George Boole emerged from the British tradition of the “New Analytic”, known for the view that the laws of logic are laws of thought. Logicians in the New Analytic tradition were influenced by the work of Immanuel Kant, and by the German logicians Wilhelm Traugott Krug and Wilhelm Esser, among others. In his 1854 work An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Boole argues that (...) the laws of thought acquire normative force when constrained to mathematical reasoning. Boole’s motivation is, first, to address issues in the foundations of mathematics, including the relationship between arithmetic and algebra, and the study and application of differential equations (Durand-Richard, van Evra, Panteki). Second, Boole intended to derive the laws of logic from the laws of the operation of the human mind, and to show that these laws were valid of algebra and of logic both, when applied to a restricted domain. Boole’s thorough and flexible work in these areas influenced the development of model theory (see Hodges, forthcoming), and has much in common with contemporary inferentialist approaches to logic (found in, e.g., Peregrin and Resnik). (shrink)

This paper is a contribution to the long-standing debate over the coherence of Charles Sanders Peirce’s overall system of philosophy. It approaches that issue through the lens of a contemporary debate over the notion of metaphysical grounding, or more broadly, the nature of metaphysical explanation, employing the laws of logic as a case study. The central question concerns how we can take seriously what we shall call Peirce’s Rule—that nothing can be admitted to be absolutely inexplicable—without being vulnerable (...) to a vicious regress or equally vicious circularity. I first argue that in Peirce’s early work he offers a quietist conception of grounding that provides a persuasive and ground-breaking answer to this central question. I then raise a familiar concern, that in Peirce’s later work we find hints of a more metaphysical conception of grounding that seems unable to answer that question and is thus inconsistent with his earlier work. The paper ends with a speculative interpretation of Peirce’s approach to metaphysics and its possible role in grounding logical principles. (shrink)

Frege claims that the laws of logic are characterized by their “generality,” but it is hard to see how this could identify a special feature of those laws. I argue that we must understand this talk of generality in normative terms, but that what Frege says provides a normative demarcation of the logical laws only once we connect it with his thinking about truth and science. He means to be identifying the laws of logic (...) as those that appear in every one of the scientific systems whose construction is the ultimate aim of science, and in which all truths have a place. Though an account of logic in terms of scientific systems might seem hopelessly antiquated, I argue that it is not: a basically Fregean account of the nature of logic still looks quite promising. (shrink)

The goals of this paper are two-fold: I wish to clarify the Aristotelian conception of the law of non-contradiction as a metaphysical rather than a semantic or logical principle, and to defend the truth of the principle in this sense. First I will explain what it in fact means that the law of non-contradiction is a metaphysical principle. The core idea is that the law of non-contradiction is a general principle derived from how things are in the world. For example, (...) there are certain constraints as to what kind of properties an object can have, and especially: some of these properties are mutually exclusive. Given this characterisation, I will advance to examine what kind of challenges the law of non-contradiction faces; the main opponent here is Graham Priest. I will consider these challenges and conclude that they do not threaten the truth of the law of non-contradiction understood as a metaphysical principle. (shrink)

Abstract. As a general theory of reasoning—and as a general theory of what holds true under every possible circumstance—logic is supposed to be ontologically neutral. It ought to have nothing to do with questions concerning what there is, or whether there is anything at all. It is for this reason that traditional Aristotelian logic, with its tacit existential presuppositions, was eventually deemed inadequate as a canon of pure logic. And it is for this reason that modern quantification (...) theory, too, with its residue of existentially loaded theorems and patterns of inference, has been claimed to suffer from a defect of logical purity. The law of non-contradiction rules out certain circumstances as impossible—circumstances in which a statement is both true and false, or perhaps circumstances where something both is and is not the case. Is this to be regarded as a further ontological bias? (shrink)

Some think that logic concerns the “laws of truth”; others that logic concerns the “laws of thought.” This paper presents a way to reconcile both views by building a bridge between truth-maker theory, à la Fine, and normative bilateralism, à la Restall and Ripley. The paper suggests a novel way of understanding consequence in truth-maker theory and shows that this allows us to identify a common structure shared by truth-maker theory and normative bilateralism. We can thus (...) transfer ideas from normative bilateralism to truth-maker theory, such as non-transitive solutions to paradox, and vice versa, such as notions of factual equivalence and containment. (shrink)

In this paper, I argue that Goff's view that universal consciousness grounds logical laws such as the law of non-contradiction cannot be true on the grounds that we cannot guarantee the classical logic loving nature of universal consciousness that Goff desires in order to ground logical laws. I will present three arguments to show this.

The Laws of Thought is an exploration of the deductive and inductive foundations of rational thought. The author here clarifies and defends Aristotle’s Three Laws of Thought, called the Laws of Identity, Non-contradiction and Exclusion of the Middle – and introduces two more, which are implicit in and crucial to them: the Fourth Law of Thought, called the Principle of Induction, and the Fifth Law of Thought, called the Principle of Deduction. This book is a thematic compilation (...) drawn from past works by the author over a period of twenty-three years (1990-2013). (shrink)

In this paper, we argue that a distinction ought to be drawn between two ways in which a given world might be logically impossible. First, a world w might be impossible because the laws that hold at w are different from those that hold at some other world (say the actual world). Second, a world w might be impossible because the laws of logic that hold in some world (say the actual world) are violated at w. We (...) develop a novel way of modelling logical possibility that makes room for both kinds of logical impossibility. Doing so has interesting implications for the relationship between logical possibility and other kinds of possibility (for example, metaphysical possibility) and implications for the necessity or contingency of the laws of logic. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

Amongst the entities making up social reality, are there necessary relations whose necessity is not a mere reflection of the logical connections between corresponding concepts? We distinguish three main groups of answers to this question, associated with Hume and Adolf Reinach at opposite extremes, and with Searle who occupies a position somewhere in the middle. We first set forth Reinach’s views on what he calls ‘material necessities’ in the realm of social entities. We then attempt to show that Searle has (...) not identified a sustainable position somewhere between the Humean and the Reinachian extremes. This is because Searle’s position is threatened by circularity, and to steer clear of that danger it must incorporate at least some elements of Reinach’s essentialism. (shrink)

We intend at this article to show some reasons to think the laws of nature as metaphysically necessary: to distinguish the metaphysical modality from the epistemical modality, and to have an absolute modality to face the relative physical and logical modalities. Lately, we indicate what does it mean to talk about metaphysically necessary laws, distinguishing two kinds of metaphysical modalities, and we account for the question about if the laws of nature are metaphysically necessary. The conclusion we (...) get is that the laws, whichever they are, are metaphysically necessary. (shrink)

The starting point of this paper concerns the apparent difference between what we might call absolute truth and truth in a model, following Donald Davidson. The notion of absolute truth is the one familiar from Tarski’s T-schema: ‘Snow is white’ is true if and only if snow is white. Instead of being a property of sentences as absolute truth appears to be, truth in a model, that is relative truth, is evaluated in terms of the relation between sentences and models. (...) I wish to examine the apparent dual nature of logical truth (without dwelling on Davidson), and suggest that we are dealing with a distinction between a metaphysical and a linguistic interpretation of truth. I take my cue from John Etchemendy, who suggests that absolute truth could be considered as being equivalent to truth in the ‘right model’, i.e., the model that corresponds with the world. However, the notion of ‘model’ is not entirely appropriate here as it is closely associated with relative truth. Instead, I propose that the metaphysical interpretation of truth may be illustrated in modal terms, by metaphysical modality in particular. One of the tasks that I will undertake in this paper is to develop this modal interpretation, partly building on my previous work on the metaphysical interpretation of the law of non-contradiction (Tahko 2009). After an explication of the metaphysical interpretation of logical truth, a brief study of how this interpretation connects with some recent important themes in philosophical logic follows. In particular, I discuss logical pluralism and propose an understanding of pluralism from the point of view of the metaphysical interpretation. (shrink)

This paper contributes to explaining the rise of logical empiricism in mid-twentieth century (North) America and to a better understanding of American philosophy of science before the dominance of logical empiricism. We show that, contrary to a number of existing histories, philosophy of science was already a distinct subfield of philosophy, one with its own approaches and issues, even before logical empiricists arrived in America. It was a form of speculative philosophy with a concern for speculative metaphysics, normative issues relating (...) to science and society and issues which later were associated with logical empiricist philosophy of science, issues such as confirmation, scientific explanation, reductionism and laws of nature. Further, philosophy of science was not primarily pragmatist in orientation. We also show, with the help of our historical characterization, that a recent account of the emergence of analytic philosophy applies to the rise of logical empiricism. It has been argued that the emergence of American analytic philosophy is partly explained by analytic philosophers’ use of key institutions, including of journals, to marginalize speculative philosophy and promote analytic philosophy. We argue that this use of institutions included the marginalization of speculative and value-laden philosophy of science and the promotion of logical empiricism. (shrink)

The purpose of this paper is to examine the status of logic from a metaphysical point of view – what is logic grounded in and what is its relationship with metaphysics. There are three general lines that we can take. 1) Logic and metaphysics are not continuous, neither discipline has no bearing on the other one. This seems to be a rather popular approach, at least implicitly, as philosophers often skip the question altogether and go about their (...) business, be it logic or metaphysics. However, it is not a particularly plausible view and it is very hard to maintain consistently, as we will see. 2) Logic is prior to metaphysics and has metaphysical implications. The extreme example of this kind of approach is the Dummettian one, according to which metaphysical questions are reducible to the question of which logic to adopt. 3) Metaphysics is prior to logic, and your logic should be compatible with your metaphysics. This approach suggests an answer to the question of what logic is grounded in, namely, metaphysics. Here I will defend the third option. (shrink)

In this paper, the relationship between the e-value of a complex hypothesis, H, and those of its constituent elementary hypotheses, Hj, j = 1… k, is analyzed, in the independent setup. The e-value of a hypothesis H, ev, is a Bayesian epistemic, credibility or truth value defined under the Full Bayesian Significance Testing mathematical apparatus. The questions addressed concern the important issue of how the truth value of H, and the truth function of the corresponding FBST structure M, relate to (...) the truth values of its elementary constituents, Hj, and to the truth functions of their corresponding FBST structures Mj, respectively. (shrink)

This comprehensive volume features the proceedings of the Third International Workshop on Logics for New-Generation Artificial Intelligence and the International Workshop on Logic, AI and Law, held in Hangzhou, China on September 8-9 and 11-12, 2023. The collection offers a diverse range of papers that explore the intersection of logic, artificial intelligence, and law. With contributions from some of the leading experts in the field, this volume provides insights into the latest research and developments in the applications of (...) logic in these areas. It is an essential resource for researchers, practitioners, and students interested in the latest advancements in logic and its applications to artificial intelligence and law. (shrink)

The question of whether the Pyrrhonist adheres to certain logical principles, criteria of justification, and inference rules is of central importance for the study of Pyrrhonism. Its significance lies in that, whereas the Pyrrhonist describes his philosophical stance and argues against the Dogmatists by means of what may be considered a rational discourse, adherence to any such principles, criteria, and rules does not seem compatible with the radical character of his skepticism. Hence, if the Pyrrhonist does endorse them, one must (...) conclude that he is inconsistent in his outlook. Despite its import, the question under consideration has not received, in the vast literature on Pyrrhonism of the past three decades, all the attention it deserves. In the present paper, I do not propose to provide a full examination of the Pyrrhonist’s attitude towards rationality, but to focus on the question of whether he endorses the law of non-contradiction (LNC). However, I will also briefly tackle the question of the Pyrrhonist’s outlook on both the canons of rational justification at work in the so-called Five Modes of Agrippa and the logical rules of inference. In addition, given that the LNC is deemed a fundamental principle of rationality, determining the Pyrrhonist’s attitude towards it will allow us to understand his general attitude towards rationality. (shrink)

Putnam, Hilary FPhilosophy of logic. Harper Essays in Philosophy. Harper Torchbooks, No. TB 1544. Harper & Row, Publishers, New York-London, 1971. v+76 pp. The author of this book has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is largely (...) correct as far as it goes, or at least that it is rational to believe in it. The main thesis of the book is that consistent acceptance of contemporary science requires the acceptance of some sort of Platonistic idealism affirming the existence of abstract, non-temporal, non-material, non-mental entities (numbers,scientific laws, mathematical formulas, etc.). The author is thus in direct opposition to the extreme materialism which had dominated philosophy of science in the first three quarters of this century. the book can be especially recommended to the scientifically literate, general reader whose acquaintance with these areas is limited to the earlier literature of when it had been assumed that empiricistic materialism was the only philosophy compatible with a scientific outlook. To this group the book presents an eye-opening challenge fulfilling the author’s intention of “shaking up preconceptions and stimulating further discussion”. (shrink)

Karl Popper discovered in 1938 that the unconditional probability of a conditional of the form ‘If A, then B’ normally exceeds the conditional probability of B given A, provided that ‘If A, then B’ is taken to mean the same as ‘Not (A and not B)’. So it was clear (but presumably only to him at that time) that the conditional probability of B given A cannot be reduced to the unconditional probability of the material conditional ‘If A, then B’. (...) I describe how this insight was developed in Popper’s writings and I add to this historical study a logical one, in which I compare laws of excess in Kolmogorov probability theory with laws of excess in Popper probability theory. (shrink)

Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article does (...) not discuss many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)

The conception of a ‘law of nature’ is a human product. It was created to play a role in natural philosophy, in the Cartesian tradition. In light of this, philosophers and scientists must sort out what they mean by a law of nature before evaluating rival theories and approaches. If one’s conception of the laws of nature is yoked to metaphysical notions of truth and explanation, that connection must be made explicit and defended. If, on the other hand, one’s (...) aim is to disentangle laws from truth or from explanation, that must be stated and defended as well. If philosophers do not make such assumptions, intuitions, and methodological commitments clear, then it will be impossible to identify the source of disagreement in debates about the laws of nature. Are the conflicts rooted in disagreement about the conclusions reached, or do the background commitments of the combatants block any resolution to the dispute in principle or in practice? (shrink)

There is a widespread view that Buddhist philosophy embodies logical contradictions such that there would be 'true' contradictions, This article explains that this is not the case and that Buddhist philosophy, more generally the Perennial philosophy, denies all contradictions for the sake of a doctrine of Unity.

In this paper I defend the traditional interpretations of Kant’s Formula of a Law of Nature from recent attacks leveled by Faviola Rivera-Castro, James Furner, Ido Geiger, Pauline Kleingeld and Sven Nyholm. After a short introduction, the paper is divided into four main sections. In the first, I set out the basics of the three traditional interpretations, the Logical Contradiction Interpretation, the Practical Contradiction Interpretation and the Teleological Contradiction Interpretation. In the second, I examine the work of Geiger, Kleingeld and (...) Nyholm: these three commentators reject the traditional interpretations entirely, but I argue that this rejection is ill-founded. In the third and fourth, I take a detailed look at Furner’s work, work in which he seeks to revise (rather than reject) the traditional interpretations. I argue that, despite his more modest aims, Furner’s revision is also ill-founded. (shrink)

The traditional possible-worlds model of belief describes agents as ‘logically omniscient’ in the sense that they believe all logical consequences of what they believe, including all logical truths. This is widely considered a problem if we want to reason about the epistemic lives of non-ideal agents who—much like ordinary human beings—are logically competent, but not logically omniscient. A popular strategy for avoiding logical omniscience centers around the use of impossible worlds: worlds that, in one way or another, violate the (...) class='Hi'>laws of logic. In this paper, we argue that existing impossible-worlds models of belief fail to describe agents who are both logically non-omniscient and logically competent. To model such agents, we argue, we need to ‘dynamize’ the impossible-worlds framework in a way that allows us to capture not only what agents believe, but also what they are able to infer from what they believe. In light of this diagnosis, we go on to develop the formal details of a dynamic impossible-worlds framework, and show that it successfully models agents who are both logically non-omniscient and logically competent. (shrink)

After a brief survey of the literature on ceteris paribus clauses and ceteris paribus laws (1), the problem of exceptions, which creates the need for cp laws, is discussed (2). It emerges that the so-called skeptical view of laws of nature does not apply to laws of any kind whatever. Only some laws of physics are plagued with exceptions, not THE laws (3). Cp clauses promise a remedy, which has to be located among the (...) further reactions to the skeptical view (4). After inspecting various translations of the Latin term 'ceteris paribus' (5), the paper arrives at the conclusion that, on the most reasonable translation, there are no such things as cp laws, for reasons of logical form. Cp clauses have an indexical content, so that they need singular propositions as their habitat, not general ones. Cp clauses and the universal generalizations they are supposed to modify are not fit for each other (6). (shrink)

A critical discussion of Lu-Adler's chapter on Kant's mature view of pure general logic. I sketch an alternative interpretation of its formality on which Kant would hold no deduction is possible of this logic's laws.

This paper is on Descartes’ account of modality and, in particular, his account of the necessity of the laws of nature. He famously argues that the necessity of the “eternal truths” of logic and mathematics depends on God’s will. Here I suggest he has the same view about the necessity of the laws of nature. Further, I argue, this is a plausible theory of laws. For philosophers often talk about something being nomologically or physically necessary because (...) of the laws of nature, but this necessity is thought to be metaphysically contingent. However, they struggle to explain how the laws could be genuinely necessary while being metaphysically contingent. The chief advantage of Descartes view, I argue, is that God’s will can plausibly explain both the necessity of the laws and the contingency of the laws. So, Descartes’ theistic account of laws provides a plausible explanation, perhaps the best explanation, of the contingent-necessity of laws of nature. (shrink)

1994.Επιστημολογία της Λογικής. Συγγραφέας Επαμεινώνδας Ξενόπουλος Μοναδική μελέτη και προσέγγιση της θεωρίας της γνώσης, για την παγκόσμια βιβλιογραφία, της διαλεκτικής πορείας της σκέψης από την λογική πλευρά της και της μελλοντικής μορφής που θα πάρουν οι διαλεκτικές δομές της, στην αδιαίρετη ενότητα γνωσιοθεωρίας, λογικής και διαλεκτικής, με την «μέθοδο του διαλεκτικού υλισμού». Έργο βαρύ με θέμα εξαιρετικά δύσκολο διακατέχεται από πρωτοτυπία και ζωντάνια που γοητεύει τον κάθε ανήσυχο στοχαστή από τις πρώτες γραμμές.

My analysis here is an attempt to bring out the main through-line in the development of Bulgarian philosophy of law today. A proper account of Bulgarian philosophy of law in the 20th century requires an attempt to find, on the one hand, a solution to epistemological and methodological problems in law and, on the other, a clear-cut influence of the Kantian critical tradition. Bulgarian philosophy of law follows a complicated path, ranging from acceptance and revision of Kantian philosophy to the (...) development of interesting theories on the logic of legal reasoning. (shrink)

This article challenges the prevailing interpretations of Hegel's account of the concept "contradiction" in the Science of Logic by arguing that it is concerned with the principle of Hegel's method rather than with the classical law of non-contradiction. I first consider Hegel's Doctrine of Essence in view of Kant's discussion of the concepts of reflection in the first Critique. On this basis, I examine Hegel's account of the logical principles based on the concepts "identity," "opposition," and "contradiction." Finally, I (...) point out how the principle Hegel derives from the concept of contradiction actually informs his own method. (shrink)

PUTNAM has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is largely correct as far as it goes, or at least that it is rational to believe in it. The main thesis of the book is that consistent acceptance of (...) contemporary science requires the acceptance of some sort of Platonistic idealism affirming the existence of abstract, non-temporal, non-material, non-mental entities (numbers, scientific laws, mathematical formulas, etc.). The author is thus in direct opposition to the extreme materialism which had dominated philosophy of science in the first three quarters of this century. The book can be recommended to the scientifically literate, general reader whose acquaintance with these areas is limited to the literature of the 1950’s and before, when it had been assumed that empiricistic materialism was the only philosophy compatible with a scientific outlook. To this group the book presents an eye-opening challenge fulfilling the author’s intention of “shaking up preconceptions and stimulating further discussion”. QUINE’S book is not easy to read, partly because the level of sophistication fluctuates at high frequency between remote extremes and partly because of convoluted English prose style and devilish terminology. Almost all of the minor but troublesome technical errata in the first printing have been corrected [see reviews, e.g., the reviewer, Philos. Sci. 39 (1972), no. 1, 97–99]. In the opinion of the reviewer the book is not suitable for undergraduate instruction, and without external motivation few mathematicians are likely to have the patience to appreciate it. Nevertheless, a careful study of the book will more than repay the effort and one should expect to find frequent references to this book in coming years. (shrink)

Many Kantian scholars have debated what normative guidance the formula of the law of nature provides. There are three ways of understanding the role of FLN in Kant’s ethics. The first line of interpretation claims that FLN and FLU are logically equivalent. The second line claims that there are only subjective differences, meaning that FLN is easier to apply than the abstrct method of FUL. The third line of interpretation claims that there are objective differences between FLN and FUL in (...) the sense that each formula has an irreducible role in Kant’s ethics. In this article I will show that the first and second lines of interpretation cannot fully explain Kant’s account of FLN and I will propose a new interpretation which pertains to the third type. I will explore the schematism model to understand the role of FLN and argue that it is an intermediary principle that fills in a practical gap between the moral law and action. In the end, I will consider a possible objection against this understanding which claims that the schematism model is not applicable to practical judgment since nothing is given in experience. (shrink)

Normativism is the view that logic provides rules for correct reasoning. Some influential critics of normativism, such as Gilbert Harman, claim that logical rules provide reasoners with bad or misleading standards. Others, such as Gillian Russell, claim that logic is a descriptive subject and thus cannot, given Hume’s law, provide rules for reasoning. We think these critics are mistaken. Our aim in this paper is to defend normativism by sketching an alternative way of thinking about the normative force (...) of logical rules. On our view, logical rules are best characterized, in a broadly Rossian manner, as intellectual prima facie duties; i.e., they provide standards for evaluating reasoning and reasoners that are universal and authoritative, but not absolute. This position accommodates the inherent normativity of logic, contra Russell, while circumventing the challenges to normativism raised by Harman. (shrink)

Many philosophers have criticized John Rawls’s Law of Peoples. However, often these criticisms take it for granted that the moral conclusions drawn in A Theory of Justice are superior to those in the former book. In my view, however, Rawls comes to many of his 'conclusions' without too many actual inferences. More precisely, my argument here is that if one takes Rawls’s premises and the assumptions made about the original position(s) seriously and does in fact think them through to their (...) logical conclusions, both 'A Theory of Justice' and 'The Law of Peoples' have abysmally counterintuitive and immoral implications. To wit, if the members in the original position think, as Rawls suggests,that their society is closed and they will have no interaction with outsiders, and if, furthermore, they are self-interested and concerned with the basic structure of their own society, than there is absolutely no reason for them to use the terms “persons” or “least advantaged” in the formulation of the two principles. Rather, they will use the terms “citizens of our society” and “least advantaged of our society” instead. But thus revised, the principles of justice imply that the genocide or the enslavement of outsiders is unobjectionable. I will consider attempts to block this conclusion and demonstrate that they all fail. The Law of Peoples, moreover, faces similar problems. (shrink)

I show that intuitive and logical considerations do not justify introducing Leibniz’s Law of the Indiscernibility of Identicals in more than a limited form, as applying to atomic formulas. Once this is accepted, it follows that Leibniz’s Law generalises to all formulas of the first-order Predicate Calculus but not to modal formulas. Among other things, identity turns out to be logically contingent.

In this paper we offer a new argument for the existence of God. We contend that the laws of logic are metaphysically dependent on the existence of God, understood as a necessarily existent, personal, spiritual being; thus anyone who grants that there are laws of logic should also accept that there is a God. We argue that if our most natural intuitions about them are correct, and if they are to play the role in our intellectual (...) activities that we take them to play, then the laws of logic are best construed as necessarily existent thoughts -- more specifically, as divine thoughts about divine thoughts. We conclude by highlighting some implications for both theistic arguments and antitheistic arguments. (shrink)

We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models (...) to show that some of these logics are non-degenerate. (shrink)

Wittgenstein taught us that there could not be a logically private language— a language on the proper speaking of which it was logically impossible for there to be more than one expert. For then there would be no difference between this person thinking she was using the language correctly and her actually using it correctly. The distinction requires the logical possibility of someone other than her being expert enough to criticize or corroborate her usage, someone able to constitute or hold (...) her to a standard of proper use. I shall explore the possibility of something opposite- sounding about laws, namely, that there could in principle be laws whose existence, legitimacy, goodness, and efficacy depend upon their being private, in this sense: their existence is kept secret from those who legitimately benefit from the laws and yet who would misguidedly destroy them were they to come to know of them; and it is kept secret from those who would illegitimately benefit from being able to circumvent the laws, and who could circumvent them if they knew of them. The secrecy of the laws increases their efficacy against bad behavior, and since were the public to come to know of these laws the public would lose its nerve and demand that the laws be rescinded, it prevents the public from destroying laws that are in fact in the public interest. These laws are therefore in a way logically private: they cannot at the same time exist, have the foregoing virtues, and be public. After proposing conditions under which such laws ought to be enacted, I moot logical objections to the very idea that there could be such laws, practical objections to their workability, and moral objections to their permissibility. I conclude by suggesting that, while we normally think of secret laws as creatures of the executive branch, things functionally equivalent to secret laws could also be created by other branches of government and societal institutions, and that all of this would be compatible with the form of sovereignty that is democratically grounded in the will and interests of the people. (shrink)

Let (leeway) incompatibilism be the thesis that causal determinism is incompatible with the freedom to do otherwise. Several prominent authors have claimed that incompatibilism alone can capture, or at least best captures, the intuitive appeal behind Jorge Luis Borges's famous “Garden of Forking Paths” metaphor. The thought, briefly, is this: the “single path” leading up to one's present decision represents the past; the forking paths that one must decide between represent those possible futures consistent with the past and the (...) class='Hi'>laws of nature. But if determinism is true, there is only one possible future consistent with the past and the laws and, hence, only one path to choose from. That is, if determinism is true, then we are not free to do otherwise. In this paper, I argue that this understanding of the Garden of Forking Paths faces a number of problems and ought to be rejected even by incompatibilists. I then present an alternative understanding that not only avoids these problems but still supports incompatibilism. Finally, I consider how various versions of (leeway) compatibilism fit with the Garden of Forking Paths as well as the broader question of whether metaphors, however intuitive, have any dialectical force in the debates over freedom. (shrink)

There is an old and powerful argument for the claim that divine foreknowledge is incompatible with the freedom to do otherwise. A recent response to this argument, sometimes called the “dependence response,” centers around the claim that God’s relevant past beliefs depend on the relevant agent’s current or future behavior in a certain way. This paper offers a new argument for the dependence response, one that revolves around different cases of time travel. Somewhat serendipitously, the argument also paves the way (...) for a new reply to a compelling objection to the dependence response, the challenge from prepunishment. But perhaps not so serendipitously, the argument also renders the dependence response incompatible with certain views of providence. (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, (...) clarity, observationalism, contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. One rather conspicuous example is that the process of refining logical terminology has been productive. Future logic students will no longer be burdened by obscure terminology and they will be able to read, think, talk, and write about logic in a more careful and more rewarding manner. Closely related is increased use and study of variable-enhanced natural language as in “Every proposition x that implies some proposition y that is false also implies some proposition z that is true”. Another welcome development is the culmination of the slow demise of logicism. No longer is the teacher blocked from using examples from arithmetic and algebra fearing that the students had been indoctrinated into thinking that every mathematical truth was a tautology and that every mathematical falsehood was a contradiction. A fifth welcome development is the separation of laws of logic from so-called logical truths, i.e., tautologies. Now we can teach the logical independence of the laws of excluded middle and non-contradiction without fear that students had been indoctrinated into thinking that every logical law was a tautology and that every falsehood of logic was a contradiction. This separation permits the logic teacher to apply logic in the clarification of laws of logic. This lecture expands the above points, which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

This paper attempts to illuminate and interpret the contradictory portrait of God as both seen and unseen in the Torah. Thus Moses is commanded not to look on the face of God yet also praised for having spoken to God “face to face". We seek ways to reconcile the contradictory portraits of God through the use of the term “doubled-mindedness” in the theology of Jerome Gellman, in the logic of “thirdness” in C.S. Peirce’s semiotics, and in the use of (...) both particle and wave models in Einstein’s physics of light. The paper concludes by disusing the practical consequences of theological double-mindedness for the religious life and the philosophical meaning of redemption as the time when the contradiction of the unseen and seen God is resolved. (shrink)

This doctoral dissertation investigates the notion of physical necessity. Specifically, it studies whether it is possible to account for non-accidental regularities without the standard assumption of a pre-existent set of governing laws. Thus, it takes side with the so called deflationist accounts of laws of nature, like the humean or the antirealist. The specific aim is to complement such accounts by providing a missing explanation of the appearance of physical necessity. In order to provide an explanation, I recur (...) to fields that have not been appealed to so far in discussions about the metaphysics of laws. Namely, I recur to complex systems’ theory, and to the foundations of statistical mechanics. The explanation proposed is inspired by how complex systems’ theory has elucidated the way patterns emerge, and by the probabilistic explanations of the 2nd law of thermodynamics. More specifically, this thesis studies how some constraints that make no direct reference to the dynamics can be a sufficient condition for obtaining in the long run, with high probability, stable regular behavior. I hope to show how certain metaphysical accounts of laws might benefit from the insights achieved in these other fields. According to the proposal studied in this thesis, some regularities are not accidental not in virtue of an underlying physical necessity. The non-accidental character of certain regular behavior is only due to its overwhelming stability. Thus, from this point of view the goal becomes to explain the stability of temporal patterns without assuming a set of pre-existent guiding laws. It is argued that the stability can be the result of a process of convergence to simpler and stable regularities from a more complex lower level. According to this project, if successful, there would be no need to postulate a (mysterious) intermediate category between logical necessity and pure contingency. Similarly, there would be no need to postulate a (mysterious) set of pre-existent governing laws. Part I of the thesis motivates part II, mostly by arguing why further explanation of the notions of physical necessity and governing laws should be welcomed (chapter 1), and by studying the plausibility of a lawless fundamental level (chapters 2 and 3). Given so, part II develops the explanation of formation of simpler and stable behavior from a more complex underlying level. (shrink)

Suppose Molinism is true and God placed Adam in the garden because God knew Adam would freely eat of the fruit. Suppose further that, had it not been true that Adam would freely eat of the fruit, were he placed in the garden, God would have placed someone else there instead. When Adam freely eats of the fruit, is he free to do otherwise? This paper argues that there is a strong case for both a positive and a negative answer. (...) Assuming such cases are possible under Molinism, we are left with a puzzling question: if Molinism is true, what can you do? (shrink)

Intermediary metabolism molecules are orchestrated into logical pathways stemming from history (L-amino acids, D-sugars) and dynamic constraints (hydrolysis of pyrophosphate or amide groups is the driving force of anabolism). Beside essential metabolites, numerous variants derive from programmed or accidental changes. Broken down, variants enter standard pathways, producing further variants. Macromolecule modification alters enzyme reactions specificity. Metabolism conform thermodynamic laws, precluding strict accuracy. Hence, for each regular pathway, a wealth of variants inputs and produces metabolites that are similar to but (...) not the exact replicas of core metabolites. As corollary, a shadow, paralogous metabolism, is associated to standard metabolism. We focus on a logic of paralogous metabolism based on diversion of the core metabolic mimics into pathways where they are modified to minimize their input in the core pathways where they create havoc. We propose that a significant proportion of paralogues of well-characterized enzymes have evolved as the natural way to cope with paralogous metabolites. A second type of denouement uses a process where protecting/deprotecting unwanted metabolites - conceptually similar to the procedure used in the laboratory of an organic chemist - is used to enter a completely new catabolic pathway. (shrink)

Is the virtue of the good citizen the same as the virtue of the good man? Aristotle addresses this in Politics 3.4. His answer is twofold. On the one hand, (the account for Difference) they are not the same both because what the citizen’s virtue is depends on the constitution, on what preserves it, and on the role the citizen plays in it, and because the good citizens in the best constitution cannot all be good men, whereas the good man’s (...) virtue is uniform. On the other hand, (the account for Identity) the two virtues are identical in the good men in the best constitution, in which all are good citizens, each possessing the ability for ruling and for being ruled. This nuanced answer can be seen as Aristotle’s synthesis of a Periclean view (contribution to state blots out personal wrongs) and a Socratic view (no good citizen is without justice). Its nuances reflect the extent to which Aristotle’s conceptions of good citizenship and the best constitution accommodate deviations of what is probable from the human ideal set out in his ethical writings. In the present chapter, I will first address three puzzles regarding the nuances of Aristotle’s answer. Second, I will consider one question about its implication: Does Aristotle’s account for Difference turn out to entail something ultra-Periclean: that no one can simultaneously be a good man and a good citizen in any constitution other than the best? I shall argue the negative. (shrink)

We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where one (...) can get an analogue of Diaconescu’s result, but also can disentangle the roles of certain other assumptions that are hidden in mathematical presentations. It is our view that these results have not received the attention they deserve: logicians are unlikely to read a discussion because the results considered are “already well known,” while the results are simultaneously unknown to philosophers who do not specialize in what most philosophers will regard as esoteric logics. This is a problem, since these results have important implications for and promise signif i cant illumination of contem- porary debates in metaphysics. The point of this paper is to make the nature of the results clear in a way accessible to philosophers who do not specialize in logic, and in a way that makes clear their implications for contemporary philo- sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism. Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett, realism, antirealism. (shrink)

This article reviews John Wood’s Truth in Fiction: Rethinking its Logic.

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical (...) interpretation of the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators. (shrink)