This volume presents a collection of refereed papers reflecting the state of the art in the area of over-constrained systems. Besides 11 revised full papers, selected from the 24 submissions to the OCS workshop held in conjunction with the First International Conference on Principles and Practice of Constraint Programming, CP '95, held in Marseilles in September 1995, the book includes three comprehensive background papers of central importance for the workshop papers and the whole field. Also included is an introduction (...) by one of the volume editors together with a bibliography listing 243 entries. All in all this is a very useful reference book relevant for all researchers and practitioners interested in hierarchical, partial, and over-constrained systems. (shrink)

Logic arguably plays a role in the normativity of reasoning. In particular, there are plausible norms of belief/disbelief whose antecedents are constituted by claims about what follows from what. But is logic also relevant to the normativity of agnostic attitudes? The question here is whether logical entailment also puts constraints on what kinds of things one can suspend judgment about. In this paper I address that question and I give a positive answer to it. In particular, I advance two (...) logical norms of agnosticism, where the first one allows us to assess situations in which the subject is agnostic about the conclusion of a valid argument and the second one allows us to assess situations in which the subject is agnostic about one of the premises of a valid argument. (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)

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

Anti-exceptionalism about logic is the thesis that logic is not special. In this paper, I consider, and reject, a challenge to this thesis. According to this challenge, there are basic logical principles, and part of what makes such principles basic is that they are epistemically exceptional. Thus, according to this challenge, the existence of basic logical principles provides reason to reject anti-exceptionalism about logic. I argue that this challenge fails, and that the exceptionalist positions motivated (...) by it are thus unfounded. I make this case by disambiguating two senses of ‘basic’ and showing that, once this disambiguation is taken into account, the best reason we have for thinking that there are basic principles actually implies that those principles do not require a special epistemology. Consequently, the existence of basic logical principles provides reason to accept, rather than reject, anti-exceptionalism concerning the epistemology of logic. I conclude by explaining how an abductivist, anti-exceptionalist approach to the epistemology of logic can accommodate the notion of basic logical principles. (shrink)

The workshop will discuss new insights in the interaction between logic and law, and more precisely the study of different answers to the question: What role does logic play in legal reasoning? It will present both current challenges and historical perspectives in the relation between logic and law. The perspectives to be discussed involve the interface of the following studies.

Una interpretación particularmente influyente de la teoría de la implicación lógica de Bertrand Russell y Clarence I. Lewis es la propuesta por Quine en su artículo “Reply to Professor Marcus”. Allí Quine sostiene que la lógica modal de Lewis nació en pecado: el pecado de confundir uso con mención, ya que cuando se afirma que una oración implica lógicamente a otra, estas oraciones no están siendo usadas sino mencionadas. Según la interpretación de Quine, Clarence I. Lewis persistió en el error (...) de Russell, que consistió en confundir la implicación material con la implicación lógica, y confundió la implicación estricta con la implicación lógica. Estos ‘pecados’ de Russell y Lewis pueden entenderse mejor si se tiene en cuenta que tanto Lewis como Russell sostienen que los axiomas y los teoremas en un cálculo lógico se usan de dos maneras: (a) como premisas a partir de las cuales se obtienen nuevos teoremas, y (b) como reglas de inferencia mediante las cuales se obtienen nuevos teoremas. En efecto, estos autores suscriben la teoría del doble uso de los principios lógicos que parece originarse en la obra de Peano y que ha sido casi completamente ignorada en la literatura acerca de la historia de la lógica. -/- A particularly influential interpretation of Bertrand Russell’s and Clarence I. Lewis’s theory of logical implication is the one proposed by Quine in his article "Reply to Professor Marcus". In that article Quine argues that Lewis's modal logic was born in sin: the sin of confusing use with mention, since when it is stated that a sentence logically implies another sentence, these sentences are not being used but mentioned. According to Quine's interpretation, Clarence I. Lewis persisted in Russell's error, which consisted in confusing material implication with logical implication, and confused strict implication with logical implication. These 'sins' of Russell and Lewis can be better understood if we take into account that both logicians argue that the axioms and theorems in a logical calculus are used in two ways: (a) as premises from which new theorems are obtained, and (b) as rules of inference by which new theorems are obtained. In fact, these authors subscribe to the theory of the double use of logical principles that seems to originate in Peano's work and that has been almost completely ignored in the literature about the history of logic. (shrink)

The paper generalizes Van McGee's well-known result that there are many maximal consistent sets of instances of Tarski's schema to a number of non-classical theories of truth. It is shown that if a non-classical theory rejects some classically valid principle in order to avoid the truth-theoretic paradoxes, then there will be many maximal non-trivial sets of instances of that principle that the non-classical theorist could in principle endorse. On the basis of this it is argued that the idea of classical (...) recapture, which plays such an important role for non-classical logicians, can only be pushed so far. (shrink)

We present a new English translation of L.E.J. Brouwer's paper ‘De onbetrouwbaarheid der logische principes’ of 1908, together with a philosophical and historical introduction. In this paper Brouwer for the first time objected to the idea that the Principle of the Excluded Middle is valid. We discuss the circumstances under which the manuscript was submitted and accepted, Brouwer's ideas on the principle of the excluded middle, its consistency and partial validity, and his argument against the possibility of absolutely undecidable propositions. (...) We note that principled objections to the general excluded middle similar to Brouwer's had been advanced in print by Jules Molk two years before. Finally, we discuss the influence on George Griss' negationless mathematics. (shrink)

There are two main objections against epistemological foundation of logical principles: 1. Every argument for them must necessarily make use of them. 2. Logical principles cannot be abstracted from experience because they imply elements of meaning that exceed in principle our finite experience (like universality & necessity). In opposition to these objections I argue for Husserl's thesis that logic needs a theory of experience as a foundation. To show the practicability of his attempt I argue that (...) he is able to avoid the two circles mentioned. Motivated by this investigation the Appendix presents some mathematical doubts concerning the proofs by Cantor of the transfiniteness of the set of real numbers. (shrink)

The first principle of Fichte’s philosophy, the Wissenschaftslehre, is “I am,” whereas the logical principle of identity is “A is A.” The relationship of Fichte’s philosophy to logic helps us understand the relation between the principles of these two disciplines. The first principle of the Wissenschaftslehre as an Act (Thathandlung) is the ground of consciousness and therefore renders logic possible as a science. Even though logic is grounded on the Wissenschaftslehre, the form of logic is used by the (...) latter to comprehend itself. The difference between logic and the Wissenschaftslehre consists in the fact that the former merely supplies a form and the latter contains both the form and the content. Logic arises from the Wissenschaftslehre through the free acts termed “abstraction” and “reflection.” In the Wissenschaftslehre, the form of logical propositions as equality, affirmation and substitution, are derived from the primordial act of the self-positing of the I. (shrink)

We aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic $\mathbf {IQC}$ and first-order arithmetic $\mathbf {HA}$ from a Kripke model for intuitionistic propositional logic $\mathbf {IPC}$. To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata (...) in $\mathbf {IQC}$ and a separation theorem of a sentence from a set of schemata in $\mathbf {HA}$. We see several examples which give us separations among omniscience principles. (shrink)

Artificial intelligence is nowadays a reality. Setting rules on the potential outcomes of intelligent machines, so that no surprise can be expected by humans from the behavior of those machines, is becoming a priority for policy makers. In its recent Communication “Artificial Intelligence for Europe”, for instance, the European Commission identifies the distinguishing trait of an intelligent machine in the presence of “_a certain degree of autonomy_” in decision making, in the light of the context. The crucial issue to be (...) addressed is, therefore, whether it is possible to identify a set of rules for data use by intelligent machines so that the decision-making autonomy of machines can allow for humans’ traditional informational self-determination, as enshrined in many existing legal frameworks and can actually turn out to be further beneficial to individuals. Governing the autonomy of machines can be a very ambitious goal for humans since machines are geared first to the principles of formal logic and then—possibly—to ethical or legal principles. This introduces an unprecedented degree of complexity in how a norm should be engineered, which requires, in turn, an in-depth reflection in order to prevent conflicts between the legal and ethical principles underlying humans’ civil coexistence and the rules of formal logic upon which the functioning of machines is based. (shrink)

Aristotle shares with Plato the attitude that the world, ‘the all,’ is a kosmos, a well-ordered and beautiful whole which, as such, can be rendered intelligible, or understood, by the intellect. One understands things, generally speaking, by tracing them back to their sources, origins or principles and causes or explanatory factors, and seeing in what manner they are related to these principles. We know, or understand, a thing when we grasp ‘the why’ or cause. Consequently, understanding is systematic. (...) Some things we understand through themselves - these are the first principles and as such are not understood by tracing them back to causes. We understand other things by systematically relating them in appropriate ways to what is known through itself. These other things are known through, or by means of, their causes and principles. (shrink)

The abstract status of Kant's account of his ‘general logic’ is explained in comparison with Gödel's general definition of a formal logical system and reflections on ‘abstract’ (‘absolute’) concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction (...) that tolerates a sort of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system. (shrink)

The Principles of Logic by Herbert Austin Aikins, first published in 1902, is a rare manuscript, the original residing in one of the great libraries of the world. This book is a reproduction of that original, which has been scanned and cleaned by state-of-the-art publishing tools for better readability and enhanced appreciation. Restoration Editors' mission is to bring long out of print manuscripts back to life. Some smudges, annotations or unclear text may still exist, due to permanent damage to (...) the original work. We believe the literary significance of the text justifies offering this reproduction, allowing a new generation to appreciate it. (shrink)

This paper offers a critical analysis of the logical structure of principles proposed by Robert Alexy and, in particular, of their structure as optimisation commands. Its first part dwells on the question whether the optimisation element in the logical structure should be understood as part of modalisation, as part of the consequent, or as an independent element. In the second part, the author analyses possible forms of inter-definability of deontic operators. Finally, some questions are raised on the (...) conditional structure proposed by Alexy for principles. (shrink)

Hegel's Science of Logic is a unique intellectual center of the philosopher's development. In a sense everything that he had done previously was used in his search for a logical principle and in shaping a system of logic and everything he wrote later brought out the theoretical elements that it implicitly contained.

In this paper, I seek to undermine G.A. Cohen ’s polemical use of a metaethical claim he makes in his article, ‘ Facts and Principles’, by arguing that that use requires an unsustainable equivocation between epistemic and logical grounding. I begin by distinguishing three theses that Cohen has offered during the course of his critique of Rawls and contractualism more generally, the foundationalism about grounding thesis, the justice as non-regulative thesis, and the justice as all-encompassing thesis, and briefly (...) argue that they are analytically independent of each other. I then offer an outline of the foundationalism about grounding thesis, characterising it, as Cohen does, as a demand of logic. That thesis claims that whenever a normative principle is dependent on a fact, it is so dependent in virtue of some other principle. I then argue that although this is true as a matter of logic, it, as Cohen admits, cannot be true of actual justifications, since logic cannot tell us anything about the truth as opposed to the validity of arguments. Facts about a justification cannot then be decisive for whether or not a given argument violates the foundationalism about grounding thesis. As long as, independently of actual justifications, theorists can point to plausible logically grounding principles, as I argue contractualists can, Cohen ’s thesis lacks critical bite. (shrink)

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by ) that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise (...) to a notion of logical consequence: we study the corresponding theory of validity by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth. Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence. (shrink)

THE PURPOSE OF THE PAPER IS TO CRITICALLY EXAMINE TWO METAPHYSICAL PRINCIPLES ADVOCATED BY PROFESSOR MORRIS IN HIS BOOK "THE LOGIC OF GOD INCARNATE", NAMELY (I) THE DISTINCTION BETWEEN "COMMON" HUMAN PROPERTIES AND "ESSENTIAL" HUMAN PROPERTIES; (II) THE DISTINCTION BETWEEN BEING "MERELY" X AND "FULLY" X. THE FIRST DISTINCTION IS BOTH DEFENDED AND EXPANDED ON; THE SECOND IS REJECTED ON THE GROUNDS THAT IT INVOKES AN IMPOSSIBLE COMPARISON; A COMPARISON BETWEEN A QUANTITATIVE ASSESSMENT ON THE ONE HAND AND THE (...) QUALITATIVE ASSESSMENT ON THE OTHER. IN THE LIGHT OF THIS IT IS CLAIMED THAT MORRIS’S DEFENCE OF THE ORTHODOX DOCTRINE OF THE INCARNATION FAILS. (shrink)

Doubts concerning the validity of logic are as old as the empirical criticism of science. In the last two decades the idea that truth is relative to given sets of basic assumptions has been prominent; and the controversy about the principle of excluded middle has focussed renewed attention upon the nature of logic and its fundamental principles.Recent investigations in formal logic have contributed greatly to the understanding of the principles of logic. It is simply a misunderstanding to conclude (...) from them that the logical principles have been suspended, despite the various formal structures which are called “logics.” The purely formal treatment of problems concerning the foundations of logic, while necessarily undertaken as far as it can go, has essential limitations. (shrink)

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order (...) language, taken together, have substantial logical strength, amounting to a kind of abstract consistency statement. (shrink)

Bertrand Russell presented three systems of propositional logic, one first in Principles of Mathematics, University Press, Cambridge, 1903 then in “The Theory of Implication”, Routledge, New York, London, pp. 14–61, 1906) and culminating with Principia Mathematica, Cambridge University Press, Cambridge, 1910. They are each based on different primitive connectives and axioms. This paper follows “Peirce’s Law” through those systems with the aim of understanding some of the notorious peculiarities of the 1910 system and so revealing some of the early (...) history of classical propositional logic. “Peirce’s Law” is a valid formula of elementary propositional logic: [ ⊃ p] ⊃ p This sentence is not even a theorem in the 1910 system although it is one of the axioms in 1903 and is proved as a theorem in 1906. Although it is not proved in 1910, the two lemmas from the proof in 1906 occur as theorems, and Peirce’s Law could have been derived from them in a two step proof. The history of Peirce’s Law in Russell’s systems helps to reconstruct some of the history of axiomatic systems of classical propositional logic. (shrink)

Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic.

Second edition of the introductory guidebook to the basic principles of constructing sound arguments and criticising bad ones. Non-technical in approach, it is based on 186 examples, which Douglas Walton, a leading authority in the field of informal logic, discusses and evaluates in clear, illustrative detail. Walton explains how errors, fallacies, and other key failures of argument occur. He shows how correct uses of argument are based on sound strategies for reasoned persuasion and critical responses. This edition takes into (...) account many developments in the field of argumentation study that have occurred since 1989, many created by the author. Drawing on these developments, Walton includes and analyzes 36 new topical examples and also brings in work on argumentation schemes. Ideally suited for use in courses in informal logic and introduction to philosophy, this book will also be valuable to students of pragmatics, rhetoric, and speech communication. (shrink)

This two-volume work, first published in 1843, was John Stuart Mill's first major book. It reinvented the modern study of logic and laid the foundations for his later work in the areas of political economy, women's rights and representative government. In clear, systematic prose, Mill disentangles syllogistic logic from its origins in Aristotle and scholasticism and grounds it instead in processes of inductive reasoning. An important attempt at integrating empiricism within a more general theory of human knowledge, the work constitutes (...) essential reading for anyone seeking a full understanding of Mill's thought. Continuing the discussion of induction, Volume 2 concludes with Book VI, 'On the Logic of the Moral Sciences', in which Mill applies empirical reasoning to human behaviour. A crucial early formulation of his thinking regarding free will and necessity, this book establishes the centrality of 'the social science' to Mill's philosophy. (shrink)

This two-volume work, first published in 1843, was John Stuart Mill's first major book. It reinvented the modern study of logic and laid the foundations for his later work in the areas of political economy, women's rights and representative government. In clear, systematic prose, Mill (1806–73) disentangles syllogistic logic from its origins in Aristotle and scholasticism and grounds it instead in processes of inductive reasoning. An important attempt at integrating empiricism within a more general theory of human knowledge, the work (...) constitutes essential reading for anyone seeking a full understanding of Mill's thought. Volume 1 contains Mill's introduction, which elaborates upon his definition of logic as 'not the science of Belief, but the science of Proof, or Evidence'. It also features discussions of the central components of logical reasoning - propositions and syllogisms - in relation to Mill's theories of inductive reasoning and experimental method. (shrink)

I consider the question of how literally one can construe the “gauge argument,” which is the canonical means of understanding the putatively central import of local gauge symmetry principles for fundamental physics. As I argue, the gauge argument must be afforded a heuristic reading. Claims to the effect that the argument reflects a deep “logic of nature” must, for numerous reasons I discuss, be taken with a grain of salt.

F. H. Bradley was the foremost philosopher of the British Idealist school, which came to prominence in the second half of the nineteenth century and remained influential into the first half of the twentieth. Bradley, who was educated at Oxford, and spent his life as a fellow of Merton College, was influenced by Hegel, and also reacted against utilitarianism. He was recognised during his lifetime as one of the greatest intellectuals of his generation and was the first philosopher to receive (...) the Order of Merit, in 1924. In this major work, originally published in 1883, Bradley discusses the basic principles of logic: judgment and inference. He rejects the idea of a separation between mind and body, arguing that human thought cannot be separated from its worldly context. In the second edition, published in 1922 and reissued here, Bradley added a commentary and essays, but left the text largely unaltered. (shrink)

F. H. Bradley was the foremost philosopher of the British Idealist school, which came to prominence in the second half of the nineteenth century and remained influential into the first half of the twentieth. Bradley, who was influenced by Hegel and also reacted against utilitarianism, was recognised during his lifetime as one of the greatest intellectuals of his generation, and was the first philosopher to receive the Order of Merit, in 1924. In this major work, originally published in 1883, Bradley (...) discusses the basic principles of logic: judgment and inference. He rejects the idea of a separation between mind and body, arguing that human thought cannot be separated from its worldly context. In the second edition, published in 1922 and reissued here, Bradley added a commentary and essays, but left the text largely unaltered. Volume 1 contains Book 1 on judgment and Book 2 on inference. (shrink)

F. H. Bradley (1846–1924) was the foremost philosopher of the British Idealist school, which came to prominence in the second half of the nineteenth century and remained influential into the first half of the twentieth. Bradley, who was influenced by Hegel and also reacted against utilitarianism, was recognised during his lifetime as one of the greatest intellectuals of his generation, and was the first philosopher to receive the Order of Merit, in 1924. In this major work, originally published in 1883, (...) Bradley discusses the basic principles of logic: judgment and inference. He rejects the idea of a separation between mind and body, arguing that human thought cannot be separated from its worldly context. In the second edition, published in 1922 and reissued here, Bradley added a commentary and essays, but left the text largely unaltered. Volume 1 contains Book 1 on judgment and Book 2 on inference. (shrink)

In his recent work Professor Morris writes: ‘I am suggesting that, armed with a couple of fairly simple metaphysical distinctions we can begin to see how the doctrine of the Incarnation can possibly be true.’ What are these ‘metaphysical distinctions’ and do they stand up to critical examination? My answer to the latter part of this question in regard to the first distinction is a reserved ‘Yes’; in regard to the second distinction a definite ‘No’. If my criticism of the (...) second distinction holds good, then Morris's defence of the orthodox doctrine of the Incarnation fails, for his defence crucially relies on the validity of that distinction. (shrink)

This books states, as clearly and concisely as possible, the most fundamental principles of set-theory and mathematical logic. Included is an original proof of the incompleteness of formal logic. Also included are clear and rigorous definitions of the primary arithmetical operations, as well as clear expositions of the arithmetic of transfinite cardinals.

The present paper investigates why logical empiricists remained silent about one of the most philosophy-laden matters of theoretical physics of their day, the principle of least action (PLA). In the two decades around 1900, the PLA enjoyed a remarkable renaissance as a formal unification of mechanics, electrodynamics, thermodynamics, and relativity theory. Taking Ernst Mach's historico-critical stance, it could be liberated from much of its physico-theological dross. Variational calculus, the mathematical discipline on which the PLA was based, obtained a new (...) rigorous basis. These three developments prompted Max Planck to consider the PLA as formal embodiment of his convergent realist methodology. Typically rejecting ontological reductionism, David Hilbert took the PLA as the key concept in his axiomatizations of physical theories. It served one of the main goals of the axiomatic method: ''deepening the foundations.'' Although Moritz Schlick was a student of Planck's, and Hans Hahn and Philipp Frank enjoyed close ties to Gottingen, the PLA became a veritable Shibboleth to them. Rather than being worried by its historical connections with teleology and determinism, they erroneously identified Hilbert's axiomatic method tout court with Planck's metaphysical realism. Logical empiricists' strict containment policy against metaphysics required so strict a separation between physics and mathematics to exclude even those features of the PLA and the axiomatic method not tainted with metaphysics. (shrink)

In a definition (∀ x )(( x є r )↔D[ x ]) of the set r, the definiens D[ x ] must not depend on the definiendum r . This implies that all quantifiers in D[ x ] are independent of r and of (∀ x ). This cannot be implemented in the traditional first-order logic, but can be expressed in IF logic. Violations of such independence requirements are what created the typical paradoxes of set theory. Poincaré’s Vicious Circle Principle (...) was intended to bar such violations. Russell nevertheless misunderstood the principle; for him a set a can depend on another set b only if ( b є a ) or ( b ⊆ a ). Likewise, the truth of an ordinary first-order sentence with the Gödel number of r is undefinable in Tarki’s sense because the quantifiers of the definiens depend unavoidably on r. (shrink)