The paper offers a historical survey of the emergence of logical formalization in twentieth-century analytically oriented philosophy of religion. This development is taken to have passed through three main ?stages?: a pioneering stage in the late nineteenth and early twentieth centuries (led by Frege and Russell), a stage of crisis in the 1920s and early 1930s (occasioned by Wittgenstein, logical positivists such as Carnap, and neo-Thomists such as Maritain), and a stage of rehabilitation in the 1930s, 1940s, and 1950s (led (...) by the Cracow Circle and Quine). (shrink)

To define new property terms, we combine already familiar ones by means of certain logical operations. Given suitable constraints, these operations may presumably include the resources of first-order logic: truth-functional sentence connectives and quantification over objects. What is far less clear is whether we can also use modal operators for this purpose. This paper clarifies what is involved in this question, and argues in favor of modal property definitions.

This paper concerns the recent debate on the nature and motivations of the epistemological project advanced in Rudolf Carnap's (1891-1970) Aufbau. Much of this debate has been initiated by Michael Friedman and Alan Richardson who argue (against the received view of the Aufbau as a foundationalist defense of empiricism) that Carnap's epistemological project is located in the tradition of neo-Kantian epistemology. On this revisionist reading of the Aufbau, Carnap's project is not motivated to address traditional empiricist problems regarding the justification (...) of knowledge, but rather to show how objective knowledge is possible. A central aspect of the Aufbau that is neglected in the revisionists' analysis is the role of epistemic justification in Carnap's project. The aim of the present study is to argue that although the general epistemology in the Aufbau can be cast as neo-Kantian, Carnap's method of construction theory (or rational reconstruction) is formulated precisely as an empiricist method for the justification of conceptual knowledge. Construction theory radically redefines `empirical justification' into a formal-conventional notion, and is part of Carnap's more general agenda of redefining epistemology as a purely formal discipline. (shrink)

ABSTRACT I argue that some of the central doctrines of Wittgenstein's Tractatus Logico-Philosophicus can be seen as addressing the twin problems of semantic unity and...

The reflections recorded in this paper arise from three moments in the theory of definition and of conceptual analysis. The moments are: Frege’s review of Husserl’s Philosophy of Arithmetic, the discussion there of the paradox of analysis, and the division that Frege marks, ensuing upon his distinction of Sinn/sense from Bedeutung/reference, between two different conceptions of definition; Leibniz’s still serviceable account of a distinction between the clarity and the distinctness of ideas---a distinction that prompts the suggestion that the guiding purpose (...) of lexical definition is Leibnizian clarity whereas that of real definition is inseparable from the pursuit of Leibnizian distinctness; Leibniz’s speculations concerning the limit or terminus of analysis. The apparent failure of these speculations, casting doubt as it does upon the aspirations that give rise to them, points to the long-standing need to reconfigure the philosophical business of enquiry into concepts. (shrink)

This paper is concerned with De Morgan’s explanation of the validity of arguments that involve relational notions. It discusses De Morgan’s expansion of traditional logic aimed at accommodating those inferences, and makes the point that his endeavour is not successful in that the rules that made up his new logic are not sound. Nevertheless, the most important scholarly work on De Morgan’s logic, and contrary to that De Morgan’s mistake is not beyond repair. The rules that determine his new logic (...) are in fact monotonie replacement rules. And provided they are restricted in the correct way, these rules are demonstrably sound. (shrink)

Ronald Dworkin, Law's Empire Cambridge: The Belknap Press of Harvard University Press, 1986, xiii + 470 pp.

In a recent book, Robert Trueman develops a version of the identity theory of truth, the theory that true propositions are not in some kind of correspondence with, but are rather identical with, facts. He claims that this theory ‘collapses the gap between mind and world’. Whether it does so will obviously depend on how the theory is to be understood, which in turn depends on the argumentative route to it. Trueman’s route is clear, rigorous, and free of extravagant assumptions. (...) Perhaps because of those merits it seems obvious that it falls short of the claim he makes for it. But there are difficult questions about the nature of the shortfall and about what in the character of Trueman’s philosophical approach prevents him from appreciating it. The paper explores those questions through a comparison with Moore’s ‘original’ identity theory and the Idealist philosophy he directed it against. (shrink)

Reply to Beaney: the closing of the historical mindIn his comments, Michael Beaney sets himself up as the arbiter of what is genuine history and what isn’t. While celebrating the outpouring of specialized scholarship on Frege, he has no patience with the enterprise outlined in the Précis, which attempts to construct a large-scale picture of the richness of the analytic tradition. That enterprise is one in which great figures of our recent past are challenged by aspects of contemporary thought, and (...) our current struggles are enriched by insights of theirs that haven’t been fully absorbed. Although some work of this sort can be done piecemeal, one historical figure at a time, there is much to be learned from a more encompassing perspective. While no picture constructed by a single author can be authoritative about everything, the attempt to give informative, connected assessments of major milestones in the tradition is our best hope of understanding who we are and where we have come from. (shrink)

Never shall this be proved, that things that are not are. ParmenidesTo say that something does not exist, or that there is something which is not, is clearly a contradiction in terms; hence “ ” must be true. Moreover, we should certainly expect leave to put any primitive name of our language for the “x” of any matrix “ … x … ”, and to infer the resulting singular statement from “ ”; it is difficult to contemplate any alternative logical (...) rule for reasoning with names. (shrink)

The metaphysics of relations is still in its infancy. We use the idea of truthmaking to gain purchase on this metaphysics. Assuming a modest supervenience conception of truthmaking, where true relational predications require multiply dependent truthmakers, these are indispensable relations. Though some such relations are required, none are needed for internal relatedness, nor for several other kinds of relational predication. Discerning the metaphysically basic kinds of relations is fraught with uncertainties, but must be tackled if progress is to be made.

In this essay, I first solve solve a conundrum and then deal with criteria of identity, Leibniz's definition of identity and Frege's adoption of it in his (failed) attempt to define the cardinality operator contextually in terms of Hume's Principle in Die Grundlagen der Arithmetik. I argue that Frege could have omitted the intermediate step of tentatively defining the cardinality operator in the context of an equation of the form ‘NxF(x) = NxG(x)'. Frege considers Leibniz's definition of identity to be (...) purely logical, although without saying why it is in line with his logicist project. I argue that the universal criterion of identity that Frege takes from Leibniz's definition and the specific criterion of identity for cardinal numbers embodied in Hume’s Principle (namely equinumerosity) work hand in hand in the tentative contextual definition of the cardinality operator. Yet the interplay between the two criteria is powerless to prevent the emergence of the Julius Caesar problem, let alone suggests how it could be solved. The final explicit definition of the cardinality operator that Frege sets up still rests on the identity criterion of equinumerosity since cardinal numbers are defined as equivalence classes of that relation. (shrink)

The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ?theory of quantity? (?Größenlehre?) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section (...) 2, I first analyze Frege's use of the term ?source of knowledge? (?Erkenntnisquelle?) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege 1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of real numbers in Die Grundlagen der Arithmetik (Frege 1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege 1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section 6, I return to Frege's own constructive treatment of analysis in Frege 1903 and succinctly describe what I take to be the quintessence of his account. (shrink)

Fundamental changes in sciences offer new perspectives for the management of complexity. Increased complexity in society, economics, and technology requires a new and suitable organization and management. What are the consequences and results for project management? That is the theme of this article. First of all it will given a short introduction to project management, which will be later called "traditional project management" or "project management 1st order (PM-1)." Then, the challenges by the fundamental changes in sciences and the increased (...) complexity in society, economics, and technology will be discussed. It will state that traditional project management cannot solve these challenges. The widespread working-themes and results of the research program "Beyond Frontiers of Traditional Project Management" as an answer to these challenges will be presented at a glance. Subsequently, it will discuss some selected results of the research program: The principle-definition and foundation of "Evolution of 1st and 2nd Order." The Evolution of 1st Order and the impact on Project Management methods and processes. Evolution of 2nd Order and the Grand Evolutionary Systems Theory (GEST) of E. Laszlo as also the impact on Project Management methods and processes. Management of crisis: turn a change to advantage or risk-assurance? Thereafter, the concept of "Project Management Second Order (PM-2)" is presented as a highlighted result of the research program, as a new paradigm in project management, and as an answer to the challenges, described earlier will be explained in detail. Finally, a real example of transfer evolutionary and self-organizational management principles in a real project life will be demonstrated. (shrink)

The paper reviews some conceptions of logical form in the light of Andrea Iacona’s book Logical Form. I distinguish the following: logical form as schematization of natural language, provided by, for example, Aristotle’s syllogistic; the relevance to logical form of formal languages like those used by Frege and Russell to express and prove mathematical theorems; Russell’s mid-period conception of logical form as the structural cement binding propositions; the conceptions of logical form discussed by Iacona; and logical form regarded as an (...) empirical hypothesis about the psychology of language processing, as in the Discourse Representation Theory tradition. Whereas neither schematization, nor the use of special languages for mathematics, raise general methodological or empirical difficulties, other conceptions of logical form raise at least apparent problems. (shrink)

In Russell's Problems of Philosophy (PP), acquaintance is the basis of thought and also the basis of empirical knowledge. Thought is based on acquaintance, in that a thinker has to be acquainted with the basic constituents of his thoughts. Empirical knowledge is based on acquaintance, in that acquaintance is involved in perception, and perception is the ultimate source of all empirical knowledge.

The paper provides a proof theoretic characterization of the Russellian theory of definite descriptions (RDD) as characterized by Kalish, Montague and Mar (KMM). To this effect three sequent calculi are introduced: LKID0, LKID1 and LKID2. LKID0 is an auxiliary system which is easily shown to be equivalent to KMM. The main research is devoted to LKID1 and LKID2. The former is simpler in the sense of having smaller number of rules and, after small change, satisfies cut elimination but fails to (...) satisfy the subformula property. In LKID2 an additional analysis of different kinds of identities leads to proliferation of rules but yields the subformula property. This refined proof theoretic analysis leading to fully analytic calculus with constructive proof of cut elimination is the main contribution of the paper. (shrink)

Symbolic logic is sited at intersection of philosophy, mathematics, linguistics and computer science. It deals with the structure of reasoning, and the formal features of information. Work in symbolic logic has almost exclusively treated the deductive validity of arguments: those arguments for which it is impossible for the premises to be true and the conclusion false. However, techniques from twentieth-century logic have found a place in the study of inductive or probabilistic reasoning, in which premises need not render their conclusions (...) certain. (shrink)

Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory that distinguishes between ordinary and abstract objects.This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is (...) simply adjoined to AOT’s specially formulated comprehension principle for relations. This result constitutes a new and important paradox, given how much expressive and analytic power is contributed by having the two kinds of complex terms in the system. Its discovery is the highlight of our joint project and provides strong evidence for a new kind of scientific practice in philosophy, namely, computational metaphysics.Our results were made technically possible by a suitable adaptation of Benzmüller’s metalogical approach to universal reasoning by semantically embedding theories in classical higher-order logic. This approach enables one to reuse state-of-the-art higher-order proof assistants, such as Isabelle/HOL, for mechanizing and experimentally exploring challenging logics and theories such as AOT. Our results also provide a fresh perspective on the question of whether relational type theory or functional type theory better serves as a foundation for logic and metaphysics. (shrink)

This paper gives a survey of David Hilbert's (1862–1943) changing attitudes towards logic. The logical theory of the Göttingen mathematician is presented as intimately linked to his studies on the foundation of mathematics. Hilbert developed his logical theory in three stages: (1) in his early axiomatic programme until 1903 Hilbert proposed to use the traditional theory of logical inferences to prove the consistency of his set of axioms for arithmetic. (2) After the publication of the logical and set-theoretical paradoxes by (...) Gottlob Frege and Bertrand Russell it was due to Hilbert and his closest collaborator Ernst Zermelo that mathematical logic became one of the topics taught in courses for Göttingen mathematics students. The axiomatization of logic and set-theory became part of the axiomatic programme, and they tried to create their own consistent logical calculi as tools for proving consistency of axiomatic systems. (3) In his struggle with intuitionism, represented by L. E. J. Brouwer and his advocate Hermann Weyl, Hilbert, assisted by Paul Bemays, created the distinction between proper mathematics and meta-mathematics, the latter using only finite means. He considerably revised the logical calculus of thePrincipia Mathematica of Alfred North Whitehead and Bertrand Russell by introducing the ε-axiom which should serve for avoiding infinite operations in logic. (shrink)

This essay accounts for the notion of _Lebensform_ by assigning it a _logical _role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the _PI_ occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise _The Brown Book_. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later (...) remarks on _Lebensformen_ is offered which factors in his reading of Alan Turing’s “On computable numbers, with an application to the _Entscheidungsproblem_“, as well as his discussions with Turing 1937-1939. An interpretation of the five occurrences of _Lebensform_ in the PI is then given in terms of a logical “regression” to _Lebensform_ as a fundamental notion. This regression characterizes Wittgenstein’s mature answer to the question, “What is the nature of the logical?”. (shrink)

We prove the independence of some weakenings of the axiom of choice related to the question if the unions of wellorderable families of wellordered sets are wellorderable.

It is argued that the finitist interpretation of wittgenstein fails to take seriously his claim that philosophy is a descriptive activity. Wittgenstein's concentration on relatively simple mathematical examples is not to be explained in terms of finitism, But rather in terms of the fact that with them the central philosophical task of a clear 'ubersicht' of its subject matter is more tractable than with more complex mathematics. Other aspects of wittgenstein's philosophy of mathematics are touched on: his view that mathematical (...) propositions are 'grammatical propositions' (and so, Strictly speaking, Not genuine propositions at all) and his view that in mathematics 'everything is algorithm, Nothing meaning'. His views on consistency and his anti-Foundationalism are linked with this central thesis. (shrink)

Cet article s'ajoute à la liste déjà longue de ceux qui traitent des rapports entre la théorie russellienne des descriptions définies et la théorie ramifiée des types. Seule la prétention d'aborder cette question dans une perspective nouvelle justifie ici sa présence: d'une part, la théorie des descriptions définies sera resituée dans le contexte initial et souvent méconnu de la théorie des expressions dénotantes et d'autre part, c'est à la logique intensionnelle de Russell, objet d'une méconnaissance au moins égale, que nous (...) nous intéresserons. De plus, c'est du point de vue de la théorie de la signification et non pas du point de vue ontologique, génético-historique ou logico-mathematique que nous nous proposons de traiter de ces rapports. (shrink)

There is the idea that rational belief for a single individual can be constructed via a process of unilateral argument. To preempt antipathy between the AI communities that can claim the idea that rational belief can be so constructed, we trace the idea to the beginning of this century, to Keynes' dispute with Russell over logic and probability. We review how Keynesian ideas were revived in AI's work on non-monotonic reasoning and parallel developments in philosophical logic.

This article discusses the argument we cannot have knowledge of abstract entities because they are not part of the causal order. The claim of this article is that the argument fails because of equivocation. Assume that the “causal order” is concerned with contingent facts involving time and space. Even if the existence of abstract entities is not contingent and does not involve time or space it does not follow that no truths about abstract entities are contingent or involve time or (...) space. I argue that it is the latter which is required to obtain the desired conclusion. (shrink)

Some interpreters have ascribed to Wittgenstein the view that mathematical statements must have an application to extra-mathematical reality in order to have use and so any statements lacking extra-mathematical applicability are not meaningful (and hence not bona fide mathematical statements). Pure mathematics is then a mere signgame of questionable objectivity, undeserving of the name mathematics. These readings bring to light that, on Wittgenstein’s offered picture of mathematical statements as rules of description, it can be difficult to see the role of (...) mathematical statements which relate to concepts that are not employed in empirical propositions e.g. set-theoretic concepts. I will argue that Wittgenstein’s picture is more flexible than might at first be thought and that Wittgenstein sees such statements as serving purposes not directly related to empirical description. Whilst this might make such systems fringe cases of mathematics, it does not bring their legitimacy as mathematical systems into question. (shrink)

In this paper is used to denote Jensen's modification of Quine's ‘new foundations’ set theory () fortified with a type‐level pairing function but without the axiom of choice. The axiom is the variant of the axiom of counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that proves the consistency of the simple theory of types with infinity (). This result implies that proves that consistency of, and that proves the consistency (...) of. (shrink)

This paper is dedicated to Alonzo Church, who died in August 1995 after a long life devoted to logic. To Church we owe lambda calculus, the thesis bearing his name and the solution to the Entscheidungsproblem.His well-known book Introduction to Mathematical LogicI, defined the subject matter of mathematical logic, the approach to be taken and the basic topics addressed. Church was the creator of the Journal of Symbolic Logicthe best-known journal of the area, which he edited for several decades This (...) paper is in three sections. The first is written in journalistic style:the story of the life of AlonzoChurch is told, including some of the many anecdotes I have collected from different sources. The secondpart is devoted to his work, but is far from being exhaustive. The last part is more original; in it I attempto show that Church’s great discovery was lambda calculus and that his remaining contributions weremainly inspired afterthoughts in the sense that most of his contributions as well as some of his pupils derivefrom that initial achievement. Included are Kleene’s Recursion Theory and the completeness proof ofHenkin. I have added an appendix in which is presented the typed lambda calculus and a proof of theundecidability of first-order logic. (shrink)

I offer you some theories of intellectual obligations and rights (virtue Ethics): initially, RBT (a Right to Believe Truth, if something is true it follows one has a right to believe it), and, NDSM (one has no right to believe a contradiction, i.e., No right to commit Doxastic Self-Mutilation). Evidence for both below. Anthropology, Psychology, computer software, Sociology, and the neurosciences prove things about human beliefs, and History, Economics, and comparative law can provide evidence of value about theories of rights. (...) However, insofar as we have methods within Philosophy to help us formulate precise concepts of 'belief' and 'rights', methods that also help us to prove links (or absence thereof) amongst families of concepts of rights and belief, our discipline is in and of itself capable of sound reasoning about issues as puzzling as the following. Suppose a Jane who does not believe in God yet who believes she ought to so believe: Jane is undergoing doxastic moral regret (moral regret for lack of faith). We have all known such Janes and perhaps at one time or another even been one. Paradox: given RBT and NDSM, Jane as described not only does not exist, Jane cannot exist. Thus, to enrich the ways in which Philosophy need not get all its evidence from other academic disciplines, I present a brief introduction to what I call Neutral Universal Frames (NUFs). NUFs solve hard puzzles about interactions among modal concepts of belief and rights, concepts that occur in RBT, NDSM and the description of our Janes. NUFs for theories precisely articulated via any two or more modal concepts are a powerful and immensely general set of tools enabling us to define rich theories of truth ("models on frames") to test philosophical theories for internal consistency and to prove the existence of connections (or absence thereof) amongst alternative articulations of philosophical theories. NUFs thereby add to the constructive knowledge producing way Logics intersect with Philosophy of Religion. And we will soon see why Jane, be she named 'Jane' or known simply as you: cannot exist. Read on at your own risk. (shrink)

The paper presents a formal explication of the early Wittgenstein's views on ontology, the syntax and semantics of an ideal logical language, and the propositional attitudes. It will be shown that Wittgenstein gave a language of thought analysis of propositional attitude ascriptions, and that his ontological views imply that such ascriptions are truth-functions of (and supervenient upon) elementary sentences. Finally, an axiomatization of a quantified doxastic modal logic corresponding to Tractarian semantics will be given.

Several difficulties, concerning the individuation and the variation of tropes, beset the initial classic version of trope theory. K. Campbell (Abstract particulars, Oxford, Basil Blackwell, 1990) presented a modified version that aims to avoid those difficulties. Unfortunately, the revised theory cannot make the case that one of the fundamental tropes, space-time, is a genuine particular.

If quantum mechanics (QM) is to be taken as an atomistic theory with the elementary particles as atoms (an ATEP), then the elementary particlcs must be individuals. There must then be, for each elementary particle a, a property being identical with a that a alone has. But according to QM, elementary particles of the same kind share all physical properties. Thus, if QM is an ATEP, identity is a metaphysical but not a physical property. That has unpalatable consequences. Dropping the (...) assumption that QM is an ATEP makes it possible to replace the assumption that elementary particles are individuals with the assumption that there are various kinds of elementary stuff that have smallest quantities — the smallest quantity of light, for example, is a photon. The problems about identity disappear, and the explanatory virtues of an ATEP are maintained. (shrink)

This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so that (...) nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown, it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here. (shrink)

This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: ZFCAσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathsf {ZFCA}_{\sigma }$$\end{document} with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents.

In this article, we evaluate the Compositionality Argument for structured propositions. This argument hinges on two seemingly innocuous and widely accepted premises: the Principle of Semantic Compositionality and Propositionalism (the thesis that sentential semantic values are propositions). We show that the Compositionality Argument presupposes that compositionality involves a form of building, and that this metaphysically robust account of compositionality is subject to counter-example: there are compositional representational systems that this principle cannot accommodate. If this is correct, one of the most (...) important arguments for structured propositions is undermined. (shrink)

Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. (...) Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)