams as samples by means of an analysis of Saccheri’s quadrilateral. (shrink)

St. Thomas’s Third Way to prove the existence of God, “Of Possibility and Necessity” (ST 1, q.2, art. 3, response) is one of the most controverted passages in the entire Thomistic corpus. The central point of dispute is that if there were only possible beings, each at some time would cease to exist and, therefore, at some point in time nothing would exist, and because something cannot come from nothing, in such an eventuality, nothing would exist now—a reductio ad (...) absurdum conclusion. Therefore, at least one necessary being must exist. Generations of critics and defenders have contended over St. Thomas’s proof. This article argues that the principle of pros hen analogy is implicit in the Third Way and that once identified explains the ontological dependency of possible beings, as secondary analogates, on the first necessary being, as primary analogate. Thus, without the necessary being as primary analogate, possible beings simply could not exist. The fact that they do exist is evidence for the existence of the necessary being. St. Thomas makes synthesizes the principle of pros hen analogy, as found in Aristotle’s Metaphysics, with the Neoplatonic principle of participation. Aristotle develops pros hen analogy in contradistinction to univocal and equivocal predication as well as to genus in Metaphysics 4.2, 11.3, 12.3-5. Since Scotus and re-enforced by modern analytic logic, philosophers have almost universally regarded any kind of analogical predication as a sub-category of equivocal predication and, thus, implicitly occlude the possibility of considering pros hen analogy in their readings of the Third Way. Distinction of per se and per accidens infinite regress and of radical and natural contingency are also central to understanding the Third Way. While resolving apparent problems in the Third Way, the article also seeks to rehabilitate the doctrine of pros hen analogy as a basic principle in Thomistic and, indeed, Aristotelian metaphysics. (shrink)

1. Despite the plain fact that there is nothing in this world that can be proved without reliance on some assumption or another, there is an inalienable difference between an argument that begins by assuming what it is designed to establish and one that begins by assuming the contradictory of what it is designed to establish. Arguments of the first kind are uncontroversially acknowledged to be circular, or question-begging; though valid they achieve nothing. Those of the second kind conform to (...) the classical pattern of reductio ad absurdum or indirect proof. A typical example is the proof of the irrationality of √2, which begins by assuming that for some integral m, n the identity m2/n2 = 2, and arrives at a contradiction. Nothing whatever is established or justified by arguments of the first kind, even granted the truth of all the other assumptions, and the validity of the rules of inference; while the second may establish that, if those assumptions are true, then the additional assumption is false. (shrink)

Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and negation in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical (...) Refutations—goes far beyond any of these. (shrink)

Eduardo García-Ramírez has offered a reductio of the counterfactual analysis of causation. The argument purportedly shows that, given a natural generalization of Lewis’ semantics for counterfactuals, statements expressing the existence of causal dependence across worlds are satisfiable. The aim of the present paper is twofold. In the first part, I show that the purported reductio is flawed, as it relies on an overly strong construal of the semantics for counterfactuals. In particular, it is assumed that we can assign (...) a degree of similarity to any given pair of possible worlds. As it turns out, that assumption reduces to the thesis that the relations of comparative similarity featured in the standard semantics for counterfactuals define an interval scale of measurement on the set of all possible worlds. It will be argued that such a thesis is incompatible with a viable understanding of comparative similarity. The second part of the paper is devoted to a new proof of the possibility of trans-world causation. Nevertheless, the new proof does not amount to a reductio of Lewis’ account of causation per se, but rather of the conjunction of several substantive theses (the counterfactual analysis of causation, modal plenitude, the existence of mereological sums and the best theory account of natural laws). (shrink)

Mariusz Grygianiec has criticized the so called "proofs of nonexistence of general objects" as based on a wrong definition. In this paper one of his arguments is shown to depend on an unsatisfiable condition (contradictory to some basic ontological intuitions) without which, however, it is inconclusive as a reductio ad absurdum. Furthermore, it is suggested that even if the argument was sound, it could by no means be counted - contrary to the Author's intention - as a counterargument to (...) Stanisław Leśniewski and Tadeusz Kotarbiński. (shrink)

Ghazālī’s The Incoherence of the Philosophers is an unusual philosophical work for a number of reasons, not the least of which is the author’s explicit disavowalof any of the conclusions contained within it. The present essay examines some of the hermeneutical challenges that face readers of the work and offers anexegetical account of the much-neglected Third Discussion, which examines a key point of Neoplatonic metaphysics. The paper argues that Ghazālī’s maintaining of the incompatibility of metaphysical creationism and Neoplatonic emanationism should (...) not be viewed as simply a rhetorical or dialectical argument, but rather is best understood, to use Ghazālī’s words, as a philosophical “proof.” Essential to this proof in the solution to the argument of the Third Discussion is an implicit theory of metaphorical predication that can be pieced together from several of Ghazālī’s remarks as well as a reductio ad absurdum argument about the very possibility of ethical discourse. (shrink)

Human rights law has been called upon to help with the problem of persistently high greenhouse gas emissions. An obligation on states and other legal entities to lower their emissions (mitigation) is said to be deducible from that body of law. I refute this thesis. First, I consider two practical difficulties—causality and non-triviality—that face a plaintiff who, with emission mitigation as the objective, attempts to prove a human rights violation using the regular pattern of proof for a violation. Proponents (...) of the “human rights approach” to mitigation have held that proof of an emission “contribution” by the defendant together with proof of an “impact” by climate change on the plaintiff’s human rights are sufficient to discharge the evidentiary burden for the proof of causation. The rest of the causation chain is simply presumed. Thus, the original proof pattern for a human rights violation is abandoned. The proponents’ answer to the triviality difficulty has been to aggregate emitters into very large entities and sue _them_. However, aggregation can be shown to lead to a _reductio ad absurdum_. In my argument’s second part, I identify a more fundamental difficulty with the human rights approach to mitigation: The defendant’s emissions do not amount to a norm violation. Everyone contributes emissions without legislative or other prohibition. Treaty law on climate change itself recognizes emitting behaviour as lawful and permits the continuation of state emissions through to at least 2050. A rise in global warming from preindustrial levels to 1.5° Celsius with room for an even greater rise to close to 2° Celsius has been _budgeted for_ by the Paris Agreement on Climate Change. The setting up of a budget affirms the normalized status of within-budget emissions. This universal license to emit denies the human rights approach to mitigation the very conditions of application of human rights law. (shrink)

We extend the language of the classical syllogisms with the sentence-forms “At most 1 p is a q” and “More than 1 p is a q”. We show that the resulting logic does not admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed.

I argue that Moore's arguments have anti-skeptical force even though they beg the question against skepticism because they target the skeptic rather than skepticism directly. Moore offers two arguments which are usually conflated by his interpreters, namely, his proof of an external world and a reductio argument. I explain why the anti-skeptical force of the latter has to be derived from that of the former. I consider an objection to Moore that is based on distinguishing between the everyday (...) and the philosophical contexts. I argue that the objection fails even on the most plausible understanding of the distinction. (shrink)

Frederic Fitch’s celebrated reasoning to the conclusion that all truths are known can be interpreted as a reductio of the claim that all truths are knowable. Given this, nearly all of the proof’s reception has involved canvassing the prospects for some form of verificationism. Unfortunately, debates of this sort discount much of the philosophical import of the proof. In addition to its relevance for verificationism, Fitch’s proof is also an argument for the existence of God, one (...) at least as strong as the traditional demonstrations. Perhaps unlike other such proofs, Fitch’s also operates as a key lemma in a proof that establishes that God can’t exist. (shrink)