It is argued that contraposition is valid for a class of natural language conditionals, if some modifications are allowed to preserve the meaning of the original conditional. In many cases, implicit temporal indices must be considered, making a change in verb tense necessary. A suitable contrapositive for implicative counterfactual conditionals can also usually be found. In some cases, the addition of certain words is necessary to preserve meaning that is present in the original sentence and would be lost or (...) changed in the contrapositive without them. A distinction is made between adding new meaning and adding new words to preserve existing meaning. For concessive conditionals and relevance conditionals, however, no valid contrapositive can be found. They do not belong to the class of contraposable conditionals, which can be independently defined. Difficult cases are also discussed in which the contradictory of the consequent semantically entails the truth of the antecedent. In such cases the content of the antecedent is implicit in the meaning of the consequent. Contraposition becomes possible if what is implicit in the original consequent is made explicit in the contrapositive antecedent. (shrink)

Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally |$\vee $| complemented Heyting algebra (c|$\vee $|cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension |${\textrm {ILM}}$|-|${\vee }$| for c|$\vee (...) $|cHas are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce’s logic and Vakarelov’s logic MIN. With a focus on properties of the two negations, relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$| are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and |${\textrm {ILM}}$|-|${\vee }$| are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn’s logical framework for the purpose, two logics |$K_{im}$| and |$K_{im-{\vee }}$| are discussed, where the language does not include implication. The |$K_{im}$|-algebras are reducts of ccHas and are different from relevant algebraic structures having two negations. The negations in the |$K_{im}$|-algebras and |$K_{im-{\vee }}$|-algebras are shown to occupy distinct positions in an enhanced form of Dunn’s kite of negations. Relational semantics for |$K_{im}$| and |$K_{im-{\vee }}$| is provided by a class of frames that are based on Dunn’s compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$|⁠. (shrink)

RésuméCet article démontre qu'un exemple cité par Ernest Adams pour montrer que l'implication matérielle n'est pas l'interprétation correcte de la sémantique de la conjonction de subordination si, n'est rien d'autre qu'un corollaire d'une observation d'jà faite par Lewis Carroll, il y a cent ans, dans l'exposition de son paradoxe du salon de coiffure.

The article studies two related issues. First, it introduces the notion of the contraposition of quantifiers which is a “dual” notion of symmetry and has similar relations to co-intersectivity as symmetry has to intersectivity. Second, it shows how symmetry and contraposition can be generalised to higher order type quantifiers, while preserving their relations with other notions from generalized quantifiers theory.

The concept of an evidential conditional If A then C that can be defined by the conjunction of A>C and ¬C>¬A, where > is a conditional of the kind introduced by Stalnaker and Lewis, has recently been studied in a series of papers by Vincenzo Crupi and Andrea Iacona. In this paper I argue that Crupi and Iacona’s central idea that contraposition captures the idea of evidential support cannot be maintained. I give examples showing that contraposition is neither (...) necessary nor sufficient for a conditional’s antecedent supporting its consequent. Crupi and Iacona’s alternative account of evidential conditionals that is based on a probabilistic measure of evidential support cannot add to the credentials of their modal account, because both the theoretical role of contraposition and the resulting logic are different in this account. (shrink)

We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_{2}^{0}}$$\end{document}-formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.

The rule of contraposition has been investigated thoroughly by Arabic logicians. In this paper, we study the work done by Fārābā and Avicenna, the fathers of Arabic logic. Fārābā studied contraposition of universal affirmatives, discussed its four forms, and discovered a relation between one form and the conversion of negative universals. Although Fārābā and logicians before him have used contraposition only for conditionals, as well as for indefinite and universal affirmative categorical propositions, Avicenna generalized the rule to (...) all the four Aristotelian quantified categorical propositions. However, many of his ideas on contraposition were opposed by his 12th and 13th century followers, which are to be investigated later. (shrink)

Many logics in the relevant family can be given a proof theory in the style of Belnap's display logic. However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which more closely models relevant logics. In addition, we use this proof theory to show decidability for a large range of substructural logics.

The logical rule of contraposition as applied to a particular affirmative proposition (I-contraposition), despite its rejection in the medieval Latin logic, had a different history in the medieval Arabic logic, varying from common acknowledgement to total dismissal (it was accepted by Avicenna and by all of his followers in the eleventh and twelfth centuries and rejected by all of Arabic logicians in the late thirteenth century onwards). This paper is a narrative of the fate of I-contraposition in (...) the early Arabic logic. I will study Avicenna’s, al-Suhrawardī’s, al-Rāzī’s, al-Kashshī’s, and al-Khūnajī’s views on this rule. Although Avicenna explicitly acknowledged I-contraposition in a brief note, and in an apparently Meinongian language, the other four figures found I-contraposition problematic and listed some counterexamples to it. Nonetheless, they all attempted to justify Avicenna’s acceptance of the rule by restricting its application conditions. These efforts led to interesting syntactic and semantic analyses but gathered little attention from later medieval logicians. Finally, in the middle Arabic logic, the revised versions of I-contraposition were also declared invalid. (shrink)

We investigate the relationship between M. De's empirical negation in Kripke and Beth Semantics. It turns out empirical negation, as well as co-negation, corresponds to different logics under different semantics. We then establish the relationship between logics related to these negations under unified syntax and semantics based on R. Sylvan's CCω.

We continue the investigation of the first paper where we studied logics with various negations including empirical negation and co-negation. We established how such logics can be treated uniformly with R. Sylvan's CCω as the basis. In this paper we use this result to obtain cut-free labelled sequent calculi for the logics.

To Lévi-Strauss’s anthropological eye, comparison with others is the condition for discovering one own’s identity; but only by avoiding the two extremes of total closure or indiscriminate openness does it become a positive element of growth and inner evolution. In this essay I consider how this assumption could help us today in dealing with the delicate topic of the relation between human rights culture (which is indeed universalist in character, but developed in the West), and Islamic culture (geographically and historically (...) relative, and yet to a certain extent also universalist). (shrink)

I develop and defend a truthmaker semantics for the relevant logic R. The approach begins with a simple philosophical idea and develops it in various directions, so as to build a technically adequate relevant semantics. The central philosophical idea is that truths are true in virtue of specific states. Developing the idea formally results in a semantics on which truthmakers are relevant to what they make true. A very natural notion of conditionality is added, giving us relevant implication. I then (...) investigate ways to add conjunction, disjunction, and negation; and I discuss how to justify contraposition and excluded middle within a truthmaker semantics. (shrink)

We analyze the variety of A. Monteiro’s tetravalent modal algebras under the perspective of two logic systems naturally associated to it. Taking profit of the contrapositive implication introduced by A. Figallo and P. Landini, sound and complete Hilbert-style calculi for these logics are presented.

This paper presents the first compositional semantics for if then conditionals. The semantics of each element are first examined separately. The meaning of if is modeled according to a possible worlds semantics. The particle then is analyzed as an anaphoric word that places its focused element inside the context settled by a previous element. Their meanings are subsequently combined in order to provide a formal semantics of if A then C conditionals, which differs from the simple if A, C form. (...) This semantics has the particularity of validating contraposition for the first type but invalidating it for the second type. Finally, a detailed examination of the sentences presented in the literature opposing this schema of reasoning shows that these counterexamples do not generally concern if then conditionals but, rather, even if conditionals and that contraposition is therefore a valid means of reasoning with regard to if then conditionals in natural language, as this system predicts. (shrink)

Ātman (soul) and Nairātmya (no soul) are, for the Brahmanical schools and the Buddhists respectively, equally fundamental tenets which neither side can concede to the other. Among the 16 formulations presented by Uddyotakara, the fifteenth, which is a proof of Ātman and is originally an indirect proof ( avīta/āvīta ), is presented in a prasaṅga -style, and contains double negation ( na nairātmyam ) in the thesis. However, it is perhaps Dharmakīrti who first transformed it into a normal style ( (...) sātmakam ). He is well aware of the law of excluded middle, and insisits that the negation is paryudāsa . On the Nyāya side, Uddyotakara at least seems to be unaware of the law of the logical equivalence of contraposition concerning pervasion ( vyāpti ). After Uddyotakara, however, Vyoman (Vyomaśiva), Bhāsarvajña and Vācaspatimiśra, all seem to be well aware of it. Dharmakīrti, in his conter-argument against the proof of ātman , discusses the negative expressions ‘‘ nairātmya ” and ‘‘ a-nairātmya ” Dharmakīrti here uses two logical arguments skillfully and tactically. As a critic of both the authenticity of the Veda and the existence of ātman , he insists on the theory of dichotomy and the equivalence of anvaya and vyatireka , whereas as an apologist he denies the application of these theories to the relation between the existence of ātman and the concept of nairātmya , because for him as a Buddhist the latter is not a negative but essentially positive state of affairs. (shrink)

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture the following (...) results: nothing can be called a negation properly if it does not satisfy Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps, and other notations in the work.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely (...) copy and distribute this work, as no entity has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

World food production is facing exorbitant challenges like climate change, use of resources, population growth, and dietary changes. These, in turn, raise major ethical and political questions, such as how to uphold the right to adequate nutrition, or the right to enact a gastronomic culture and to preserve the conditions to do so. Proposals for utopic solutions vary from vertical farming and lab meat to diets filled with the most fanciful insects and seaweeds. Common to all proposals is a polarized (...) understanding of food and diets, famously captured by Warren Belasco in the contraposition between technological fixes and anthropological fixes. According to the first, technology will deliver clean, just, pleasurable, affordable food; future generations will not need to adjust much of their dietary cultures. According to the second, future generations should dramatically change their dietary habits (what they eat and how they eat it) to achieve a sustainable diet. The two fixes found remarkably distinct perspectives over dietary politics and the ethics of food production and consumption. In this paper we argue that such polarized thinking rests on a misrepresentation of the ontological status of food, which in turn affects the underlying ethical and political issues. Food is a socially constructed object that draws in specific ways on habits, norms, traditions, geographical, and climatic conditions. Although this thesis seems somewhat obvious, its consequences on the ethical and political perspectives on the future of food have not been derived properly. After introducing the issue at stake (¤1), we point out the polarities that characterize food utopias (¤2) and their ontological faults (¤3). We hence suggest that a socio-ontological analysis of food can better deliver the principles for a foundation of food utopias (¤4). (shrink)

Those who want to deny the ‘ought implies can’ principle often turn to weakened views to explain ‘ought implies can’ phenomena. The two most common versions of such views are that ‘ought’ presupposes ‘can’, and that ‘ought’ conversationally implicates ‘can’. This paper will reject both views, and in doing so, present a case against any pragmatic view of ‘ought implies can’. Unlike much of the literature, I won't rely on counterexamples, but instead will argue that each of these views fails (...) on its own terms. ‘Ought’ and ‘can’ do not obey the negation test for presupposition, and they do not obey the calculability or the cancelability tests for conversational implicature. I diagnose these failures as partly a result of the importance of the contrapositive of ‘ought implies can’. I end with a final argument emphasizing the role the principle plays in moral thinking, and the fact that no pragmatic account can do it justice. (shrink)

The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn't possibly be known. More specifically, if p is a truth that is never known then it is unknowable that p is a truth that is never known. The proof has been used to argue (...) against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge. Still others have taken the proof to be fallacious, since it collapses an apparently moderate brand of anti-realism into an obviously implausible and naive idealism. (shrink)

Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ST and TS, along with two structural cousins, LP and K3. It is well known that LP and K3 are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are (...) tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, ST and TS are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning ST which are being intensively discussed in the current literature: whether ST preserves classical logic and whether it is LP in sheep’s clothing. Explaining in what way ST and TS are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference. (shrink)

This entry traces the historical development of the Square of Opposition, a collection of logical relationships traditionally embodied in a square diagram. This body of doctrine provided a foundation for work in logic for over two millenia. For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms. Certain additional principles (" (...) class='Hi'>contraposition" and "obversion") were sometimes adopted along with the Square, and they genuinely yielded inconsistency. By the nineteenth century an inconsistent set of doctrines was widely adopted. Strawson's 1952 attempt to rehabilitate the Square does not apply to the traditional doctrine; it does salvage the nineteenth century version but at the cost of yielding inferences that lead from truth to falsity when strung together. (shrink)

In contemporary Western moral philosophy literature that discusses the Chinese ethical tradition, it is a commonplace practice to use the Chinese term daode 道德 as a technical translation of the English term moral. The present study provides some empirical evidence showing a discrepancy between the terms moral and daode. There is a much more pronounced difference between prototypically immoral and prototypically uncultured behaviors in English (USA) than between prototypically bu daode 不道德 and prototypically bu wenming 不文明 behaviors in Mandarin Chinese (...) (Mainland China). If the Western concept of immorality is defined in contraposition to things that are matters of etiquette or conventional norms and thus tied to a more or less tangible moral / conventional distinction, then we are dealing with a different structure in Mandarin Chinese – the prototypically bu daode and bu wenming behaviors seem to largely overlap. We also discuss whether bu lunli 不倫理 and bu hefa 不合法 can be considered adequate candidates for translation of immorality and we answer in the negative. (shrink)

Drawing on the puzzling behavior of ordinary knowledge ascriptions that embed an epistemic (im)possibility claim, we tentatively conclude that it is untenable to jointly endorse (i) an unfettered classical logic for epistemic language, (ii) the general veridicality of knowledge ascription, and (iii) an intuitive ‘negative transparency’ thesis that reduces knowledge of a simple negated ‘might’ claim to an epistemic claim without modal content. We motivate a strategic trade-off: preserve veridicality and (generalized) negative transparency, while abandoning the general validity of (...) class='Hi'>contraposition. We criticize various approaches to incorporating veridicality into domain semantics, a paradigmatic ‘information-sensitive’ framework for capturing negative transparency and, more generally, the non-classical behavior of sentences with epistemic modals. We then present a novel information-sensitive semantics that successfully executes our favored strategy: stable acceptance semantics, extending a vanilla bilateral state-based semantics for epistemic modals with a knowledge operator loosely inspired by the defeasibility theory of knowledge. (shrink)

Nancy Cartwright (1983, 1999) argues that (1) the fundamental laws of physics are true when and only when appropriate ceteris paribus modifiers are attached and that (2) ceteris paribus modifiers describe conditions that are almost never satisfied. She concludes that when the fundamental laws of physics are true, they don't apply in the real world, but only in highly idealized counterfactual situations. In this paper, we argue that (1) and (2) together with an assumption about contraposition entail the opposite (...) conclusion — that the fundamental laws of physics do apply in the real world. Cartwright extracts from her thesis about the inapplicability of fundamental laws the conclusion that they cannot figure in covering-law explanations. We construct a different argument for a related conclusion — that forward-directed idealized dynamical laws cannot provide covering-law explanations that are causal. This argument is neutral on whether the assumption about contraposition is true. We then discuss Cartwright's simulacrum account of explanation, which seeks to describe how idealized laws can be explanatory. (shrink)

This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent. $${p\Rightarrow q}$$ p ⇒ q is thus defined as $${\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}$$ ¬ ◊ (...) ( p ∧ ¬ q ) ∧ ◊ p ∧ ◊ ¬ q. We explore the logical properties of this conditional in a reflexive normal Kripke semantics, provide an axiomatic system and prove it to be sound and complete for our semantics. The implicative conditional validates transitivity and contraposition, which we take to be integral parts of reasoning and communication. But it only validates restricted versions of strengthening the antecedent, right weakening, simplification, and rational monotonicity. Apparent counterexamples to some of these properties are explained as due to contextual factors. Finally, the implicative conditional avoids the paradoxes of material and strict implication, and validates some connexive principles such as Aristotle’s theses and weak Boethius’ thesis, as well as some highly entrenched principles of conditionals, such as conjunction of consequents, disjunction of antecedents, modus ponens, cautious monotonicity and cut. (shrink)

The ambition of the paper is to provide a solution to the problem posed by Von Wright (1999): how is it possible that the two actions, one of producing P and the other of preventing P can have different deontic status, the former being obligatory and the latter being forbidden. The solution for the problem is sought for by an investigation into connections between imperative and deontic logic. First, it is asked whether a solution could be found in Lemmon's (1965) (...) system of "change logic", using his idea on connection between logic of orders being in force and deontic logic. The answer is the negative one. Next, the connection between Lemmon's imperative logic and deontic logic given in Aqvist's paper - "Next" and "Ought" (1965) - is analyzed. Than, the Lemmon's treatment of imperatives is restricted to the natural language imperatives and Aqvist's way of connecting imperative and deontic logic is modified accordingly. Some principles for the natural language imperatives are established (the negation rule ; the law of contraposition for imperative conditionals) and a simple "global" semantics is developed. The notion of "opposite action" is introduced and it is given an important role in semantics. Finally, a solution for von Wright's problem is given. In the closing sections some further topics for investigation are hinted: one of them being the connection between Aqvist's epistemic- imperative conception of interrogatives and "epistemic obligations", the other being formalization of the idea that imperatives create and re-create obligation patterns that can be described in deontic terms. (shrink)

This paper is about four inferences patterns governing conditionals: Transitivity, Simplification, Contraposition, and Antecedent Strengthening. Transitivity, Simplification, and Contraposition are intuitively compelling. Although Antecedent Strengthening may seem less attractive at first, close attention to the full range of data reveals that it too has considerable appeal. An adequate theory of conditionals should account for these facts. The strict theory does so by validating them. But the variably strict theory invalidates them. So the variably strict theorist faces a question: (...) why do we find these inferences so compelling? In this paper, I suggest an answer on her behalf: that they are reasonable inferences in the sense introduced by Stalnaker (1975). -/- . (shrink)

It is widely agreed that contraposition, strengthening the antecedent and hypothetical syllogism fail for subjunctive conditionals. The following putative counter-examples are frequently cited, respectively.

Structured argumentation formalisms, such as ASPIC +, offer a formal model of defeasible reasoning. Usually such formalisms are highly parametrized and modular in order to provide a unifying framework in which different forms of reasoning can be expressed. This generality comes at the price that, in their most general form, formalisms such as ASPIC + do not satisfy important rationality postulates, such as non-interference. Similarly, links to other forms of knowledge representation, such as reasoning with maximal consistent sets of rules, (...) are insufficiently studied for ASPIC + although such links have been established for other, less complex forms of structured argumentation where defeasible rules are absent. Clearly, for a formal model of defeasible reasoning it is important to understand for which range of parameters the formalism displays a behavior that adheres to common standards of consistency, logical closure and logical relevance and can be adequately described in terms of other well-known forms of knowledge representation. In this paper we answer this question positively for a fragment of ASPIC + without the attack form undercut by showing that it satisfies all standard rationality postulates of structured argumentation under stable and preferred semantics and is adequate for reasoning with maximal consistent sets of defeasible rules. The study is general in that we do not impose any other requirements on the strict rules than to be contrapositable and propositional and in that we also consider priorities among defeasible rules, as long as they are ordered by a total preorder and lifted by weakest link. In this way we generalize previous similar results for other structured argumentation frameworks and so shed further light on the close relations between assumption-based argumentation and ASPIC +. (shrink)

This collection offers a synoptic view of current philosophical debates concerning the relationship between facts and values, bringing together a wide spectrum of contributors committed to testing the validity of this dichotomy, exploring alternatives, and assessing their implications. The assumption that facts and values inhabit distinct, unbridgeable conceptual and experiential domains has long dominated scientific and philosophical discourse, but this separation has been seriously called into question from a number of corners. The original essays here collected offer a diversity of (...) responses to fact-value dichotomy, including contributions from Hilary Putnam and Ruth Anna Putnam who are rightly credited with revitalizing philosophical interest in this alleged opposition. Both they, and many of our contributors, are in agreement that the relationship between epistemic developments and evaluative attitudes cannot be framed as a conflict between descriptive and normative understanding. Each chapter demonstrates how and why contrapositions between science and ethics, between facts and values, and between objective and subjective are false dichotomies. Values cannot simply be separated from reason. _Facts and Values_ will therefore prove essential reading for analytic and continental philosophers alike, for theorists of ethics and meta-ethics, and for philosophers of economics and law. (shrink)

Selection explanations explain some non-accidental generalizations in virtue of a selection process. Such explanations are not particulaizable - they do not transfer as explanations of the instances of such generalizations. This is unlike many explanations in the physical sciences, where the explanation of the general fact also provides an explanation of its instances (i.e. standard D-N explanations). Are selection explanations (e.g. in biology) therefore a different kind of explanation? I argue that to understand this issue, we need to see that (...) a standard D-N explanation of some non-accidental generalization (al Fs are Gs) may also ipso facto explain its contrapositive (all non-Gs are non-Fs), but the explanation is particularizable with respect to the former but not to the latter. This can be seen by noting that the Raven Paradox counterexample to the H-D model of confirmation also generates a counterexample to the D-N model of explanation (all ravens are black does not explain why the non-black shoe is a non-raven). In such cases it is natural to take the generalization with the positive predicates to have a particularizable explanation. However, this need not be the case, and in selection explanations it is the generalization with the positive predicates whose explanation is no particularizable. Thus there is no need to suppose that selection explanations are fundamentally different. (shrink)

This article approaches the largely known four objections that Hegel held against Kant: the formalism of moral principle, the criticism of autonomy and formalism, the objection of the abstract universalism in Kant′s ethics, and the terrorism of the pure reason. The purpose is to determine the extent of these objections in front of Kant′s critical thought. At the end of the text, I will support that in spite of different misunderstandings about Kant′s ethics; however Hegel inherited us a great lesson (...) that even crosses the contemporary ethical thought: the radical contraposition between justice (virtue) and the Sittlichkeit (the concrete and effective ethics) that leads us to the loss of the basic moral phenomenon. (shrink)

This mathematics textbook covers the fundamental ideas used in writing proofs. Proof techniques covered include direct proofs, proofs by contrapositive, proofs by contradiction, proofs in set theory, proofs of existentially or universally quantified predicates, proofs by cases, and mathematical induction. Inductive and deductive reasoning are explored. A straightforward approach is taken throughout. Plenty of examples are included and lots of exercises are provided after each brief exposition on the topics at hand. The text begins with a study of symbolic logic, (...) deductive reasoning, and quantifiers. Inductive reasoning and making conjectures are examined next, and once there are some statements to prove, techniques for proving conditional statements, disjunctions, biconditional statements, and quantified predicates are investigated. Terminology and proof techniques in set theory follow with discussions of the pick-a-point method and the algebra of sets. Cartesian products, equivalence relations, orders, and functions are all incorporated. Particular attention is given to injectivity, surjectivity, and cardinality. The text includes an introduction to topology and abstract algebra, with a comparison of topological properties to algebraic properties. This book can be used by itself for an introduction to proofs course or as a supplemental text for students in proof-based mathematics classes. The contents have been rigorously reviewed and tested by instructors and students in classroom settings. (shrink)

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their own. The (...) first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples. (shrink)

Sarah Moss’ thesis that we have probabilistic knowledge is from some perspectives unsurprising and from other perspectives hard to make sense of. The thesis is potentially transformative, but not yet elaborated in sufficient detail for epistemologists. This paper interprets Mossean probabilistic knowledge in a suitably-modified Kripke framework, thus filling in key details. It argues that probabilistic knowledge looks natural and plausible when so interpreted, and shows how the most pressing challenges to the thesis can be overcome. Most importantly, probabilistic knowledge (...) can satisfy factivity in the framework, though we are not forced to accept a specific account of probabilistic “facts”. The framework also reflects Moss’ claim that old-fashioned propositional knowledge is just a limiting case of probabilistic knowledge, and all knowledge is fundamentally probabilistic. Finally, Moss endorses a failure of contraposition: for example, p implies probably p, but not probably p does not imply not p. The framework makes clear the sense in which the valid inferences regarding probably p are as Moss claims. (shrink)