In an earlier article (s. J Gen Philos Sci 40:341-355, 2009), I have rejected an interpretation of Aristotle's syllogistic which (since Patzig) is predominant in the literature on Aristotle, but wrong in my view. According to this interpretation, the distinguishing feature of perfect syllogisms is their being evident. Theodor Ebert has attempted to defend this interpretation by means of objections (s. J Gen Philos Sci 40:357-365, 2009) which I will try to refute in part [1] of the following article. (...) I want to show that (1) according to Aristotle's Prior Analytics perfect and imperfect syllogisms do not differ by their being evident, but by the reason for their being evident, (2) Aristotle uses the same words to denote proofs of the validity of perfect and imperfect syllogisms („apodeixis”, "deiknusthai" etc.), (3) accordingly, Aristotle defines perfect syllogisms not as being evident, but as "requiring nothing beyond the things taken in order to make the necessity evident", i.e. as not "requiring one or more things that are necessary because of the terms assumed, but that have not been taken among the propositions" (APr. I. 1), (4) the proofs by which the validity of perfect assertoric syllogisms can be shown according to APr. I. 4 are based on the Dictum de omni et nullo, (5) the fact that Aristotle describes these proofs only in rough outlines corresponds to the fact that his proofs of the validity of other fundamental rules are likewise produced in rough outlines, e.g. his proof of the validity of conversio simplex in APr. I. 2, which usually has been misunderstood (also by Ebert): (6) Aristotle does not prove the convertibility of E-sentences by presupposing the convertibility of I-sentences; only the reverse is true. (shrink)

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on: admissible or permissible inference rules the derivability of the admissible inference rules the structural completeness of logics the bases for admissible and valid inference rules. There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered. The book (...) is basically self-contained and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included. The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook. (shrink)

The form of the conditional syllogism resembles that of the categorical syllogism, while its subject matter is at least a conditional premise, but its conclusion is always conditional conjunctive or disjunctive. This mixed structure to which we apply the rules of the categorical syllogism, is a structure of which Aristotle did not have an idea, and which the Stoics did not conceive, and which the non-Arabian logicians did not know until in modern times. But what we have to notice (...) here is the putting of a conditional matter in the form of the categorical syllogism, and it is this kind of hybridization, if we dare to say, which generated this mixed structure which appeared for the first time in the history of logic in the treatise on the logic of Ibn Sina and which can be considered a discovery by this author until proof to the contrary, and that the ancient Arabian logicians have taken the habit of exhibiting in their treatises. (shrink)

The admissibility of Ackermann's rule γ is one of the most important problems in relevant logics. The admissibility of γ was first proved by an algebraic method. However, the development of Routley-Meyer semantics and metavaluational techniques makes it possible to prove the admissibility of γ using the method of normal models or the method using metavaluations, and the use of such methods is preferred. This paper discusses an algebraic proof of the admissibility of γ in relevant modal logics based (...) on modern algebraic models. (shrink)

Two claims have been explored, the first, that fool-proof proofs of the sort that there could be if there were a God like the God of Abraham, Isaac and Jacob are not to be expected, on good religious grounds (a claim I found wanting); and second, that there cannot be philosophical proofs of God which work beyond reasonable doubt.The argument that there cannot be philosophical proofs beyond a reasonable doubt is supported by an examination of some of the (...) class='Hi'>fundamental issues in the traditional discussions of proofs for God's existence, and by claims about the relativity of methodological rules to world-views which, I maintain, the traditional discussions indicate. I do not claim to have proved that relativity, only to have illustrated the claim that it is there.It is my further opinion, but I do not claim really to have proved it, that the failure of religious excuses for the lack of public demon strations constitutes a good reason for concluding that there is no God of the sort described as the God of Abraham, Isaac and Jacob; hence that if there is a God, it must be the God of the philosophers. However I admit that there might be sufficient hidden reasons which would offer persuasive excuses for the God of the ordinary believer.Lastly, I have made some comments about what I think is the more valuable way to view the “proofs for God.” Such an interpretation does justice to the otherwise baffling and continual philosophical disagreements better than rival theories. It is time we take these disagreements with utmost seriousness, and one can hardly do that while treating basic metaphysical arguments as fool-proof proofs. (shrink)

Different natural deduction proof systems for intuitionistic and classical logic —and related logical systems—differ in fundamental properties while sharing significant family resemblances. These differences become quite stark when it comes to the structural rules of contraction and weakening. In this paper, I show how Gentzen and Jaśkowski’s natural deduction systems differ in fine structure. I also motivate directed proof nets as another natural deduction system which shares some of the design features of Genzen and Jaśkowski’s systems, (...) but which differs again in its treatment of the structural rules, and has a range of virtues absent from traditional natural deduction systems. (shrink)

The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.

I see the question what it is that makes an inference valid and thereby gives a proof its epistemic power as the most fundamental problem of general proof theory. It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen’s remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic tradition about what a proof is. They are reviewed in the paper and I discuss (...) to what extent they succeed in answering what a proof is. Gentzen’s ideas are shown to give rise to a new notion of valid argument. At the end of the paper I summarize and briefly discuss an approach to the problem that I have proposed earlier. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Logic Pamphlets of Charles Lutwidge Dodgson and Related PiecesIrving H. AnellisFrancine F. Abeles, editor. The Logic Pamphlets of Charles Lutwidge Dodgson and Related Pieces. The Pamphlets of Lewis Carroll, 4. New York-Charlottesville-London: Lewis Carroll Society of North America-University Press of Virginia, 2010. Pp. xx + 271. Cloth, $75.00.Until William Bartley’s rediscovery and reconstruction of Dodgson’s lost Part II of Symbolic Logic, Lewis Carroll’s reputation in logic, when (...) taken seriously, rested upon his few published articles on logical paradoxes and puzzles, while his published books, The Game of Logic and Symbolic Logic, were regarded as suitable perhaps as pedagogical tools, but dismissed otherwise as amusements, on a par with the Alice works. Richard Braithwaite asserted (“Lewis Carroll as Logician”) that “Carroll regarded formal and symbolic logic not as a corpus of systematic knowledge about valid thought nor yet as an art for teaching a person to think correctly.” He suggested that Dodgson used his pseudonym for his popular and whimsical writings, reserving his legal name only for his serious mathematical products. Philosophers practicing linguistic analysis in their assault on nonsense deriving from misuse of language meanwhile developed an interest in Carroll (e.g. George Pitcher, “Wittgenstein, Nonsense, and Lewis Carroll”). Abeles, on the contrary, stresses the seriousness of Dodgson’s pedagogical and popularizing mission (195–99).Bartley’s discovery opened the door for reevaluating Dodgson as a serious logician, and Abeles has been in the forefront of that reevaluation. Her general introduction and introductions to the main divisions of this collection of Carroll’s publications (and additional material from his Nachlass and a handful of publications of others, composed in response to Carroll’s published articles) place his serious logical work and its significance in historical perspective, providing a deeper, more sophisticated view than found in Bartley’s “Editor’s Introduction” to Lewis Carroll’s Symbolic Logic. Abeles also offers brief introductions to the individual entries, which describe their physical characteristics and circumstances of composition.Dodgson aligned with the majority of those British logicians of his day, starting with George Boole, who understood formal logic to be Aristotelian syllogistic, and symbolic logic to be syllogistic logic expressed algebraically. His notational innovation was employing indices to terms so that, independently of the arrangement of terms, propositions could easily be read; thus, “xy0” can be read as “no x are y” or “no y are x” and “x1y0” as “all x are not y” or “no x are y”. In dealing with the classical elimination problem in the class calculus—to determine the maximum information, without duplications, obtainable from a given set of premises—Dodgson in his later work and unpublished letters anticipated several concepts of automated theorem proving.Central to both geometry and to formal logic are the rules of logical inference that establish the validity of arguments and guarantee that conclusions inferred from true [End Page 506] premises are true. The modern analytic tableaux, or tree method, is a fundamental tool in this regard, both for deriving theorems and for determining the validity or invalidity of the proofs for propositional calculus and first-order predicate calculus.Introducing his reconstruction of Part II of Symbolic Logic, Bartley noted that, along with the logic diagrams that Carroll devised, he also devised a tableau method that strongly resembled Beth’s semantic tableaux. Abeles went more deeply and convincingly into that resemblance in her earlier research. Beth’s tableaux, together with Hintikka’s model sets, were the direct ancestors of Smullyan’s analytic tableaux, which in turn are the theoretical basis for the Robinson resolution method that continues to be central to much work in programming logic.Abeles has also shown that the work in Studies in Logic by Charles Peirce and his logic students at Johns Hopkins University served as a source of inspiration for Dodgson. More critically, she demonstrated that Dodgson devised and employed the tree method for carrying out proofs of syllogisms and chains of syllogisms, or soriteses, in Part II of Symbolic Logic (see her “Lewis Carroll’s Method of Trees: Its Origins in Studies in Logic,” Modern Logic 1 [1990]: 25–34). The tree... (shrink)

What is morally permissible, and what is morally obligatory? These questions form the core of a vast amount of philosophical reasoning. In his Fundamental Principles of the Metaphysics of Morals, Immanuel Kant developed a basis for the answers. In this landmark work, the German philosopher asks what sort of maxim might function as a guide to appropriate action under a given set of circumstances. By universalizing such a maxim, would morally permissible behavior not become clear? Suppose that everyone were (...) to behave in accordance with this maxim. If everyone followed the maxim in the same way without harm to civilized culture, then the behavior would be morally permissible. But what if no one followed the maxim? Would civilization thereby be at risk? In such a case, the behavior would be morally obligatory. Kant's test, known as the Categorical Imperative, is a logical proof of the Golden Rule and the centerpiece of this work. It constitutes his best-known contribution to ethical discussion, and a familiarity with his reasoning in this book is essential to students of philosophy, religion, and history. (shrink)

I develop the decision-theoretic approach to quantum probability, originally proposed by David Deutsch, into a mathematically rigorous proof of the Born rule in (Everett-interpreted) quantum mechanics. I sketch the argument informally, then prove it formally, and lastly consider a number of proposed ``counter-examples'' to show exactly which premises of the argument they violate. (This is a preliminary version of a chapter to appear --- under the title ``How to prove the Born Rule'' --- in Saunders, Barrett, Kent and Wallace, (...) "Many worlds? Everett, quantum theory and reality", forthcoming from Oxford University Press.). (shrink)

The Born’s rule to interpret the square of wave function as the probability to get a specific value in measurement has been accepted as a postulate in foundations of quantum mechanics. Although there have been so many attempts at deriving this rule theoretically using different approaches such as frequency operator approach, many-world theory, Bayesian probability and envariance, literature shows that arguments in each of these methods are circular. In view of absence of a convincing theoretical proof, recently some researchers (...) have carried out experiments to validate the rule up-to maximum possible accuracy using multi-order interference (Sinha et al, Science, 329, 418 [2010]). But, a convincing analytical proof of Born’s rule will make us understand the basic process responsible for exact square dependency of probability on wave function. In this paper, by generalizing the method of calculating probability in common experience into quantum mechanics, we prove the Born’s rule for statistical interpretation of wave function. (shrink)

By considering the new notion of theinversesof syllogisms such asBarbaraandCelarent, we show how the rule ofIndirect Proof, in the form (no multiple or vacuous discharges) used by Aristotle, may be dispensed with, in a system comprising four basic rules of subalternation or conversion and six basic syllogisms.

What is morally permissible, and what is morally obligatory? These questions form the core of a vast amount of philosophical reasoning. In this landmark work, Kant proposes the concept of a maxim, which functions as a guide to appropriate action under a given set of circumstances. By universalizing the maxim, morally permissible and obligatory behavior becomes clear. The German philosopher's test, known as the Categorical Imperative, is a logical proof of the Golden Rule and the centerpiece of this work. (...) It constitutes Kant's best-known contribution to ethical discussion, and a familiarity with its constituents is essential to students of philosophy, religion, and history. (shrink)

I present a proof of the quantum probability rule from decision-theoretic assumptions, in the context of the Everett interpretation. The basic ideas behind the proof are those presented in Deutsch's recent proof of the probability rule, but the proof is simpler and proceeds from weaker decision-theoretic assumptions. This makes it easier to discuss the conceptual ideas involved in the proof, and to show that they are defensible.

In Prior Analytics 1.15 Aristotle undertakes to establish certain modal syllogisms of the form XQM. Although these syllogisms are central to his modal system, the proofs he offers for them are problematic. The precise structure of these proofs is disputed, and it is often thought that they are invalid. We propose an interpretation which resolves the main difficulties with them: the proofs are valid given a small number of intrinsically plausible assumptions, although they are in tension with some claims found (...) elsewhere in Aristotle’s modal syllogistic. The proofs make use of a rule of propositional modal logic which we call the possibility rule. We investigate how this rule interacts and coheres with the core elements of the modal syllogistic. (shrink)

To solve the probability problem of the Many Worlds Interpretation of Quantum Mechanics, D. Wallace has presented a formal proof of the Born rule via decision theory, as proposed by D. Deutsch. The idea is to get subjective probabilities from rational decisions related to quantum measurements, showing the non-probabilistic parts of the quantum formalism, plus some rational constraints, ensure the squared modulus of quantum amplitudes play the role of such probabilities. We provide a new presentation of Wallace’s proof, (...) reorganized to simplify some arguments, and analyze it from a formal perspective. Similarities with classical decision theory are made explicit, to clarify its structure and main ideas. A simpler notation is used, and details are filled in, making it easier to follow and verify. Some problems have been identified, and we suggest possible corrections. (shrink)

Having analyzed the formal aspects of Wallace’s proof of the Born rule, we now discuss the concepts and axioms upon which it is built. Justification for most axioms is shown to be problematic, and at times contradictory. Some of the problems are caused by ambiguities in the concepts used. We conclude the axioms are not reasonable enough to be taken as mandates of rationality in Everettian Quantum Mechanics. This invalidates the interpretation of Wallace’s result as meaning it would be (...) rational for Everettian agents to decide using the Born rule. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Convention for Protection of Human Rights and Dignity of the Human Being with Regard to the Application of Biology and Biomedicine: Convention on Human Rights and BiomedicineCouncil of EuropePreambleThe Member States of the Council of Europe, the other States and the European Community signatories hereto,Bearing in mind the Universal Declaration of Human Rights proclaimed by the General Assembly of the United Nations on 10 December 1948;Bearing in mind the (...) Convention for the Protection of Human Rights and Fundamental Freedoms of 4 November 1950;Bearing in mind the European Social Charter of 18 October 1961;Bearing in mind the International Covenant on Civil and Political Rights and the International Covenant on Economic, Social and Cultural Rights of 16 December 1966;Bearing in mind the Convention for the Protection of Individuals with Regard to Automatic Processing of Personal Data of 28 January 1981;Bearing also in mind the Convention on the Rights of the Child of 20 November 1989;Considering that the aim of the Council of Europe is the achievement of a greater unity between its members and that one of the methods by which that aim is to be pursued is the maintenance and further realisation of human rights and fundamental freedoms;Conscious of the accelerating developments in biology and medicine;Convinced of the need to respect the human being both as an individual and as a member of the human species and recognising the importance of ensuring the dignity of the human being; [End Page 277]Conscious that the misuse of biology and medicine may lead to acts endangering human dignity;Affirming that progress in biology and medicine should be used for the benefit of present and future generations;Stressing the need for international co-operation so that all humanity may enjoy the benefits of biology and medicine;Recognising the importance of promoting a public debate on the questions posed by the application of biology and medicine and the responses to be given thereto;Wishing to remind all members of society of their rights and responsibilities;Taking account of the work of the Parliamentary Assembly in this field, including Recommendation 1160 (1991) on the preparation of a Convention on bioethics;Resolving to take such measures as are necessary to safeguard human dignity and the fundamental rights and freedoms of the individual with regard to the application of biology and medicine;Have agreed as follows:Chapter IGeneral provisionsArticle 1. (Purpose and object)Parties to this Convention shall protect the dignity and identity of all human beings and guarantee everyone, without discrimination, respect for their integrity and other rights and fundamental freedoms with regard to the application of biology and medicine.Each Party shall take in its internal law the necessary measures to give effect to the provisions of this Convention.Article 2. (Primacy of the human being)The interests and welfare of the human being shall prevail over the sole interest of society or science.Article 3. (Equitable access to health care)Parties, taking into account health needs and available resources, shall take appropriate measures with a view to providing, within their jurisdiction, equitable access to health care of appropriate quality. [End Page 278]Article 4. (Professional standards)Any intervention in the health field, including research, must be carried out in accordance with relevant professional obligations and standards.Chapter IIConsentArticle 5. (General rule)An intervention in the health field may only be carried out after the person concerned has given free and informed consent to it.This person shall beforehand be given appropriate information as to the purpose and nature of the intervention as well as on its consequences and risks.The person concerned may freely withdraw consent at any time.Article 6. (Protection of persons not able to consent)1. Subject to Articles 17 and 20 below, an intervention may only be carried out on a person who does not have the capacity to consent, for his or her direct benefit.2. Where, according to law, a minor does not have the capacity to consent to an intervention, the intervention may only be carried out with the authorisation of his or her representative or an authority or a person or... (shrink)

Hilbert arithmetic in a wide sense, including Hilbert arithmetic in a narrow sense consisting by two dual and anti-isometric Peano arithmetics, on the one hand, and the qubit Hilbert space (originating for the standard separable complex Hilbert space of quantum mechanics), on the other hand, allows for an arithmetic version of Gentzen’s cut elimination and quantum measurement to be described uniformy as two processes occurring accordingly in those two branches. A philosophical reflection also justifying that unity by quantum neo-Pythagoreanism links (...) it to the opposition of propositional logic, to which Gentzen’s cut rule refers immediately, on the one hand, and the linguistic and mathematical theory of metaphor therefore sharing the same structure borrowed from Hilbert arithmetic in a wide sense. An example by hermeneutical circle modeled as a dual pair of a syllogism (accomplishable also by a Turing machine) and a relevant metaphor (being a formal and logical mistake and thus fundamentally inaccessible to any Turing machine) visualizes human understanding corresponding also to Gentzen’s cut elimination and the Gödel dichotomy about the relation of arithmetic to set theory: either incompleteness or contradiction. The metaphor as the complementing “half” of any understanding of hermeneutical circle is what allows for that Gödel-like incompleteness to be overcome in human thought. (shrink)

I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain (...) how the syllogistic is a fragment of my system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle. (shrink)

In ancient philosophy, Logic was seen as the instrument and method of philosophy. However, sometimes detailed and profound discussions have been made about the demonstrativeness of philosophical sciences. Most philosophers have accepted that the mathematical sciences were especially demonstrative and likewise, most of the natural sciences are demonstrative for them. But can metaphysics be demonstrative or not? This is one of the fundamental issues around which the great debates were made in Islamic philosophy. While these issues are known to (...) specialists in this field, we can hardly find studies and research that deal with this issue to a sufficient degree and precise discussions in our day, I mean the issue of demonstrativeness of art of demonstration, art of dialectic, art of sophistry, art of rhetoric and art of poetry. It is clear that the peripatetic philosophers did not write a separate book or treatise on the issue of the demonstrativeness of the logical arts, but we can claim that they believed that all these arts were demonstrative and that they were extracted/abstracted using the art of demonstration. We can come across important phrases and statements that prove this claim between the lines of the books that these philosophers wrote in the science of logic. And while Aristotle is the founder of the science of logic, then how is it to be understood that there are practical particular examples and some easy rules of these arts? According to their texts, we can say that they believed that the actual performance of these arts by their actors in materials by particular and practical actions differs from extracting their universal rules by abstracting them from their materials through theoretical and analytical research. For example, the poems of the great poets; Homer and al-Mutanabbi, are particular works in the matter of the art of poetry performed by poets. The books written by Aristotle and Ibn Rushd in the art of poetry contain universal rules extracted abstractly from the materials of their particular poems. As for understanding the demonstrativeness of the remaining four arts, it is easier, but how should we understand the demonstrativeness of the art of demonstration itself? It seems that the peripatetic logicians also distinguished between applying demonstrative inferences to significant materials and extracting the universal rules of the art of demonstration from these materials. These five logical arts are not particular and practical things, but they are philosophical, demonstrative, and analytical studies for the peripatetic logicians in Islam. In this paper, we will try to draw attention to the issue of the demonstrativeness of the five arts and to open a new discussion about it based on some important sentences and phrases expressed by al-Fārābī, Ibn Zur‘a, Ibn Sīnā, Ibn Bājja and Ibn Rushd between the lines in their books, which they wrote in the field of logic and within different and multiple contexts. (shrink)

If we consider Wittgenstein's career as a whole, it appears that he wrote more on the philosophy of logic and mathematics than any other subject. Yet his writings on these subjects have exerted little influence. Indeed, the tide of response to Remarks on the Foundations of Mathematics, which contains the bulk of his latest views of mathematics, has been for the most part overwhelmingly negative. Given his later emphasis on the context-bound character of language, mathematics and logic--where language apparently operates (...) in an maximally precise, clear and general way--represent two of the most difficult cases for Wittgenstein to confront. My thesis aims to defend Wittgenstein from the charges of benighted arrogance traditionally levelled against him. I argue that Wittgenstein's later discussions of mathematics form a central part of a larger philosophical project, internally related to Philosophical Investigations and shaping in specific ways Wittgenstein's reaction to both scepticism and accounts of the nature of logical and mathematical truth. I see Wittgenstein criticizing the unreflective use of mathematical tools in philosophy, not offering a competing philosophy of mathematics. ;Too few of Wittgenstein's readers have been willing to offer detailed exegesis of his writing . This has made it difficult to understand how the aphoristic style of his late writing bears on the philosophical problems being discussed. On my view, the quality of Wittgenstein's writing is intrinsic to his later conception of the nature of logic, mathematics and philosophy. Line by line engagement with his texts is thus imperative in order to achieve an understanding of his philosophical objectives and criticisms. My dissertation offers as a paradigm of such reading a detailed exegesis and criticism of the opening five sections of Remarks on the Foundations of Mathematics. ;I briefly explore in some detail the origin and genesis of Wittgenstein's interest in rule-following, from its roots in the Tractatus's conception of logical syntax. This background illuminates Wittgenstein's later conception of logical truth and his criticisms of Frege's and Russell's arguments for logicism. To explore these criticisms, I focus on Wittgenstein' s discussions of the Frege-Russell definition of "Number" in "logical" terms. Placed in their appropriate philosophical and historical context, Wittgenstein's seemingly outrageous remarks about the nature of proof, mathematical logic and the foundations of mathematics do not simply betray his ignorance. Nor do they commit him to a revisionist attitude toward mathematical practice. Rather, they raise fundamental questions about the philosophical presuppositions lying behind attempts to bring particular mathematical results to bear in philosophy. (shrink)

Large portions of mathematics such as algebra and geometry can be formalized using first-order axiomatizations. In many cases it is even possible to use a very well-behaved class of first-order axioms, namely, what are called coherent or geometric implications. Such class of axioms can be translated to inference rules that can be added to a sequent calculus while preserving its structural properties. In this work, this fundamental result is extended to their infinitary generalizations as extensions of sequent calculi (...) for both classical and intuitionistic infinitary logic. As an application, a simple proof of the infinitary Barr’s theorem without the axioms of choice is shown. (shrink)

We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational account of proof-theoretic harmony. With every set of (...) introduction rules a canonical elimination rule, and with every set of elimination rules a canonical introduction rule is associated in such a way that the canonical rule is in harmony with the set of rules it is associated with. An example given by Hazen and Pelletier is used to demonstrate that there are significant connectives, which are characterized by their elimination rules, and whose introduction rule is the canonical introduction rule associated with these elimination rules. Due to the availabiliy of higher-level rules and propositional quantification, the means of expression of the framework developed are sufficient to ensure that the construction of canonical elimination or introduction rules is always possible and does not lead out of this framework. (shrink)

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

One lists the distinct pairs of categorical premises formulable via only the positive terms, S,P,M, by constructing a six by six matrix obtained by pairing the six categorical P-premises, A, O, A, O, where P* ∈ {P,P′}, with the six, similar, categorical S-premises. One shows how five rules of valid syllogism, select only 15 distinct PCPs that entail logical consequences belonging to the set L+: = {A, O, A, E, O, I}. The choice of admissible LCs can be regarded (...) as a condition separated from the conditions contained in the RofVS: the usual eight PCPs that generate valid syllogisms are obtained when the only admissible LCs belong to the set L: = {A, E, O, I} and no existential imports are addressed. A 64 PCP-matrix obtains when both PCPs and LCs may contain indefinite terms—the positive, S,P,M, terms, and their complementary sets, S′,P′, M′, in the universe of discourse, U, called the negative terms. Now one can accept eight LCs: A, I, where P* ∈ {P,P′}, S* ∈ {S,S′}, and there are 32 conclusive PCPs, entailing precise, “one partitioning subset of U” LCs. The four rules of conclusive syllogisms predict the less precise LCs, left after eliminating the middle term from the exact LCs. The RofCS also predict that the other 32 PCPs of the 64 PCP-matrix are non-conclusive. The RofVS and the RofCS are generalized, and arguments are given, for also accepting as valid syllogisms the conclusive syllogisms formulable via positive terms which entail the LCs A and O. (shrink)

Frank Ramsey in his paper ‘Truth and Probability’ was the first to develop a theory of utility based on a representation theorem, and a theory of partial belief based on utility-valued odds. But his proof of the multiplication theorem, on which in his system the law of addition depends, contains a step for which there seems to be no justification, and Ramsey provided no clue as to how to supply one. I conjecture that the missing justification appeals naturally to (...) a three-valued logic that de Finetti later described in his own discussion of the multiplication theorem. If this is the case then Ramsey’s paper is even more strikingly original than has been thought. (shrink)

Consider syllogisms in which fraction (percentage) quantifiers are permitted in addition to universal and particular quantificrs, and then include further quantifiers which are modifications of such fractions (such as "almost ½ the S are P" and "Much more than ½ the S are P"). Could a syllogistic system containing such additional categorical forms be coherent? Thompson's attempt (1986) to give rules for determining validity of such syllogisms has failed; cf. Carnes & Peterson (forthcoming) for proofs of the unsoundness (...) and incompleteness of Thompson's rules. Building on Peterson (1985), the coherence of such a syllogistic can, however, be demonstrated with an algebra which provides its semantics; e.g., "almost ¾ the S arc P'" is represented as " -(3(SP) » SP)". (shrink)

The relation between ex falso and disjunctive syllogism, or even the justification of ex falso based on disjunctive syllogism, is an old topic in the history of logic. This old topic reappears in contemporary logic since the introduction of minimal logic by Johansson. The disjunctive syllogism seems to be part of our general non-problematic inferential practices and superficially it does not seem to be related to or to depend on our acceptance of the frequently disputable ex falso rule. We know (...) that the acceptance of the ex falso is a sufficient condition for the acceptance of the disjunctive syllogism, but the interesting question is: is the ex falso a necessary condition for the acceptance of the disjunctive syllogism? The aim of the present paper is to discuss some possible ways to define systems that combines the preservation of the disjunctive syllogism with the rejection of the ex falso. (shrink)

Using Heijenoort’s unpublished generalized rules of quantification, we discuss the proof of Herbrand’s Fundamental Theorem in the form of Heijenoort’s correction of Herbrand’s “False Lemma” and present a didactic example. Although we are mainly concerned with the inner structure of Herbrand’s Fundamental Theorem and the questions of its quality and its depth, we also discuss the outer questions of its historical context and why Bernays called it “the central theorem of predicate logic” and considered the form (...) of its expression to be “concise and felicitous”. (shrink)

Another textbook of traditional logic, subject to the characteristic limitations of that tradition. Propositional logic receives scant attention, and polyadic predicates are ignored. Propositions are in places confused with terms, as when the transitivity of implication is analyzed in terms of the Barbara syllogism. Although professedly Aristotelian, the treatment departs from Aristotle on a number of points: syllogisms are presented as inference rules rather than as logical theses; singular statements are assimilated to universal ones; and modal syllogisms are not (...) covered. Only about one-third of the book treats formal logic; the remainder includes discussion of basic semiotic, definition and division, induction, and fallacies, along with the rudiments of Thomistic epistemology and metaphysics.--J. B. B. (shrink)

This article investigates the nature of Aristotelian syllogistics and shows that the categorical syllogism is fundamentally about showing the connection, in the premises of the syllogism, between the major and minor terms as stated in the conclusion. It discusses how this is important for the use of the syllogism in scientific demonstration. The article then examines modern deductive logic with an eye to they way in which it contrasts with Aristotelian syllogistics. It shows howmodern logic is about making necessary connections (...) between each proposition by means of external or second order rules. In the syllogism, on the other hand, the necessity between the premises as a whole unit and the conclusion is based on the internal middle term. The article concludes with a discussion of Günther Patzig’s claim that Aristotelian syllogisms are best thought of as tautological propositions. If this were the case, then the differences asserted to exist between syllogistic and modern logic would not hold. However, it is shown that Patzig’s assimilation of syllogistics to modern deductive logic is illegitimate. (shrink)

The fundamental assumption of Dummett’s and Prawitz’ proof-theoretic justification of deduction is that ‘if we have a valid argument for a complex statement, we can construct a valid argument for it which finishes with an application of one of the introduction rules governing its principal operator’. I argue that the assumption is flawed in this general version, but should be restricted, not to apply to arguments in general, but only to proofs. I also argue that Dummett’s and (...) Prawitz’ project of providing a logical basis for metaphysics only relies on the restricted assumption. (shrink)

The aim of this article is to explain the proofs Aristotle calls proofs "by exposition" and to find out the rules and principles these proofs rest on. The first part shows why previous attempts to explain Aristotle's method of "exposition" have failed. The second part develops a new explanation and reconstructs all of Aristotle's expository proofs on the basis of a few simple rules. The last part shows that the new interpretation of Aristotle's method has very important consequences (...) regarding, firstly, the extent to which Aristotle's logic is valid, secondly, the relationship between syllogistic and geometric exposition and, last but not least, the correct understanding of the principles underlying Aristotle's logical system and the correct understanding of what a "complete" syllogism is. Thus, the article solves a number of fundamental problems concerning the interpretation of Aristotle's syllogistic. (shrink)

The consideration of careful reasoning can be traced to Aristotle and earlier authors. The possibility of rigorous rules for drawing conclusions can certainly be traced to the Middle Ages when types o f syllogism were studied. Shortly after the introduction of computers, the audacious scientist naturally envisioned the automation of sound reasoning—reasoning in which conclusions that are drawn follow l ogically and inevitably from the given hypotheses. Did the idea spring from the intent to emulate..

We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning (...) with continuously variable reasons. Some examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism. (shrink)

The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.

The rapidly increasing role of automation throughout the economy, culture and our personal lives has generated a large literature on the risks of algorithmic decision-making, particularly in high-stakes legal settings. Algorithmic tools are charged with bias, shrouded in secrecy, and frequently difficult to interpret. However, these criticisms have tended to focus on particular implementations, specific predictive techniques, and the idiosyncrasies of the American legal-regulatory regime. They do not address the more fundamental unease about the prospect that we might one (...) day replace judges with algorithms, no matter how fair, transparent, and intelligible they become. The aim of this paper is to propose an account of the source of that unease, and to evaluate its plausibility. I trace foundational unease with algorithmic decision-making in the law to the powerful intuition that there is a basic moral and legal difference between showing that something is true of many people just like you and showing that it is true of you. Human judgment attends to the exception; automation insists on blindly applying the rule. I show how this intuitive thought is connected to both epistemological arguments about the value of statistical evidence, as well as to court-centered conceptions of the rule of law. Unease with algorithmic decision-making in the law thus draws on an intuitive principle that underpins a disparate range of views in legal philosophy. This suggests the principle is deeply ingrained. Nonetheless, I argue that the powerful intuition is not as decisive as it may seem, and indeed runs into significant epistemological and normative challenges. At an epistemological level, I show how concerns about statistical evidence's ability to track the truth can be resolved by adopting a probabilistic, rather than modal, conception of truth-tracking. At a normative level, commitment to highly individualized decision-making co-exists with equally ingrained and competing principles, such as consistent application of law. This suggests that the “rule of law” may not identify a discrete set of institutional arrangements, as proponents of a court-centric conception would have it, but rather a more loosely defined set of values that could potentially be operationalized in multiple ways, including through some level of algorithmic adjudication. Although the prospect of replacing judges with algorithms is indeed unsettling, it does not necessarily entail unreasonable verdicts or an attack on the rule of law. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Islamic Disputation Theory: The Uses & Rules of Argument in Medieval Islam by Larry Benjamin MillerKhaled El-RouayhebLarry Benjamin Miller. Islamic Disputation Theory: The Uses & Rules of Argument in Medieval Islam. Logic, Argumentation and Reasoning 21. Cham: Springer 2020. Pp. xviii + 143. Hardback, €77.99.Very few unpublished PhD dissertations have had a formative influence on a field. One of the precious few is Larry Miller's Princeton (...) dissertation from 1984 on Islamic disputation theory. It proposed an original and compelling account of the development of the discipline, from the earliest extant Arabic works in the tenth century to the composition of classic madrasa handbooks in the late thirteenth century. It has been cited regularly in the past decades and has served as the starting point for all later studies of Arabic/Islamic dialectics in European languages. The fact that it has finally been published is a fitting tribute to a study that is even more admirable for having been written when many of the primary sources were still only available as manuscripts.Part of the explanation for why the dissertation was not published sooner may be that its author did not pursue an academic career. And as he states in a short preface, he has not kept abreast of all relevant studies and editions that have appeared since his dissertation. A thorough revision and update has therefore not been possible. Though some minor corrections and revisions have been made, the book is still by and large the influential 1984 dissertation, now made available with a short introduction by Walter Young, one of the leading younger scholars in the field, along with helpful indices of proper names and subjects.Miller's narrative begins with Arabic works on disputation (jadal) by a number of theologians from the late ninth and tenth centuries, such as Ibn al-Rāwandī (d. 911), [End Page 518] al-Ash'arī (d. 935), and the Karaite Jewish scholar al-Qirqisānī (d. ca. 960). The basic assumption was that a questioner (sā'il) would ask a claimant (mu'allil) (i) what his view is; (ii) what the ground or reason ('illa) for that view is; (iii) the appropriateness of the ground/reason for the view; and (iv) to verify the ground/reason by showing it to be commensurate with the view. The ground/reason for the theological dialecticians was standardly an entity or property, not a premise. The reasoning tended to be informal and analogical, as opposed to explicitly making syllogistic deductions from quantified premises.Parallel to this theological tradition of jadal, there was an Arabic Aristotelian tradition that took its point of departure from Aristotle's Topics, translated into Arabic more than once in the period from the late eighth to the early tenth centuries. Though also often referred to as jadal, this tradition, represented by the writings of al-Farabi (d. 950), Avicenna (d. 1037), and Averroes (d. 1198), retained distinct features. It tended to deny any epistemic value to dialectics, instead seeing it as useful only insofar as it provided logical training. For the Aristotelian/Neoplatonic philosophers, the pursuit of truth was the preserve of demonstrative syllogism (burhān). Whereas the theological dialecticians envisaged a claimant (mu'allil) and a questioner (sā'il), with the burden of proof on the former, the philosophical tradition had a questioner (sā'il) seek, in the manner of Socratic elenchus, to elicit from a respondent (mujῑb) a series of answers that would allow the construction of a syllogism whose conclusion contradicted the respondent's initial position.From the time of al-Ghazali (d. 1111), Islamic theologians tended to take over the syllogistic ideal of reasoning from the philosophers, perhaps explaining why theological writings on jadal trail off after the eleventh century. The informal, analogical jadal tradition continued, however, in the writings of Islamic jurists. From the eleventh century, scholars such as Abū Isḥāq al-Shīrāzī (d. 1083) and al-Juwaynī (d. 1085) wrote works on jadal in which the examples are disputes about questions of Islamic law and the disputants are partisans of the various legal schools. This tradition of what Miller calls "juristic dialectics" (74, 95) later adopted some... (shrink)

Another textbook of traditional logic, subject to the characteristic limitations of that tradition. Propositional logic receives scant attention, and polyadic predicates are ignored. Propositions are in places confused with terms, as when the transitivity of implication is analyzed in terms of the Barbara syllogism. Although professedly Aristotelian, the treatment departs from Aristotle on a number of points: syllogisms are presented as inference rules rather than as logical theses; singular statements are assimilated to universal ones; and modal syllogisms are not (...) covered. Only about one-third of the book treats formal logic; the remainder includes discussion of basic semiotic, definition and division, induction, and fallacies, along with the rudiments of Thomistic epistemology and metaphysics.--J. B. B. (shrink)

This paper consists primarily of a survey of results of Harvey Friedman about some proof-theoretic aspects of various forms of Kruskal's tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Verlen hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard's result, and Goodstein sequences. The central theme of this paper is a powerful theorem due (...) to Kruskal, the ‘tree theorem’, as well as a ‘finite miniaturization’ of Kruskal's theorem due to Harvey Friedman. These versions of Kruskal's theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called ‘natural independence phenomena’, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal's tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γo and Kruskal's theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence. (shrink)

Lively, comprehensive, and contemporary, The Voice of Reason: Fundamentals of Critical Thinking covers three principal areas: thought and language, systematic reasoning, and modes of proof. It employs highly accessible explanations and a multitude of examples drawn from social issues and various academic fields, showing students and other readers how to construct and criticize arguments using the techniques of sound reasoning. The Voice of Reason examines the traditional elements of the field and also explores new ground. The first section of (...) the book elucidates the relationship between thought and language, explaining how words function. It discusses meaning, connotation, vagueness, ambiguity, and definition, identifying the linguistic elements that can produce mistakes in thinking. The next section describes the rules of systematic reasoning, examining such topics as truth, relevance, and adequacy; deductive logic (categorical, hypothetical, and disjunctive); and induction (cause and effect, analogy, generalization, and hypothesis). Sixteen fallacies in thinking are also described through extensive illustrations and applications. The final section of the book offers a unique study of what constitutes proof in several different areas--including politics, advertising, law, and social issues--as well as in the academic disciplines of literature, science, history, and ethics. The author describes the various rules of evidence, using essays by major figures in each field as examples. An ideal text for courses in critical thinking, informal logic, and reasoning and writing, The Voice of Reason offers numerous pedagogical features including a host of examples; assignments, exercises, and puzzles at both the halfway point and at the end of each chapter; cartoons and quotations throughout; and practical applications of theoretical concepts. An extensive Instructor's Manual contains answers to the exercises that appear throughout the text. (shrink)

This paper urges the importance of including conditional proof as an inference rule in the teaching of elementary symbolic logic. The paper explains how to make clear to students that conditional proof is valid. This is done by a little proof that shows that hypothetical syllogism (or the chain rule) is both intuitively valid yet redundant. Teaching conditional proof not only aids in a deeper understanding of the meaning of “if” but also provides a strong reminder (...) to the student that they have not proved that the conclusion is true but instead have shown that the conclusion follows from the premises. (shrink)