This article examines the traditional and modern doctrines of categorical propositions and argues that both doctrines have serious problems. While the doctrines disagree about existential imports...

In seiner Dissertano de arte combinatoria entwickelt G. W. Leibniz eine Methode, um Prädikate von einem Subjekt und Subjekte zu einem Prädikat zu finden, und er stellt eine Formel auf, um deren Anzahl zu berechnen. Ich möchte im Folgenden erläutern, wie diese Methode funktioniert und warum sie sowohl einen Teil der, wie Leibniz es nennt, logica inventiva als auch eine direkte Folge seiner Variationslehre bildet, die den zentralen Teil seiner Dissertano ausmacht. Im letzten Abschnitt dieses Beitrages werde ich auf einige (...) theoretische und historische Zusammenhänge zwischen der logica inventiva und der aristotelischen Logik eingehen. Insbesondere werde ich erläutern, in welcher Weise Leibniz' Methode mit einigen traditionellen Versuchen in Verbindung steht, die tota minora eines gegebenen Terms zu bestimmen, und ob die aristotelische Theorie der kategorischen Propositionen in der Lage war, Leibniz' Methode zum Auffinden von Subjekten und Prädikaten innerhalb der logica inventiva zu begründen. (shrink)

The aim of this paper is to make sense of the Keynes–Johnson octagon of oppositions. We will discuss Keynes' logical theory, and examine how his view is reflected on this octagon. Then we will show how this structure is to be handled by means of a semantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra.

The aim of this paper is to analyse categorical propositions and their oppositional relations in Avicenna’s frame. For Avicenna’s expression and conception of categorical propositions is different from those of the authors who preceded him, due to the various conditions he adds to these categorical propositions. These additions make the oppositional relations richer and give rise to many more figures than a simple square. Our analysis exhibits some of these figures by relating all kinds (...) of quantified propositions in various ways. We thus find many squares of oppositions, several octagons, two hexagons and many complex figures which are different from each other and have specific and original structures, due to the propositions they contain. The hexagons are of Blanché’s kind, but one of them is asymmetric. Some octagons are of Buridan’s kind, but one of them is very unusual and seems to be a reversal of Buridan’s octagon. These octagons can also be replaced by cubes, which are three dimensional figures having the same number of vertices. (shrink)

Journal Name: Apeiron Issue: Ahead of print.

E.E.C. Jones’s early logical writings have recently been rescued from obscurity and it has been claimed that, in her works dating from the 1890s, she anticipated Frege’s distinction between sense and reference. This claim is challenged on the ground that it is based on a common but inadequate reading of Frege, which runs together his concept/object and sense/reference distinctions. It is admitted that a case can be made for Jones having anticipated something very like Frege’s analysis of categorical (...) class='Hi'>propositions, and that she offered a sound rebuttal of Russell’s objection to Frege’s account of the informativeness of identity statements. However, these significant achievements should not be misrepresented as an anticipation of Frege on sense and reference, a claim that encourages a defective reading of both philosophers. (shrink)

The essay first makes some observations on the general interrelationship between the logical writings of Albert and Buridan. Second, it gives an account of a ‘semantic logical model’ for analyzing complex subject terms in some basic categorical propositions which is defended by Albert of Saxony, and briefly recounts Buridan's criticisms of that model. Finally, the essay maintains that the Albertian model is typically compatible with, and a further development of, what is called by a late-fourteenth century anonymous scholar (...) ‘the English-Rule’ but the ‘determinable/determinate’ grammatical model defended by Buridan is not. (shrink)

According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done (...) within first-order logic, but it can be done within second-order one. Now let us consider the following question: can we axiomatically capture the semantic concept of conjunction? Such question, to be sure, does not make sense within the standard framework: we cannot construe it as asking whether we can form a first-order (or, for that matter, whatever-order) theory with an extralogical binary propositional operator so that its only model (up to isomorphism) maps the operator on the intended binary truth-function. The obvious reason is that the framework of standard logic does not allow for extralogical constants of this type. But of course there is also a deeper reason: an existence of a constant with this semantics is presupposed by the very definition of the framework1. Hence the question about the axiomatic capturability of concunction, if we can make sense of it at all, cannot be asked within the framework of standard logic, we would have to go to a more abstract level. To be able to make sense of the question we would have to think about a propositional ‘proto-language’, with uninterpreted logical constants, and to try to search out axioms which would fix the denotations of the constants as the intended truth-functions. Can we do this? It might seem that the answer to this question is yielded by the completeness theorem for the standard propositional calculus: this theorem states that the axiomatic delimitation of the calculus and the semantic delimitation converge to the same result.. (shrink)

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical (...) form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

The early thirteenth century tract Ars Meliduna deals with the issue whether categorical propositions entail, or are entailed by, ‘continuing’ propositions, i.e. by implications. From the perspective of modern logic, with implication interpreted as a material, truth-functional connective, the first question has to be answered in the affirmative because, e.g. β entails (α ⊃ β). But conversely (α ⊃ β) ‘normally’ doesn’t entail the truth (or the falsity) of any of the components α, β; hence the second (...) question should be answered in the negative. These results cannot, however, be directly transferred to medieval logic because implication was there usually interpreted as a strict implication. Yet, within the framework of modal logic, one obtains parallel results. In particular, whenever β is necessary, then (α → β) must be true. Furthermore, if (α → β) is itself necessary, then it is entailed by α. Thus, in particular, ‘Socrates is a man’ entails the analytically true implication ‘If Socrates is a man, Socrates is an animal’, and the same premiss also entails the tautological consequent ‘If Socrates is white, Socrates is white’. (shrink)

The early thirteenth century tract Ars Meliduna deals with the issue whether categorical propositions entail, or are entailed by, ‘continuing’ propositions, i.e. by implications. From the perspective of modern logic, with implication interpreted as a material, truth-functional connective, the first question has to be answered in the affirmative because, e.g. β entails (α ⊃ β). But conversely (α ⊃ β) ‘normally’ doesn’t entail the truth (or the falsity) of any of the components α, β; hence the second (...) question should be answered in the negative. These results cannot, however, be directly transferred to medieval logic because implication was there usually interpreted as a strict implication. Yet, within the framework of modal logic, one obtains parallel results. In particular, whenever β is necessary, then (α → β) must be true. Furthermore, if (α → β) is itself necessary, then it is entailed by α. Thus, in particular, ‘Socrates is a man’ entails the analytically true implication ‘If Socrates is a man, Socrates is an animal’, and the same premiss also entails the tautological consequent ‘If Socrates is white, Socrates is white’. (shrink)

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms (...) by taking just one syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence. (shrink)

After people learn to sort objects into categories they see them differently. Members of the same category look more alike and members of different categories look more different. This phenomenon of within-category compression and between-category separation in similarity space is called categorical perception (CP). It is exhibited by human subjects, animals and neural net models. In backpropagation nets trained first to auto-associate 12 stimuli varying along a onedimensional continuum and then to sort them into 3 categories, CP arises as (...) a natural side-effect because of four factors: (1) Maximal interstimulus separation in hidden-unit space during autoassociation learning, (2) movement toward linear separability during categorization learning, (3) inverse-distance repulsive force exerted by the between-category boundary, and (4) the modulating effects of input iconicity, especially in interpolating CP to untrained regions of the continuum. Once similarity space has been "warped" in this way, the compressed and separated "chunks" have symbolic labels which could then be combined into symbol strings that constitute propositions about objects. The meanings of such symbolic representations would be "grounded" in the system's capacity to pick out from their sensory projections the object categories that the propositions were about. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity (...) theorem for second-order —these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

We argue that, if taken seriously, Kripke's view that a language for science can dispense with a negation operator is to be rejected. Part of the argument is a proof that positive logic, i.e., classical propositional logic without negation, is not categorical.

본 논문의 목적은 19세기와 20세기의 아리스토텔레스적 논리학에서 제시된 정언삼단논법의 타당성 규칙을 설명하고 타당성 검사 테크닉을 소개하는 것이다. 필자는 우선 ‘주연 혹은 부주연의 지위에 있는 명사’라는 말이 무엇을 의미하는지를 설명하였다. 다음으로 필자는 ‘주연의 지위에 있는 명사’라는 개념이 중세 후기의 ‘분포 상정’ 이론에서 등장하는.

In his article 'Rejection' (1996), Timothy Smiley had shown how a logical system allowing rules of rejection could provide a categorical axiomatization of the classical propositional calculus. This paper shows how rules of rejection, when placed in a multiple conclusion setting, can also provide categorical axiomatizations of a range of non-classical calculi which permit truth-value gaps, among them the calculus in Smiley's own 'Sense without denotation' (1960).

We pursue the idea that predicate logic is a “fibred algebra” while propositional logic is a single algebra; in the context of intuitionism, this algebraic understanding of predicate logic goes back to Lawvere, in particular his concept of hyperdoctrine. Here, we aim at demonstrating that the notion of monad-relativised hyperdoctrines, which are what we call fibred algebras, yields algebraisations of a wide variety of predicate logics. More specifically, we discuss a typed, first-order version of the non-commutative Full Lambek calculus, which (...) has extensively been studied in the past few decades, functioning as a unifying language for different sorts of logical systems (classical, intuitionistic, linear, fuzzy, relevant, etc.). Through the concept of Full Lambek hyperdoctrines, we establish both generic and set-theoretical completeness results for any extension of the base system; the latter arises from a dual adjunction, and is relevant to the tripos-to-topos construction and quantale-valued sets. Furthermore, we give a hyperdoctrinal account of Girard’s and Gödel’s translation. (shrink)

What is meant under the genuine title of Aristotle’s ta Analytika is rarely properly understood. Presumably, his analytics was inspired by the method of geometric analysis. For Aristotle, this was a regressive or heuristic procedure, departing from a proposed conclusion and asking which premises could be found in order to syllogize, demonstrate or explain it. The terms that form categorical and modal propositions play a fundamental role in analytics. Aristotle introduces letters in lieu of the triples of terms (...) constitut­ing the propositions and the three syllogistic figures that schematize them. His formulation of the three syllogistic figures refers to a syntacti­cal and predicative order and position of the triples of terms, arranged in some diagrammed schemata, which, regrettably, are missing from the extant text of the Prior Analytics. Considering planar and graphic arrangements, both vertical and horizontal orders as well as the posi­tion of the three terms involved, we propose a reconstruction, at least to some extent, of these probable lettered diagrams. In such reconstructed diagrams, we can appropriately capture the definition of syllogism as a predicative connexion of terms, and easier survey a synoptic account of all valid predicative relations and transpositions, and also reduce the imperfect syllogisms into the moods of the first figure. Aristotle’s syllogistic is an analytical calculation of terms, understood as predicates and subjects within the categorical propositions, and more precisely of three terms schematized in three figures in predicative links such that, by means of a middle, follows from necessity a conclusion of the extreme terms. The necessity of the consequence is not based on the implication or inference of the propositions, but on a predictive transi­tivity through the middle term within the syllogistic figures. Syllogism must draw its conclusion through the way its terms are predicated of one another. Aristotle in his Prior Analytics developed also a complex account of modal syllogisms within necessity and possibility of belonging. This account involves also such an analyti­cal reduction to the syllogistic figures. In this analytical perspective, we try to throw some light on his modal syllogisms, although this difficult and nowadays thoroughly discussed topic would require a much wider treatment. (shrink)

The categoricity problem for a system of logic reveals an asymmetry between the model-theoretic and the proof-theoretic resources of that logic. In particular, it reveals prima facie that the proof-theoretic instruments are insufficient for matching the envisaged model-theory, when the latter is already available. Among the proposed solutions for solving this problem, some make use of new proof-theoretic instruments, some others introduce new model-theoretic constrains on the proof-systems, while others try to use instruments from both sides. On the proof-theoretical side, (...) two main approaches for solving the categoricity problem for propositional classical logic consist in the enforcement of the formal systems of this logic by introducing rules of inference of a new kind. A multiple-conclusions logic allows rules of inference with more than one conclusion, while a bilateralist system contains rules of inference formulated in terms of assertion and rejection. Both these approaches reveal interesting formal features of logical reasoning. This paper analyses some of the advantages and limitations of these two approaches as solutions for a full formalization of logic, i.e., for a categorical formalization. (shrink)

Formalizing categorical propositions of traditional logic in the language of quantifiers and propositional functions is no straightforward matter, especially when modalities get involved. Starting...

Studies of the conceptual abilities of nonhuman primates demonstrate the substantial range of these abilities as well as their limitations. Such abilities range from categorization on the basis of shared physical attributes, associative relations and functions to abstract concepts as reflected in analogical reasoning about relations between relations. The pattern of results from these studies point to a fundamental distinction between monkeys and apes in both their implicit and explicit conceptual capacities. Monkeys, but not apes, might be best regarded as (...) “paleo‐logicans” in the sense that they form common class concepts of identity on the basis of identical predicates (i.e., shared features). The discrimination of presumably more abstract relations commonly involves relatively simple procedural strategies mediated by associative processes likely shared by all mammals. There is no evidence that monkeys can perceive, let alone judge, relations‐between‐relations. This analogical conceptual capacity is found only in chimpanzees and humans. Interestingly, the “analogical ape,” like the child, can make its analogical knowledge explicit only if it is first provided with a symbol system by which propositional representations can be encoded and manipulated. (shrink)

In his Mathesis rationis Leibniz discounted out of hand four categorical propositions that would have considerably broadened the resultant syllogistic logic. He did this despite the facts both that he had devised a suitable manner for expressing the latent quantification over terms, and that he had reasoned adequately to determine which of the syllogisms in the resulting broadened logic were valid. Leibniz's reasons for discounting these non-standard propositions are shown to be inadequate, and the resultant syllogistic logic (...) is outlined. (shrink)

A system of logic usually comprises a language for which a model-theory and a proof-theory are defined. The model-theory defines the semantic notion of model-theoretic logical consequence (⊨), while the proof-theory defines the proof- theoretic notion of logical consequence (or logical derivability, ⊢). If the system in question is sound and complete, then the two notions of logical consequence are extensionally equivalent. The concept of full formalization is a more restrictive one and requires in addition the preservation of the standard (...) meanings of the logical terms in all the admissible interpretations of the logical calculus, as it is proof-theoretically defined. Although classical first-order logic is sound and complete, its standard formalizations fall short to be full formalizations since they allow non-intended interpretations. This fact poses a challenge for the logical inferentialism program, whose main tenet is that the meanings of the logical terms are uniquely determined by the formal axioms or rules of inference that govern their use in a logical calculus, i.e., logical inferentialism requires a categorical calculus. This paper is the first part of a more elaborated study which will analyze the categoricity problem from its beginning until the most recent approaches. I will first start by describing the problem of a full formalization in the general framework in which Carnap (1934/1937, 1943) formulated it for classical logic. Then, in sections IV and V, I shall discuss the way in which the mathematicians B.A. Bernstein (1932) and E.V. Huntington (1933) have previously formulated and analyzed it in algebraic terms for propositional logic and, finally, I shall discuss some critical reactions Nagel (1943), Hempel (1943), Fitch (1944), and Church (1944) formulated to these approaches. (shrink)

In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic raises some technical and conceptual issues. We relate Sandqvist’s (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness (...) and completeness arguments, thereby demonstrating the naturality of Sandqvist’s constructions. This naturality includes Sandqvist’s treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist’s semantics can also be viewed as the natural disjunction in a category of sheaves. (shrink)

Error theory turns on a particular presupposition about the conceptual commitments of moral realism, namely that the moral facts posited by realists need to be categorical. True moral propositions are said to have an absolute authority in their prescriptions in the sense that an agent, regardless of her own ends, needs or desires, is categorically obligated and has reason to act in accordance with their prescriptions. But, nothing in the world has such a queer property as categoricity, and (...) therefore we ought to be suspect of the enterprise of moral discourse. Some philosophers, like Stephen Finlay, have argued that this categoricity is not a necessary feature of moral language, and in so doing hope to have shown that the error theoretic critique is thus refuted. In this article I offer a survey of the literature on this topic and then contribute independent motivations for siding with those who think moral facts need not be categorical. (shrink)

In this work we summarise the concept of bisimulation, widely used both in computational sciences and in modal logic, that characterises modal structures with the same behaviour in terms of accessibility relations. Then, we offer a sketch of categorical interpretation of bisimulation between modal structures, which comprise both the structure and the valuation from a propositional language.

This collection contains 34 essays, 23 of them previously published, written between 1939 and 1960. They are of varying lengths, generality, and polish; and they cover the wide range of Hall's philosophical interests from metaphilosophy and value theory—the subjects of his best known books—to the theory of perception and the inadequacies of the Oxford philosophy of a decade ago. For Hall the study of language was not a way of repudiating or avoiding the traditional translingual issues, but rather a method (...) for attacking them. One might say that like Leibniz, Kant, and the early Wittgenstein, in their different ways, Hall took "the general form of propositions" as a philosopher's key to the basic categories of reality. Because his idiom was unfashionable among metaphysicians and because his doctrines were alien to the so-called "analysts," his work has not been widely influential. But the many insights and suggestions in this book may yet be taken up. These essays illuminate Hall's books, What Is Value?, Our Knowledge of Fact and Value, and Philosophical Systems; and they also make up an important work on their own account.—C. T. W. (shrink)

Some versions of moral naturalism are faulted for implausibly denying that moral obligations and prescriptions entail categorical reasons for action. Categorical reasons for action are normative reasons that exist and apply to agents independently of whatever desires they have. I argue that several defenses of moral naturalism against this charge are unsuccessful. To be a tenable meta-ethical theory, moral naturalism must accommodate the proposition that, necessarily, if anyone morally ought to do something, then s/he has a categorical (...) reason to do it. Versions of moral naturalism that deny this claim would, if widely believed, disable some crucial practical uses of moral concepts. In particular, if the existence of normative reasons for action is taken to be dependent on agents’ desires, it would breed profound skepticism about the legitimacy of evaluating others’ actions from a moral point of view. Also, it would raise doubts about whether people ought to correct their own behavior in light of moral considerations. Following Richard Joyce, I take these consequences to indicate that the concept of a categorical reason is a “non-negotiable” part of moral concepts. (shrink)

According to the Lockean thesis, a proposition is believed just in case it is highly probable. While this thesis enjoys strong intuitive support, it is known to conflict with seemingly plausible logical constraints on our beliefs. One way out of this conflict is to make probability 1 a requirement for belief, but most have rejected this option for entailing what they see as an untenable skepticism. Recently, two new solutions to the conflict have been proposed that are alleged to be (...) non-skeptical. We compare these proposals with each other and with the Lockean thesis, in particular with regard to the question of how much we gain by adopting any one of them instead of the probability 1 requirement, that is, of how likely it is that one believes more than the things one is fully certain of. (shrink)

On the basis of passages from John Buridan's Summula Suppositionibus and Sophismata, E. Karger has reconstructed what could be called the 'Buridanian theory of inferential relations between doubly quantified propositions', presented in her 1993 article 'A theory of immediate inference contained in Buridan's logic'. In the reconstruction, she focused on the syntactical elements of Buridan's theory of modes of personal supposition to extract patterns of formally valid inferences between members of a certain class of basic categorical propositions. (...) The present study aims at offering semantic corroboration 'a proof of soundness' to the inferential relations syntactically identified by E. Karger, by means of the analysis of Buridan's semantic definitions of the modes of personal supposition. The semantic analysis is done with the help of some modern logical concepts, in particular that of the model. In effect, the relations of inference syntactically established are shown to hold also from a semantic point of view, which means thus that this fragment of Buridan's logic can be said to be sound. (shrink)

Is logical consistency required for a set of beliefs or propositions to be categorically coherent? An affirmative answer is often assumed by mainstream epistemologists, and yet it is unclear why. Cases like the lottery and the preface call into question the assumption that beliefs must be consistent in order to be epistemically rational. And thus it is natural to wonder why all inconsistent sets of propositions are incoherent. On the other hand, Easwaran and Fitelson have shown that particular (...) kinds of inconsistency entail the epistemically ‘irrationality’ of holding certain sets of beliefs. In cases of the latter kind of inconsistency, it seems more reasonable to insist that such sets of beliefs or propositions are categorically incoherent. What the precise relationship is between coherence and consistency depends on the nature of the coherence relation. We shall examine recent attempts to explicate the coherence relation in terms of probabilistic measures of confirmation or agreement to see what they can teach us about the relationship between coherence and consistency. We shall show that some probabilistic measures of coherence allow for inconsistent sets to be categorically coherent, while satisfying plausible epistemic rationality constraints. Other probabilistic measures of coherence impose very strong logical consistency requirements, and some measures are tolerant of most forms of inconsistency. As we try to understand what distinguishes coherence measures in this respect, we will also draw some important lessons about Bayesian confirmation measures and differences in the way that they treat contradictory propositions. (shrink)

n this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods of Barbara and Datisi. By adding first-degree propositional negation to this system, we prove that the square of opposition holds without using many of the other rules (...) of classical logic (including double negation elimination). We then show that the Propositional Substructural Logic SLe is the best logic to study Aristotelian Syllogisms. Also, based on the IFLe square of opposition, the rules of conversation and the rules of negation are completely proved in Muzaffar's logic. For this purpose, we used the monadic first-order logic with the same standard deductive apparatus of quantifiers in classical logic, plus the axioms of "some a is an a" and "some not-a is a not-a". Finally, to show that there is no existential commitment to general terms in categorical logic, the Strong Four-Valued Relevant-classical Logic KR4 was used. With the same existential interpretation of the quantifiers and the standard translation of the quarter quantified. (shrink)

Kant informs us that the categorical imperative is a synthetic a priori proposition. It might be thought that Kant means thereby to say that we have an a priori rational insight, somewhat like that which Plato claimed for the Forms, into this basic moral truth. W. D. Ross had a similar view about human rational insight into certain basic moral principles, and in his book Kant's Ethical Theory he attritubes such a view to Kant.I argue that this initially plausible (...) interpretation is not correct, and that the "rational intuitionism" that it involves goes against Kant's whole approach to philosophy, in the first Critique as well as in the moral philosophy. I develop a new alternative interpretation, as follows: the categorical imperative is synthetic because it contains a claim about the causal/motivational power we have to act purely from the motive of duty, without any assistance from "sensible" or empirical motives, such as pleasure or pain. This claim, qua motivational, goes beyond the meaning of the categorical imperative considered as a mere ethical precept. It is a priori because this motive's source is the noumenal self.Hence to justify the categorical imperative is to show that this sort of non-sensible motive is or could be active in us as agents, and Kant is an "internalist": one who believes that morality and motivation are internally, analytically connected. Nach Kant ist der kategorische Imperativ ein synthetischer Satz a priori. Kant könnte damit sagen wollen, wir hätten in die grundlegende moralische Wahrheit, die der kategorische Imperativ darstellt, eine vernünftige Einsicht a priori, die unserer Wahrnehmung der Formen, wie Plato sie sieht, vergleichbar wäre. W. D. Ross vertrat eine ähnliche Auffassung von einer vernünftigen menschlichen Einsicht in gewisse grundlegende moralische Prinzipien, und in seinem Buch "Kant's Ethical Theory" schreibt er diese Auffassung auch Kant zu.Ich behaupte, daß diese auf den ersten Blick als plausibel erscheinende Interpretation nicht richtig ist, und daß der "rationale Intuitionismus", den sie impliziert, Kants philosophischem Ansatz, sowohl in der ersten Kritik als auch in der Moralphilosophie, ganz grundsätzlich widerspricht. Ich entwickele eine neue, alternative Interpretation: Der kategorische Imperativ ist synthetisch, weil er eine Behauptung über eine kausale/motivationale Kraft enthält, die uns zukommt und die uns befähigt, ohne irgendeine Unterstützung durch bloß empirische oder Beweggründe der Zweckrationalität wie Freude oder Schmerz allein aus der Vorstellung der Pflicht heraus zu handeln. Soweit sie den vernünftigen Beweggrund zu handeln betrifft, geht die Behauptung über die Bedeutung des kategorischen Imperativs als eines rein ethischen Gebots hinaus. Sie ist a priori, da die Quelle des vernünftigen Beweggrunds das noumenale Ich ist.Jede Rechtfertigung des kategorischen Imperativs muß folglich zeigen, daß die fragliche Art vernünftiger Beweggründe, die nicht Beweggründe bloßer Zweckrationalität sind, in uns als Handelnden aktiv ist oder jedenfalls aktiv sein könnte und daß Kant insoweit ein "Internalist" ist: das ist jemand, der meint, Moral und Motivation seien innerlich, analytisch miteinander verbunden. (shrink)

This monograph explores the logical systems of early logicians in the Arabic tradition from a theoretical perspective, providing a complete panorama of early Arabic logic and centering it within an expansive historical context. By thoroughly examining the writings of the first Arabic logicians, al-Fārābī, Avicenna and Averroes, the author analyzes their respective theories, discusses their relationship to the syllogistics of Aristotle and his followers, and measures their influence on later logical systems. Beginning with an introduction to the writings of the (...) most prominent Arabic logicians, the author scrutinizes these works to determine their categorical logic, as well as their modal and hypothetical logics. Where most other studies written on this subject focus on the Arabic logicians’ epistemology, metaphysics, and theology, this volume takes a unique approach by focusing on the actual technical aspects and features of their logics. The author then moves on to examine the original texts as closely as possible and employs the symbolism of modern propositional, predicate, and modal logics, rendering the arguments of each logician clearly and precisely while clarifying the theories themselves in order to determine the differences between the Arabic logicians’ systems and those of Aristotle. By providing a detailed examination of theories that are still not very well-known in Western countries, the author is able to assess the improvements that can be found in the Arabic writings, and to situate Arabic logic within the breadth of the history of logic. This unique study will appeal mainly to historians of logic, logicians, and philosophers who seek a better understanding of the Arabic tradition. It also will be of interest to modern logicians who wish to delve into the historical aspects and progression of their discipline. Furthermore, this book will serve as a valuable resource for graduate students who wish to complement their general knowledge of Arabic culture, logic, and sciences. (shrink)

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen–Specker’s theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event algebras. We show explicitly that the latter category is equipped with an object of truth (...) values, or classifying object, which constitutes the appropriate tool for assigning truth values to propositions describing the behavior of quantum systems. Effectively, this category-theoretic representation scheme circumvents consistently the semantic ambiguity with respect to truth valuation that is inherent in conventional quantum mechanics by inducing an objective contextual account of truth in the quantum domain of discourse. The philosophical implications of the resulting account are analyzed. We argue that it subscribes neither to a pragmatic instrumental nor to a relative notion of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature. (shrink)

There are still on-going debates on what exactly is wrong with Prior’s pathological “tonk.” In this article I argue, on the basis of categorical inferentialism, that two notions of inconsistency ought to be distinguished in an appropriate account of tonk; logic with tonk is inconsistent as the theory of propositions, and it is due to the fallacy of equivocation; in contrast to this diagnosis of the Prior’s tonk problem, nothing is actually wrong with tonk if logic is viewed (...) as the theory of proofs rather than propositions, and tonk perfectly makes sense in terms of the identity of proofs. Indeed, there is fully complete semantics of proofs for tonk, which allows us to link the Prior’s old philosophical idea with contemporary issues at the interface of categorical logic, computer science, and quantum physics, and thereby to expose commonalities between the laws of Reason and the laws of Nature, which are what logic and physics are respectively about. I conclude the article by articulating the ideas of categorical logical positivism and pluralistic unified science as its goal, including the unification of realist and antirealist conceptions of meaning by virtue of the categorical logical basis of metaphysics. (shrink)

Why does Aristotle not use the copulative wording for categorical propositions, but instead the clumsier terminological formulations (e. g. the B belongs to every A) in his syllogistic? The proposed explanations by Alexander, Lukasiewicz and Patzig: Aristotle wants to make clear the difference between subject and predicate, seems to be insufficient. In quantified categorical propositions, this difference is always sufficiently clear by the use of the pronouns going with the subject expressions. Aristotle opts for the terminological (...) wording because in premiss pairs of figures two and three he can thus suppress the middle term in one of the premisses and connect the major and minor term, using connecting particles. This renders the syllogisms more transparent. Had he used the copulative wording instead, he would have run into difficulties, in particular with o-propositions among the premisses (i. e. in Baroco and Bocardo) because in these cases the pronoun expressing the quantification would have to go with the subject term, the negation with the predicate. (shrink)

Canonical extension has proven to be a powerful tool in algebraic study of propositional logics. In this paper we describe a generalisation of the theory of canonical extension to the setting of first order logic. We define a notion of canonical extension for coherent categories. These are the categorical analogues of distributive lattices and they provide categorical semantics for coherent logic, the fragment of first order logic in the connectives ∧, ∨, 0, 1 and ∃. We describe a (...) universal property of our construction and show that it generalises the existing notion of canonical extension for distributive lattices. Our new construction for coherent categories has led us to an alternative description of the topos of types, introduced by Makkai in [22]. This allows us to give new and transparent proofs of some properties of the action of the topos of types construction on morphisms. Furthermore, we prove a new result relating, for a coherent category C, its topos of types to its category of models. (shrink)

Kosta Došen argued in his papers Inferential Semantics (in Wansing, H. (ed.) Dag Prawitz on Proofs and Meaning, pp. 147–162. Springer, Berlin 2015) and On the Paths of Categories (in Piecha, T., Schroeder-Heister, P. (eds.) Advances in Proof-Theoretic Semantics, pp. 65–77. Springer, Cham 2016) that the propositions-as-types paradigm is less suited for general proof theory because—unlike proof theory based on category theory—it emphasizes categorical proofs over hypothetical inferences. One specific instance of this, Došen points out, is that the (...) Curry–Howard isomorphism makes the associativity of deduction composition invisible. We will show that this is not necessarily the case. (shrink)