It is widely believed that (1) if theological determinism were true, in virtue of God’s role in determining created agents to perform evil actions, created agents would be neither free nor morally responsible for their evil actions and God would not be perfectly good; (2) if metaphysical compatibilism were true, the free-will defense against the deductive problem of evil would fail; and (3) on the assumption of metaphysical compatibilism, God could have actualized just any one (...) of those myriad possible worlds that are populated only by compatibilist free creatures. The primary thesis of this essay is that none of these propositions is true. This thesis is defended by appealing to a recently proposed novel, acausal, composite, unified theory of free action – the Theory of Middle Freedom – that evades the central problems plaguing traditional theories of metaphysical compatibilism. (shrink)

ABSTRACT We show that there is a polynomial over the rational number field corresponding to each propositional formula in a given many-valued logic. To decide whether a propositional formula can be deduced from a finite set of such formulas (deduction problem), we only need to decide whether a polynomial vanishes on an algebraic variety. By using Wu's method, an algorithm for this problem is presented.

1. It is a fact that developing a good proof theory for modal logics is a difficult task. The problem is not in having deductive systems. In fact, all the main modal logics enjoy an axiomatic prese...

In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof? -/- The answer is an emphatic yes, as I explain in this article. I argue that non-deductive justification is in fact pervasive in mathematics, (...) and that it is in good epistemic standing. (shrink)

This book stands at the intersection of two topics: the decidability and computational complexity of hybrid logics, and the deductive systems designed for them. Hybrid logics are here divided into two groups: standard hybrid logics involving nominals as expressions of a separate sort, and non-standard hybrid logics, which do not involve nominals but whose expressive power matches the expressive power of binder-free standard hybrid logics.The original results of this book are split into two parts. This division reflects the division (...) of the book itself. The first type of results concern model-theoretic and complexity properties of hybrid logics. Since hybrid logics which we call standard are quite well investigated, the efforts focused on hybrid logics referred to as non-standard in this book. Non-standard hybrid logics are understood as modal logics with global counting operators ) whose expressive power matches the expressive power of binder-free standard hybrid logics. The relevant results comprise: 1. Establishing a sound and complete axiomatization for the modal logic K with global counting operators ), which can be easily extended onto other frame classes, 2. Establishing tight complexity bounds, namely NExpTime-completeness for the modal logic with global counting operators defined over the classes of arbitrary, reflexive, symmetric, serial and transitive frames ), MT), MD), MB), MK4) with numerical subscripts coded in binary. Establishing the exponential-size model property for this logic defined over the classes of Euclidean and equivalential frames ), MS5).Results of the second type consist of designing concrete deductive systems for standard and non-standard hybrid logics. More precisely, they include: 1. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for a rich class of binder-free standard hybrid logics. An interesting feature of indicated calculi is the nonbranching character of the rule, 2. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for non-standard hybrid logics. The internalization technique applied to a tableau calculus for the modal logic with global counting operators is novel in the literature, 3. Devising the first hybrid algorithm involving an inequality solver for modal logics with global counting operators. Transferring the arithmetical part of reasoning to an inequality solver turned out to be sufficient in ensuring termination.The book is directed to philosophers and logicians working with modal and hybrid logics, as well as to computer scientists interested in deductive systems and decision procedures for logics. Extensive fragments of the first part of the book can also serve as an introduction to hybrid logics for wider audience interested in logic.The content of the book is situated in the areas of formal logic and theoretical computer science with some elements of the theory of computational complexity. (shrink)

I discuss three elements of Dennis Schulting’s new book on the transcendental deduction of the pure concepts of the understanding, or categories. First, that Schulting gives a detailed account of the role of each individual category. Second, Schulting’s insistence that the categories nevertheless apply ‘en bloc’. Third, Schulting’s defence of Kant’s so-called reciprocity thesis that subjective unity of consciousness and objectivity in the sense of cognition’s objective purport are necessary conditions for the possibility of one another. I endorse these fascinating (...) but unfashionable claims and sketch my own version of what they amount to, which is quite different to Schulting’s own construal. I point to some fundamental limitations and problems for Schulting’s position and argue that his project needs to be reshaped or at least reconceived in the face of them. Even if Schulting’s argument is sound, it does not provide a deduction, properly speaking, of the categories. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:104 COMMENTARY ON PROFESSOR TWEYMAN ' S 'HUME ON EVIL' Philo concludes his long and celebrated debate with Cleanthes on the problem of evil (Parts X and Xl of Hume's Dialogues) with the assertion that the "true conclusion" to be drawn from the "mixed phenomena" in the world is that "the original source" of whatever order we find in the world is "indifferent" to matters of (...) good and evil. From what Philo says immediately thereafter it is clear that by the phrase "The true conclusion is..." he does not mean 'The truth is...' but rather something like 'The conclusion best supported by the available evidence is....' (D 212) Now according to both Nelson Pike and Stanley Tweyman, the claim made by Philo in these passages involves him in a shift or departure from the procedure he adopted in earlier dialogues when he examined the design argument set forth by Cleanthes. It is claimed that whereas in the earlier dialogues Philo emphatically professed a sceptical outlook — "the view requiring that one embrace no metaphysical position of one's own" (to quote Nelson Pike ) — in the quoted passage he seems to abandon this scepticism in favor of embracing what amounts to being a metaphysical position in natural theology. Pike thinks that this alleged shift is only an apparent one, that the central import of these passages is sceptical rather than metaphysical, and that here, as elsewhere in the earlier dialogues, Philo's sole purpose in offering a more plausible counter-hypothesis is to show that Cleanthes' hypothesis is false. Tweyman, on the other hand, argues that the shift is a very real one and that Hume does indeed "intend Philo's conclusion in Part XI to be a truth in natural theology [.]... Philo is 105 not arguing as a sceptic; rather he is employing the hypothetico-deductive method for testing Cleanthes' 2 hypothesis. " Let me say at once that I agree with Tweyman, as against Pike, that in these passages Philo does claim to put forward a truth Jji natural theology, and I also agree with just about everything that Tweyman says in his paper concerning the proper interpretation of Philo's views in Parts X and XI leading up to this conclusion. I follow the lead of those scholars who, at least since Kemp Smith, have argued that Philo's whole purpose in demonstrating such "truths" is to discredit natural theology in the eyes of a true Christian believer by dramatizing the enormous gulf between the very attenuated form of theism ("diaphonous deism" one scholar calls it), which is the most one can hope to get out of Cleanthes' experimental approach to religion, and the claim of Christian theism to which Cleanthes aspires. But I do not accept the view that Philo's claim in these dialogues represents any sort of sudden departure (and certainly not a "fundamental methodological difference") from his critical stance against the design argument in earlier dialogues. What I hope to show below is that not only does this view distort what Philo does in the earlier dialogues, but it also distracts our attention from the true nature of Hume's achievement in Parts X and XI of the Dialogues. (Incidentally, I will assume throughout the sequel that Kemp Smith's identification of Hume with Philo is essentially correct. ) It would indeed be extremely paradoxical, in the face of Philo's famous (or notorious) peroration at the end of Part VIII proclaiming the "complete triumph of the sceptic" in regard to "all religious 106 systems," to raise any doubts about Philo's acceptance of the sceptical import of his own earlier arguments. But this still leaves unsettled the question as to what sort of scepticism it is whose triumph Philo proclaims at this point and whether it is inconsistent with the conclusion he reaches at the end of his discussion of evil. One would have hoped that at this late stage in the scholarship on the Dialogues, if there is one issue on which all of us could agree it is that Philo at no time in the Dialogues takes the (easy) Pyrrhonian line against Cleanthes' "hypothesis of experimental theism." I mention this because at several places... (shrink)

P.J. McGrath has recently challenged the standard claim that to escape the problem of evil one need only alter one’s conception of God by limiting his power or his goodness. If we assume that God is infinitely good but not omnipotent, then God can scarcely be a proper object of worship. And if we assume that if God is omnipotent but limited in goodness, he becomes a moral monster. Either way evil remains a problem for theistic (...) belief. I argue that McGrath fails to distinguish between the deductive and inductive problem of evil and between a limitation in God’s “strength” and a limitation in God’s “ability to act”, and that once these distinctions are made, his argument fails. (shrink)

Minimalism is the view that our concept of truth is constituted by our disposition to accept instances of the truth schema ‘The proposition that p is true if and only if p’. The generalisation problem is the challenge to account for universal generalisations concerning logical truths such as ‘Every proposition of the form 〈if p, then p〉 is true’. This paper argues that such generalisations can be deduced using a single example of the logical truth in question and a (...) single corresponding instance of the truth schema, employing the logical method of reasoning with arbitrary instances of universal and existential generalisations. Suggesting an inferentialist construal of Minimalism, the paper introduces conditional and general acceptance dispositions, distinguishes inferential meaning constitution from implicit definition, highlights the inferential nature of acceptance of instances of the truth schema, sketches a suitable account of structured propositions, compares higher-order with first-order means of quantification, and argues that the conception of truth Minimalism attributes to ordinary speakers is essentially inferential. It finally applies the deductive strategy to generalisations concerning logical validity as well as more complex logical truths. (shrink)

In philosophy, an argument consists of a set of statements called premises that serve as grounds for affirming another statement called the conclusion. Philosophers typically distinguish arguments in natural languages (such as English) into two fundamentally different types: deductive and inductive. Each type of argument is said to have characteristics that categorically distinguish it from the other type. The two types of argument are also said to be subject to differing evaluative standards. Pointing to paradigmatic examples of each type (...) of argument helps to clarify their key differences. The distinction between the two types of argument may hardly seem worthy of philosophical reflection, as evidenced by the fact that their differences are usually presented as straightforward, such as in many introductory philosophy textbooks. Nonetheless, the question of how best to distinguish deductive from inductive arguments, and indeed whether there is a coherent categorical distinction between them at all, turns out to be considerably more problematic than commonly recognized. This article identifies and discusses a range of different proposals for marking categorical differences between deductive and inductive arguments while highlighting the problems and limitations attending each. Consideration is also given to the ways in which one might do without a distinction between two types of argument by focusing solely on the application of evaluative standards to arguments instead. (shrink)

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way (...) that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

Hypothetico-deductive (H-D) confirmation builds on the idea that confirming evidence consists of successful predictions that deductively follow from the hypothesis under test. This article reviews scope, history and recent development of the venerable H-D account: First, we motivate the approach and clarify its relationship to Bayesian confirmation theory. Second, we explain and discuss the tacking paradoxes which exploit the fact that H-D confirmation gives no account of evidential relevance. Third, we review several recent proposals that aim at a sounder (...) and more comprehensive formulation of H-D confirmation. Finally, we conclude that the reputation of hypothetico-deductive confirmation as outdated and hopeless is undeserved: not only can the technical problems be addressed satisfactorily, the hypothetico-deductive method is also highly relevant for scientific practice. (shrink)

This work is an introductory textbook for deductive logic being primarily concerned with truth-functional logic, but also containing an introduction to syllogisms with the application of Venn diagrams, an introduction to quantification theory, and a brief discussion of axiom systems. Harrison employs six logical operators in his truth-functional calculus, including both inclusive and exclusive disjunction. The six operators are initially defined by truth tables, but in the natural deduction presentation negation and conjunction are taken as primitive and the other (...) connectives are defined in terms of these two. The conditional and indirect methods of proof are included with the approach being essentially the same as that given in Copi’s Symbolic Logic. Categorical statements and syllogisms are analyzed from both hypothetical and existential viewpoints. The treatment of quantification theory includes two-place predicates and employs the four standard rules for generalization and instantiation. The book contains an abundance of explanations, examples, and exercises. Selected answers, usually for the odd numbered problems, are given in an appendix.—T. G. N. (shrink)

This paper describes a cubic water tank equipped with a movable partition receiving various amounts of liquid used to represent joint probability distributions. This device is applied to the investigation of deductive inferences under uncertainty. The analogy is exploited to determine by qualitative reasoning the limits in probability of the conclusion of twenty basic deductive arguments (such as Modus Ponens, And-introduction, Contraposition, etc.) often used as benchmark problems by the various theoretical approaches to reasoning under uncertainty. The probability (...) bounds imposed by the premises on the conclusion are derived on the basis of a few trivial principles such as "a part of the tank cannot contain more liquid than its capacity allows", or "if a part is empty, the other part contains all the liquid". This stems from the equivalence between the physical constraints imposed by the capacity of the tank and its subdivisions on the volumes of liquid, and the axioms and rules of probability. The device materializes de Finetti's coherence approach to probability. It also suggests a physical counterpart of Dutch book arguments to assess individuals' rationality in probability judgments in the sense that individuals whose degrees of belief in a conclusion are out of the bounds of coherence intervals would commit themselves to executing physically impossible tasks. (shrink)

By deduction, we often extend both our knowledge and our justified belief. Moreover, in achieving knowledge or justified belief of some proposition, we commonly acquire justification for believing many of its entailed consequences, such as at least some of those that self-evidently follow from it. These and related facts have led some philosophers to endorse strong closure principles, for instance: If a person, S, is justified in believing a proposition, p, and p entails q, then S is justified in believing (...) q. Others have denied such principles, whether for justification or, with the appropriate additions, for knowledge. The debate continues. There is, however, this much consensus: there are difficult problems about the scope of closure, and epistemology should develop theories that can resolve them. (shrink)

Excerpt from Logic, Vol. 1: Deduction The present work aims at embracing a full course of Logic, both Formal and Inductive. In an introductory chapter, are set forth such doctrines of psychology as have a bearing on Logic, the nature of knowledge in general, and the classification of the sciences the intention being to avoid doctrinal digressions in the course of the work. Although preparatory to the under standing of what follows, this chapter may be passed over lightly on a (...) first perusal of the work. The part on Deduction contains the usual doctrines of the Syllogism, with the additions of Hamilton, and a full abstract of the novel and elaborate schemes of De Morgan and Boole. The Inductive portion comprises the methods of induc tive research, and all those collateral topics brought for ward by Mr. Mill, as part of the problem of Induction; various modifications being made in the manner of state ment, the order of topics, and the proportion of the hand ling. The greatest innovation is the rendering of Cause by the new doctrine called the Conservation, Persistence, or Correlation of Force. Mr. Mill's view of the relation of Deduction and Induc tion is fully adopted, as being the solution of the otherwise inextricable puzzle of the syllogism, and the means of giving unity and comprehensiveness to Logic. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Deductive Logic is designed as an intermediate-level text directed at upper-division students from philosophy and the humanities. Its focus is exclusively on deductive logic, avoiding altogether topics such as informal reasoning and scientific method normally included in introductory logic courses. Its exposition of logical topics is informal, with emphasis on explaining the basic concepts and procedures of modern symbolic logic in the simplest and most intuitive manner possible rather than on developing a rigorous formal system and providing proofs (...) of its properties. The fact that the text presupposes a course offered to philosophy students and serves to introduce them to logic as the "language of philosophy" has strongly influenced the selection of topics. The topics here are controversial, and the problems not easily resolved, but this text strives to relate the formal logical structures introduced to issues of philosophic interest. (shrink)

We investigate how the perceived uncertainty of a conditional affects a person's choice of conclusion. We use a novel procedure to introduce uncertainty by manipulating the conditional probability of the consequent given the antecedent. In Experiment 1, we show first that subjects reduce their choice of valid conclusions when a conditional is followed by an additional premise that makes the major premise uncertain. In this we replicate Byrne. These subjects choose, instead, a qualified conclusion expressing uncertainty. If subjects are given (...) a third statement that qualifies the likelihood of the additional premise, then the uncertainty of the conclusions they choose is systematically related to the suggested uncertainty. Experiment 2 confirms these observations in problems that omit the additional premise and qualify the first premise directly. Experiment 3 shows that the qualifying statement also affects the perceived probability of the consequent given the antecedent of the conditional. Experiment 4 investigates the effect of suggested uncertainty on the fallacies and shows that increases in uncertainty reduce the number of certain conclusions that are chosen while affirming the consequent but have no effect on denying the antecedent. We discuss our results in terms of rule theories and mental models and conclude that the latter give the most natural account of our results. (shrink)

Most automated theorem provers are clausal-form provers based on variants of resolutionrefutation. In my [1990], I described the theorem prover OSCAR that was based instead on natural deduction. Some limited evidence was given suggesting that OSCAR was suprisingly efficient. The evidence consisted of a handful of problems for which published data was available describing the performance of other theorem provers. This evidence was suggestive, but based upon too meager a comparison to be conclusive. The question remained, “How does natural deduction (...) compare with resolution-refutation?” In the ensuing seven years, OSCAR has evolved in important ways, and other developments have made it possible to collect more accurate comparative data. Specifically, the creation of the TPTP library of problems for theorem provers,1 and the availability of important theorem provers on the world wide web, make objective comparisons easier. These developments recently inspired Geoff Sutcliffe, one of the founders of the TPTP library, to issue a challenge to OSCAR. At CADE-13, a competition was held for clausal-form theorem provers.2 Otter was one of the most successful contestants. In addition, Otter is able to handle problems stated in natural form (as opposed to clausal form), and Otter is readily available for different platforms.3 Sutcliffe selected 212 problems from the TPTP library, and suggested that OSCAR and Otter run these problems on the same hardware. This “Shootout at the ATP corral” took place, with the result that OSCAR was on the average 40 times faster than Otter. In addition, OSCAR was able to find proofs for 16 problems on which Otter failed, and Otter was able to find proofs for 3 problems on which OSCAR failed. Taking into account that Otter was written in C and OSCAR in LISP, the speed difference of the algorithms themselves could be as much as an order of magnitude greater. Apparently, natural deduction has some advantages over resolution-refutation.. (shrink)

Le but principal de la communication est d’esquisser les traits essentiels de la méthode appliquée dans les sciences déductives.1. A quoi tend la méthode déductive? Termes primitifs et définis ; axiomes et théorèmes. Les sciences antérieures à une science donnée. La méthode déductive considérée comme propriété caractéristique des mathématiques.2. Liberté dans le choix des termes primitifs et des axiomes ; notion d’équivalence de deux systèmes de termes ou de propositions.Postulats d’indépendance des termes primitifs et des axiomes.3. Postulats de la formalisation (...) des définitions et des démonstrations. Règles de définition et de démonstration ; les démonstrations complètes ; les sciences déductives formalisées. Le rôle de la logique mathématique au point de vue des nouvelles exigences méthodologiques.4. Les notions de science non-contradictoire et complète. L’importance théorique de ces notions ; leur rapport avec le problème de la vérité des mathématiques. Pourquoi les notions en question n’exercent pas en pratique une grande influence sur la construction des sciences mathématiques.5. Certaines propriétés des sciences mathématiques découlant d’une application conséquente de la méthode déductive.ion du sens des termes primitifs dans la construction de la science. Modèles et interprétations du système d’axiomes. Concept de système formel. Valeur de la méthode déductive du point de vue de l’économie de la pensée. (shrink)

Lucy Allais seeks to provide a reading of the Transcendental Deduction of the Categories which is compatible with a nonconceptualist account of Kant’s theory of intuition. According to her interpretation, the aim of the Deduction is to show that a priori concept application is required for empirical concept application. I argue that once we distinguish the application of the categories from the instantiation of the categories, we see that Allais’s reconstruction of the Deduction cannot provide an answer to Hume’s (...) class='Hi'>problem about our entitlement to use a priori concepts when thinking about the objects of empirical intuition. If the Deduction is to provide a response to Hume, Allais’s interpretation must be rejected. (shrink)

Deductive reasoning shares with other forms of thinking a reliance on strategies, as shown by the results of three experiments on the nature and development of control strategies to solve suppositional deductions. These puzzles are based on assertors who may or may not be telling the truth, and their assertions about their status as truthtellers and liars. The first experiment shows that reasoners make backward inferences as well as forward inferences, to short-cut their way through the alternatives, and the (...) generation of suppositions is a source of difficulty. The second experiment establishes that the elimination of the suppositional status of an individual does not render problems easier. The third experiment shows that reasoners can improve their reasoning accuracy and speed spontaneously, without feedback, and it clarifies the transfer of strategies and their development. We discuss the implications of these data for alternative theories of suppositional deduction and for the relationship between reasoning and other forms of thinking such as problem solving. (shrink)

In this paper, I argue that in the sense of greatest epistemological concern for Kant, empirical cognition is “rational sensory discrimination”: the identification or differentiation of sensory objects from each other, occurring through a capacity to become aware of and express judgments. With this account of empirical cognition, I show how the transcendental deduction of the first Critique is most plausibly read as having as its fundamental assumption the thesis that we have empirical cognition, and I provide evidence that Kant (...) understood Hume as granting this assumption. (shrink)

Abductive reasoning involves finding the missing premise of an “unsaturated” deductive inference, thereby selecting a possible _explanans_ for a conclusion based on a set of previously accepted premises. In this paper, we explore abductive reasoning from a structural proof-theory perspective. We present a hybrid sequent calculus for classical propositional logic that uses sequents and antisequents to define a procedure for identifying the set of analytic hypotheses that a rational agent would be expected to select as _explanans_ when presented with (...) an abductive problem. Specifically, we show that this set may not include the deductively minimal hypothesis due to the presence of redundant information. We also establish that the set of all analytic hypotheses exhausts all possible solutions to the given problem. Finally, we propose a deductive criterion for differentiating between the best _explanans_ candidates and other hypotheses. (shrink)

Propositional logic is understood as a set of theorems defined by a deductive system: a set of axioms and a set of rules. Superintuitionistic logic is a logic extending intuitionistic propositional logic \. A rule is admissible for a logic if any substitution that makes each premise a theorem, makes the conclusion a theorem too. A deductive system \ is structurally complete if any rule admissible for the logic defined by \ is derivable in \. It is known (...) that any logic can be defined by a structurally complete deductive system—its structural completion. The main goal of the paper is to study the following problem: given a superintuitionistic logic L, is the structural completion of L hereditarily structurally complete? It is shown that, on the one hand, there is continuum many of such logics, including \, and many of its standard extensions. On the other hand, there is continuum many superintutitionistic logics structural completion of which is not hereditarily structurally complete. It is observed that the class of hereditarily structurally complete superintuitionistic consequence relations does not have the smallest element and it contains continuum many members lacking the finite model property. The following statement is instrumental in obtaining negative results: if a Lindenbaum algebra of formulas on one variable is finite and has more than 15 elements, then a structural completion of such a logic is not hereditarily structurally complete. (shrink)

As the title suggests, this book contains only deductive logic exercises, 500 to be exact. There are no sections on logical theory as such. The exercises are selected from a wide variety of sources. With the exception of a few instances, the exercises are word problems; no symbolism is used. Syllogistic, propositional, and quantificational logic are covered, each relegated to its own part of the book, with the last section containing "arguments in their natural habitat." The exercises in propositional (...) logic are divided into two sections, one designed for the use of truth-tables, the other for constructing deductions. The quantificational logic sections include arguments involving both monadic and relational predication. This paperback is a supplementary problem book for introductory logic.--T. G. N. (shrink)

In a discussion of Sherlock Holmes’ “science of deduction” and the related “method of exclusion,” I show that Holmes’ claim that his inferences are deductive makes sense, if we consider his theoretical presuppositions. So, it is more accurate to say that he tries to reduce abduction to deduction than that he confuses them. His theoretical framework, albeit inadequate as a theory of empirical reasoning, can be seen as a basic model of classical (symbolic) AI. The main problems of this (...) approach are surveyed, and abduction is brought into play as both a better characterization of Holmes’ inferences and a better guide for building AI systems. This is a good background for raising the question of creativity because, according to Peirce, it relates to abduction in a substantial way. Is Sherlock Holmes creative? Can machines be creative? An affirmative answer to the second question might be given by a “logic of discovery” but the problem remains that inventing such a logic already presupposes creativity. (shrink)

This account of propositional logic concentrates on the algorithmic translation of important methods, especially of decision procedures for (subclasses of) propositional logic. Important classical results and a series of new results taken from the fields of normal forms, satisfiability and deduction methods are arranged in a uniform and complete theoretic framework. The algorithms presented can be applied to VLSI design, deductive databases and other areas. After introducing the subject the authors discuss satisfiability problems and satisfiability algorithms with complexity considerations, (...) the resolution calculus with different refinements, and special features and procedures for Horn formulas. Then, a selection of further calculi and some results on the complexity of proof procedures are presented. The last chapter is devoted to quantified boolean formulas. The algorithmic approach will make this book attractive to computer scientists and graduate students in areas such as automated reasoning, logic programming, complexity theory and pure and applied logic. (shrink)

The Justification of Deduction is the title of one of Michael Dummett’s essays. It names also an important theme in his writings to which he returned in the book The Logical Basis of Metaphysics. In the essay he distinguishes different levels of justification of increasing philosophical depth. At the third and deepest level, the focus is on explaining deduction rather than on justifying it. The task is to explain how deduction can be both legitimate and useful in giving us knowledge. (...) I suggest that it can be described as essentially being the task to say what it is that gives a deduction its epistemic force. It is a fact that deduction has such force, consisting in its capacity to provide grounds for assertions and thereby extend our knowledge, but it is a fact that has to be explained. What is it that gives a deduction this capacity? This task is more challenging than is usually assumed. Obviously, it is not the validity of its inferences, as this is usually understood, which gives a deduction its epistemic force. Truth conditional theory of meaning does not seem to have any satisfactory solution to offer, and I argue that nor have inferential theories of meaning, which take the meaning of sentences to be determined by inference rules accepted in a language. In the last part of the paper, I sketch a different approach to the problem. The main idea is here to give the concept of inference a richer content, so that to perform an inference is not only to make a speech act in which a conclusion is claimed to be supported by a number of premisses, but is in addition to operate on the grounds for the premisses with the aim of getting a ground for the conclusion. I suggest that it is thanks to such operations that deductions provide grounds for their final conclusions. (shrink)

The objective of the article is twofold: to advance an interpretation of Descartes’ position on the problem of explaining how deduction from universal propositions to their particular instances can be both legitimate and useful for discovery of truth; and to argue that his position is a valuable contribution to the philosophy of logic. In Descartes’ view. the problem in question is that syllogistic deductions from universal propositions to their particular instances is circular and hence useless as a means (...) for discovery of truth. Descartes’ solution to the problem is to claim that noncircular, useful deduction from the universal to the particular must first be based on deduction from particular truths to particular truths. I examine previous interpretations of Cartesian deduction given by E.M. Curley, Bernard Williams, and Andre Gombay. None of these interpretations fit with all of Descartes’ criticisms of syllogistic deduction and his characterization of useful and legitimate deduction (such as the cogito). I argue that the key to a correct interpretation is Descartes’ claim that implicit knowledge of universal propositions plays a crucial role in useful and legitimate deduction, and I explain how we may cash in his talk of implicit knowledge through Ryle’s notion of knowing how. Having set out a fuller explication of Descartes’ theory of deduction, I argue that it is consistent with the way people actually reason, that it helps us with problems in the philosophy of logic that have been raised by John Stuart Mill, Hilary Putnam, and Michael Dummett. (shrink)

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in formal argumentation, (...) (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity. (shrink)

The concept of evil has been an unpopular one in many recent Western political and ethical discourses. One way to justify this neglect is by pointing to the many problemswiththe concept of evil. The standard grievances brought against the very concept of evil include: that it has no proper place in secular political and ethical discourses; that it is a demonizing term of hatred that leads to violence; that it is necessarily linked with outdated notions of body (...) and sexuality; and that it only hinders rather than aids our ability to understand. I shall seek to argue in defence of the concept of evil against these charges. The upshot of this argument is that the language and concept of evil has a justified and important role to play in political and ethical discourses. (shrink)

A graphic means of representing deductive proofs in a sentential system of symbolic logic is presented. Proof construction is characterized as a domain of the cognitive theory of problem solving, and three different interface designs for supporting the working backwards method of proof construction are demonstrated. Following a description of the rule set and the working backwards method, an analysis is given of student performance data that has guided interface development during the past two years. One interface design (...) is shown to be superior to the others in respect to working backwards. Finally, some general conclusions are drawn concerning the relevance of instructional programs for empirically documenting student difficulties and for improving interface designs. (shrink)

The traditional No-Miracles Argument (TNMA) asserts that the novel predictive success of science would be a miracle, and thus too implausible to believe, if successful theories were not at least approximately true. The TNMA has come under fire in multiple ways, challenging each of its premises and its general argumentative structure. While the TNMA relies on explaining novel predictive success via the truth of the theories, we put forth a deductive version of the No-Miracles argument (DNMA) that avoids inference (...) to the best explanation entirely. Instead, a relatively simple empirical framework and a probabilistic analysis can accomplish the ambitious goals of the TNMA while entirely sidestepping its problems. This close-but-distinct argument has many independent strengths and comparatively few weaknesses. Indeed, objections tailored specifically to the DNMA reveal surprising insights into how exactly NMAs are neither circular nor question-begging, as has been widely speculated. (shrink)

Henry E. Allison presents an analytical and historical commentary on Kant`s transcendental deduction of the pure concepts of the understanding in the Critique of Pure Reason. He argues that, rather than providing a new solution to an old problem, it addresses a new problem, and he traces the line of thought that led Kant to the recognition of the significance of this problem in his 'pre-critical' period. In addition to the developmental nature of the account of Kant`s (...) views presented here, two distinctive features of Allison's reading of the deduction are a defense of Kant`s oft criticized claim that the conformity of appearances to the categories must be unconditionally rather than merely conditionally necessary and an insistence that the argument cannot be separated from Kant`s transcendental idealism. (shrink)

A philosophical argument in ordinary language is made the basis for a series of deductive logic exercises. Problems of translating the reasoning and alternative symbolizations are discussed to help guide students toward accurate charitable formalizations. Finally, the inference is critically evaluated in light of its deductive validity.

This paper is about the nature of the relationship between (1) the doctrine of Non-Conceptualism about mental content, (2) Kant's Transcendental Idealism, and (3) the Transcendental Deduction of the Pure Concepts of the Understanding, or Categories, in the B (1787) edition of the Critique of Pure Reason, i.e., the B Deduction. Correspondingly, the main thesis of the paper is this: (1) and (2) yield serious problems for (3), yet, in exploring these two serious problems for the B Deduction, we also (...) discover some deeply important and perhaps surprising philosophical facts about Kant's theory of cognition and his metaphysics. (shrink)

It is often said that in a purely formal perspective, intuitionistic logic has no obvious advantage to deal with the liar-type paradoxes. In this paper, we will argue that the standard intuitionistic natural deduction systems are vulnerable to the liar-type paradoxes in the sense that the acceptance of the liar-type sentences results in inference to absurdity (⊥). The result shows that the restriction of the Double Negation Elimination (DNE) fails to block the inference to ⊥. It is, however, not the (...) problem of the intuitionistic approaches to the liar-type paradoxes but the lack of expressive power of the standard intuitionistic natural deduction system. We introduce a meta-level negation for a given system and a meta-level absurdity, ⋏, to the intuitionistic system. We shall show that in the system, the inference to ⊥ is not given without the assumption that the system is complete. Moreover, we consider the Double Meta-Level Negation Elimination rules (DMNE) which implicitly assume the completeness of the system. Then, the restriction of DMNE can rule out the inference to ⊥. (shrink)

In "Beyond the Myth of the Myth: A Kantian Theory of Non-Conceptual Content", Robert Hanna argues for a very strong kind of non-conceptualism, and claims that this kind of non-conceptualism originally has been developed by Kant. But according to "Kant's Non-Conceptualism, Rogue Objects and the Gap in the B Deduction", Kant's non-conceptualism poses a serious problem for his argument for the objective validity of the categories, namely the problem that there is a gap in the B Deduction. This (...) gap is that the B Deduction goes through only if conceptualism is true, but Kant is a non-conceptualist. In this paper I argue, contrary to what Hanna claims, that there is not a gap in the B Deduction. (shrink)

The problem of Kant’s Neglected Alternative is that while his Aesthetic provides an argument that space and time are empirically real – in applying to all appearances – its argument seems to fall short of the conclusion that space and time are transcendentally ideal, in not applying to any things in themselves. By considering an overlooked passage in which Kant explains why his Transcendental Deduction is ‘unavoidably necessary’, I argue that it is not solely in his Aesthetic but more (...) so in his Deduction where he intends to provide his argument for the transcendental ideality of space and time. His Deduction shows that space and time do not have a valid application to any things in themselves by arguing that the categories do have a valid application to everything in space and time, but that the categories do not have a valid application to any things in themselves. (shrink)

This paper discusses the relation between the natural deduction rules of deduction in sequent format and the provability valuation starting from Garson’s Local Expression Theorem, which is meant to establish that the natural deduction rules of inference enforce exactly the classical meanings of the propositional connectives if these rules are taken to be locally valid, i.e. if they are taken to preserve sequent satisfaction. I argue that the natural deduction rules for disjunction are in no better position than the axiomatic (...) calculi in uniquely determining the intended meaning of disjunction when the local models are used, if a satisfied sequent embeds, as a logical inferentialist should require, a formal derivability relation. This happens because, when governed by these rules and without additional semantic assumptions, the disjunction sign still expresses a non-extensional connective, i.e. a connective that, properly understood, has no unique logical characteristic. However, this is not a dead end for the logical inferentialists since both a multiple conclusions formalization of the disjunction operator and a bilateralist one do succeed in restoring the standard meaning of disjunction. (shrink)

In this work, we propose a variant of so-called informational semantics, a technique elaborated by Voishvillo, for two infectious logics, Deutsch’s |${\mathbf{S}_{\mathbf{fde}}}$| and Szmuc’s |$\mathbf{dS}_{\mathbf{fde}}$|⁠. We show how the machinery of informational semantics can be effectively used to analyse truth and falsity conditions of disjunction and conjunction. Using this technique, it is possible to claim that disjunction and conjunction can be rightfully regarded as such, a claim which was disputed in the recent literature. Both |${\mathbf{S}_{\mathbf{fde}}}$| and |$\mathbf{dS}_{\mathbf{fde}}$| are formalized in (...) terms of natural deduction. This allows us to solve several problems: to develop a natural deduction calculus for |${\mathbf{S}_{\mathbf{fde}}}$| containing the standard form of disjunction elimination (in contrast to the calculus by Petrukhin), to introduce the first natural deduction calculus for |$\mathbf{dS}_{\mathbf{fde}}$| and to reflect the fundamental symmetry between |${\mathbf{S}_{\mathbf{fde}}}$| and |$\mathbf{dS}_{\mathbf{fde}}$| on proof-theoretical level forming a convenient basis for obtaining their well-known extensions |$\mathbf{K}^{\mathbf{w}}_{\mathbf{3}}$| and |$\mathbf{PWK}$|⁠. (shrink)

This article concerns the structure of defeasible arguments like: 'If Bob has red spots, Bob has the measles; Bob has red spots; therefore Bob has the measles.' The issue is whether such arguments have the form of modus ponens or not. Either way there is a problem. If they don't have the form of modus ponens, the common opinion to the contrary taught in leading logic textbooks is wrong. But if they do have the form of modus ponens, doubts (...) are raised about the conventional dogma that all arguments having the form of modus ponens are deductively valid. By carefully examining arguments on both sides of the issue, reasonable doubts are raised about the view that all arguments having a modus ponens form are valid. (shrink)

It is not uncommon for students to try, and indeed to succeed, in buying the course texts used. This often makes a great deal of sense. But for this course, you must buy the textbook new . Here's why. The textbook comes with software that you will use to submit all of the course homeworks. The problem is that only one student can register the software , per book — period. So, if you buy the textbook used, not only (...) do you run the risk that the software won't be intact, you will be unable to register the software and, therefore, unable to turn in your homework. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Kant's Transcendental Deduction by Alison LaywineKatherine DunlopAlison Laywine. Kant's Transcendental Deduction. Oxford: Oxford University Press, 2020. Pp. iv + 318. Hardback, $80.00.Alison Laywine's contribution to the rich literature on Kant's "Transcendental Deduction of the Categories" stands out for the novelty of its approach and conclusions. Laywine's declared "strategy" is "to compare and contrast" the Deduction with the Duisburg Nachlaß, an important set of manuscript jottings from the 1770s (...) (10). But her approach is also deeply informed by Kant's writings on metaphysics from the 1750s and 1760s; moreover, she gives attention to ancient Greek geometry and its importance for Kant's thought.I believe Laywine's most important interpretative claim is that the Transcendental Deduction's "final step" addresses "the question of how nature is possible" (210). Here, 'nature' is understood in what Kant calls the "formal sense," as "the totality of rules under which appearances must stand if they are to be thought as joined in one experience" (Prolegomena, §36; cf. B 165). This constitutes a novel answer to the problem posed in Dieter Henrich's landmark 1969 paper "The Proof Structure of the Transcendental Deduction" (Review of Metaphysics 22 [1969]: 640–59): how to understand §§21–26 of the (B-edition) Deduction as proving more than the thesis already stated in §20 (that the manifold in an intuition necessarily stands under the categories).Crucial support for Laywine's interpretation comes from §26 of the Deduction, which claims the possibility of cognizing objects "a priori through categories... as far as the laws of their combination are concerned, thus the possibility of as it were prescribing the law to nature and even making [nature] possible," is now "to be explained" (B 159). Kant does not use the term 'world' in this passage (as he does in Prolegomena §36). But Laywine contends that here the word 'nature' "has unmistakable, deliberate, cosmological connotations," and in fact "means 'world' in the sense of Kant's early cosmology": roughly, a whole unified by means of laws (12–13).The "cosmological" language of §26 does not recur in the following, officially concluding, section of the Deduction. So, we might ask whether Kant's explanation of how the understanding prescribes laws to nature is integral to the Deduction; Henry Allison, for instance, describes this explanation as an "appendix" (Kant's Transcendental Deduction: An Analytical-Historical Commentary [Oxford: Oxford University Press, 2015], 9). The question of whether it belongs integrally has bite because, as Laywine makes clear in her "Conclusion," the universal laws at issue are just the Analogies of Experience. Thus, the passage could be read as the "promissory note with a forward-looking reference to the System of Principles" that she finds missing from the Deduction (289). Laywine meets such worries by arguing that "the reappropriation [from the pre-Critical period] of the general cosmology is actually [End Page 162] doing... a lot of hard work" in the Deduction (13). This sustained argument comes to a head in chapter 4, which contends that each of the two steps into which Laywine divides the first half of the (B-edition) Deduction, treated in chapters 2 and 3 (respectively), relies on cosmological presuppositions.Chapter 2 analyzes the conception of knowledge as relation to an object that is asserted (in §17) to rely on the synthetic unity of apperception. Laywine traces this conception to the Duisburg Nachlaß's account of "exposition," which relates concepts a priori to appearances, in something like the way that geometrical construction relates them to pure intuition. (Laywine further connects the Latin term expositio to the Greek ekthesis, which expositio translates in the context of geometrical proof. She thereby gives the notion of ekthesis broader importance than does Jaakko Hintikka, who noted its relevance to Kant's philosophy of mathematics in "Kant on the Mathematical Method" [Monist 51 (1967): 352–75]. Laywine claims that Borelli's edition of Euclid could have been Kant's source for the term, but I doubt it was known to Kant and his contemporaries, since it is not among the editions cited by Christian Wolff; Commandino seems likelier to me. Chapter 2 is concerned to articulate the notion of... (shrink)

The confusion/non-consequential thinking explanation proposed by Newstead, Girotto, and Legrenzi (1995) for poor performance on Wason's THOG problem (a hypothetico-deductive reasoning task) was examined in three experiments with 300 participants. In general, as the cognitive complexity of the problem and the possibility of non-consequential thinking were reduced, correct performance increased. Significant but weak facilitation (33-40% correct) was found in Experiment 1 for THOG classification instructions that did not include the indeterminate response option. Substantial facilitation (up to 75% (...) correct) was obtained in Experiment 2 with O'Brien et al.'s (1990) one-other-THOG classification instruction. In Experiment 3, a revised version of O'Brien et al.'s pre-test problem format also led to substantial facilitation, even with the use of the standard three-choice THOG classification instruction. These findings are discussed in terms of Newstead et al.'s theoretical proposal and possible attentional factors. (shrink)