This paper sought to state in a concise and comparatively informal, unsystematic, and more accessible form the more technical approach the author developed during a research fellowship in 1974-75 at the Max-Planck-Institut in Starnberg, Germany. ●●●●● The ideas presented in this paper are more fully developed in later publications by the author which are listed in the two-page addendum to this paper. ●●●●● UPDATED NOTE TO THE READER - December, 2021 ●●●●● Readers will find a more fully developed position than (...) was formulated in this 1976 paper in the author's book-length study, CRITIQUE OF IMPURE REASON: Horizons of Possibility and Meaning. ●●●●● The book is available in a printed edition, ISBN 978-0-578-88646-6, from online booksellers such as Barnes & Noble, Amazon, and through independent bookstores. The book is also available as a freely downloadable _corrected second edition_ eBook, DOI: 10.5281/zenodo.5458352, through PhilSci, PhilPapers, HAL, and other online archives. ●●●●● In addition to the above-mentioned book, readers may find "A Primer on Bartlett's CRITIQUE OF IMPURE REASON" (DOI: 10.5281/zenodo.5789848) to be useful as a guide to the nearly 900-page CRITIQUE. The primer is available as a free download through PhilPapers. ●●●●●. (shrink)

Roger Vergauwen seeks to provide an answer to the question, "How does language connect to the world?" He begins with the recent developments in formal semantics and from them constructs his own 'theory of reference' with which he considers the nature of the correspondence of the world. The author locates his metalogics between the philosophy of language and epistemology while he covers a range of models from Plato to Wittgenstein. Vergauwen assumes no previous technical or logical study. Contents: Introduction: (...) The Temperature of a Hot Topic; The Tarski Program: Hijacking the Truth about Reference; Reference and Theory of Reference; Inside Out: Reference and Metalogic; Outline of a Logic with Non-Standard Meanings. (shrink)

In this paper I shall attempt to outline a nominalistic theory of mathematical truth. I call my theory nominalistic because it avoids a real (see [4]) ontological commitment to abstract entities. Traditionally, nominalists have found it difficult to justify any reference to infinite collections in mathematics. Even those who have tried to do so have typically restricted themselves to predicative and, thus, denumerable realms. I Indeed, many have linked impredicative definitions to platonism; nominalists have tended to agree with Weyl (...) that impredicative analysis is "a house built on sand" in a "logician's paradise."2 As a result, they have either worried about how much of mathematics empirical science requires (e.g., [34]) or renounced mathematical truth altogether (e.g., [12], [17]). My theory, in contrast, seeks a nominalistic interpretation of the entire body of classical mathematics. It tries to secure for the nominalist not merely the natural numbers but the reals, the ordinals, and inaccessible cardinals as well. If I am right, then nominalists can declare, with Goodman and Quine, that "any system that countenances abstract entities we deem unsatisfactory as a final philosophy" ([17],105), and, at the same time, with Hilbert, that "no one shall drive us out of the paradise which Cantor has created for us" ([20], 141). I cannot hope to convince you, in this small space, that my goal is fully achievable. Here I shall try to justify a smaller contention: that if we can explain our knowledge of logic (and of language generally), we can also explain our knowledge of mathematics. Given a nominalistic theory of logic, therefore, we can construct a nominalistic theory of mathematical truth. I shall thus argue for an epistemological thesis of logicism: any epistemology that suffices for our knowledge of logic also suffices for mathematics. (shrink)

From the Editor’s Introduction: -/- THE INTERNAL LIMITATIONS OF HUMAN UNDERSTANDING -/- We carry, unavoidably, the limits of our understanding with us. We are perpetually confined within the horizons of our conceptual structure. When this structure grows or expands, the breadth of our comprehensions enlarges, but we are forever barred from the wished-for glimpse beyond its boundaries, no matter how hard we try, no matter how much credence we invest in the substance of our learning and mist of speculation. -/- (...) The limitations in view here are not due to the mere finitude of our understanding of ourselves and of the world in which we live. They are limitations that come automatically and necessarily with any form of understanding. They are, as we shall see, part and parcel of any organization or ordering of data that we call information. -/- The consequences of these limitations are varied: As a result of them, hermeneutics cannot help but be hermetic; scientific theories of necessity are circumscribed by the boundaries of the ideas that define them; formal systems must choose between consistency and comprehensiveness; philosophical study, because it includes itself within its own proper subject matter, is forced to be reflexive in its self-enclosure. The fundamental dynamic shared by all forms of understanding testifies to an internal limitative keystone. (shrink)

From the Editor’s Introduction: "The Internal Limitations of Human Understanding." We carry, unavoidably, the limits of our understanding with us. We are perpetually confined within the horizons of our conceptual structure. When this structure grows or expands, the breadth of our comprehensions enlarges, but we are forever barred from the wished-for glimpse beyond its boundaries, no matter how hard we try, no matter how much credence we invest in the substance of our learning and mist of speculation. -/- The limitations (...) in view here are not due to the mere finitude of our understanding of ourselves and of the world in which we live. They are limitations that come automatically and necessarily with any form of understanding. They are, as we shall see, part and parcel of any organization or ordering of data that we call information. -/- The consequences of these limitations are varied: As a result of them, hermeneutics cannot help but be hermetic; scientific theories of necessity are circumscribed by the boundaries of the ideas that define them; formal systems must choose between consistency and comprehensiveness; philosophical study, because it includes itself within its own proper subject matter, is forced to be reflexive in its self-enclosure. The fundamental dynamic shared by all forms of understanding testifies to an internal limitative keystone. (shrink)

This review article provides a brief descriptive overview of past efforts to use self-referential argumentation, distinguishing pragmatical from metalogical self-referential approaches. The reviewer claims that the pragmatical self-referential argument proposed in this book is itself metalogically self-referentially inconsistent, and directs the reader to other relevant published works by the reviewer.

In lieu of an abstract, here is a brief excerpt of the content:Kennedy Institute of Ethics Journal 10.2 (2000) 175-188 [Access article in PDF] Scope Note 38 Bioethics Resources on the Web * Once described as an "enormous used book store with volumes stacked on shelves and tables and overflowing onto the floor" (Pool, Robert. 1994. Turning an Info-Glut into a Library. Science 266 (7 October): 20-22, p. 20), Internet resources now receive numerous levels of organization, from basic directory listings (...) to elaborate "links" (cross-references to other Internet documents) that flash and glow in the dark. In many ways, the Internet functions as a direct extension of the traditional library. Following linked references across various sites is an electronic version of browsing library stacks, and the serendipitous discoveries made while "surfing the Net" recall the wonders of physically going "book-to-book." Although many of the Internet's best aspects augment library strengths by eliminating physical distance from sources, Net resources pose unique problems for researchers. It is not always clear who is responsible for the content of Web documents, when the document was produced, or how often (if ever) it is updated. Internet resource tools, then, must enable researchers to identify appropriate resources and to evaluate what they find when they get there. Web Research as Taking "Snapshots" Given the Internet's visual nature and fluid content, the process of conducting research on the Web has been likened to taking an informal picture of information on a specific date at a particular time. This metaphor also conveys the "zoom-in" aspect of searching through Web pages by going from general to specific categories. In the first section of this Scope Note, National Reference Center for Bioethics Literature (NRCBL) staff present a snapshot of bioethics resources taken during the spring months of 2000. Entries are divided into the categories of directories, electronic journals, full-text documents, news/current awareness, and teaching resources. In the two subsequent sections, this series of categories is repeated for entries on a specific topic within bioethics-genetics-and again for general search and evaluation tools. Where a Web site may contain pages applicable [End Page 175] to more than one category, multiple entries are made for that site with appropriate information for each category.Much as telecommunication advances have strengthened the relationship of patients and providers as partners, the dynamic nature of the Internet acknowledges that the researcher functions as a librarian whenever accessing the Web. It is our hope that BIOETHICSLINE and the other NRCBL databases not only will enable researchers to search the Web more efficiently, but also will help them to evaluate what they find when they get there. Bioethicsline on the Web: An International Plan in Progress Once only available through libraries, the U.S. National Library of Medicine (NLM), the funding agency for the ongoing development of the BIOETHICS LINE® database from the Kennedy Institute, has provided free Web access to that database of 63,000 citations since September 1998 through Internet Grateful Med. Plans are now underway to eliminate such specialty databases and incorporate all citations into PubMed (for journal articles) or LOCATORplus (for book-like materials and chapters in books). Both of these databases are likewise available at no cost on the Web. In the transition to PubMed and LOCATORplus some features of BIOETHICSLINE that presently are available may be lost. Researchers are encouraged to contact the National Reference Center for Bioethics Literature with search requests or for search strategy assistance with either the current or the future system (medethx@georgetown.edu).The inclusion of bioethics citations in PubMed and LOCATORplus will have several advantages. First, unique keywords from the Bioethics Thesaurus will be considered for inclusion in Medical Subject Headings (MeSH), the indexing vocabulary for PubMed and LOCATORplus, thereby enhancing retrieval of relevant clinical literature by offering more precise terminology covering ethical issues. Second, it will be easier to identify foreign-language materials, which are not currently included in the scope of BIOETHICSLINE. Third, the updating schedule will become more frequent. On the other hand, there may be some loss of searching precision because certain specialized features... (shrink)

Logical form has semantic import. Logical sentences (GG: Greeks are Greeks) and their synonym interceptions (GH: Greeks are Hellenes) state the same fact but different truths with different explanations. Terms retain objectual reference but its role in explaining truth is preempted by syntax or synonymy. Church’s Test exposes puzzles. QMi sentences (GmG: ‘Greeks’ means Greeks), and QTi sentences (p≡it is true that p≡“p” is true) are metalogical necessities, true by syntax. Their interceptions alter syntax and modality, yielding contingent truths (...) (GmH: ‘Greeks’ means Hellenes, HmG: ‘Hellenes’ means Greeks). Meta-logical translation preserves syntax (GmG: ‘Greichen’ bedeutet Greichen), not necessarily objectual reference. Metalogical syntax secures truth by self-referential quotational indexing that identifies quotational referent with an intrasentential replica. (shrink)

This paper offers a novel method for nominalizing metalogic without transcending first-order reasoning about physical tokens (inscriptions, etc.) of proofs. A kind of double-negation scheme is presented which helps construct, for any platonistic statement in metalogic, a nominalistic statement which has the same assertability condition as the former. For instance, to the platonistic statement "there is a (platonistic) proof of A in deductive system D" corresponds the nominalistic statement "there is no (metalogical) proof token in (possibly informal) set (...) theory for the claim that there is no proof of A in D." And it is argued that the nominalist can use all the platonistic results by transforming them into such nominalistic correlates. (shrink)

The paper is a critique of the widespread conception of logic as a neutral arbiter between metaphysical theories, one that makes no `substantive’ claims of its own (David Kaplan and John Etchemendy are two recent examples). A familiar observation is that virtually every putatively fundamental principle of logic has been challenged over the last century on broadly metaphysical grounds (however mistaken), with a consequent proliferation of alternative logics. However, this apparent contentiousness of logic is often treated as though it were (...) neutralized by the possibility of studying all these alternative logics within an agreed metalogical framework, typically that of first-order logic with set theory. In effect, metalogic is given the role of neutral arbiter. The paper will consider a variety of examples in which deep logical disputes re-emerge at the meta-level. One case is quantified modal logic, where some varieties of actualism require a modal meta-language (as opposed to the usual non-modal language of possible worlds model theory) in order not to make their denial of the Barcan formula self-defeating. Similarly, on some views the intended model theory for second-order logic can only be given in a second-order metalanguage—this may be needed to avoid versions of Russell’s paradox when the first-order quantifiers are read as absolutely unrestricted. It can be shown that the phenomenon of higher-order vagueness eventually forces fuzzy logical treatments of vagueness to use a fuzzy metalanguage, with consequent repercussions for what first-order principles are validated. The difficulty of proving the completeness of first-order intuitionistic logic on its intended interpretation by intuitionistically rather than just classically valid means is a more familiar example. These case studies will be discussed in some detail to reveal a variety of ways in which even metalogic is metaphysically contested, substantial and non-neutral. (shrink)

A book review of _Free Choice: A Self-referential Argument_ by J. M. Boyle, Jr., G. Grisez, and O. Tollefsen. The review concerns the pragmatical self-referential argument employed in the book, and points to the fact that the argument is itself self-referentially inconsistent, but on the level of metalogical self-reference.

The problem of induction belongs to the most controversial issues in philosophy of science. If induction is understood widely, it covers every fallible inference, that is, such that its conclusion is not logically entailed by its premises. This paper analyses so-called reductive induction, that is, reasoning in which premises follow from the conclusion, but the reverse relation does not hold. Two issues are taken into account, namely the definition of reductive inference and its justification. The analysis proposed in the paper (...) employs metalogical tools. The author agrees with the view that a quantitative account of degree of confirmation for universal theories via logical probability is problematic. However, prospect for a qualitative approach look as more promising. Using the construction of maximally consistent sets allows to distinguish good and worthless induction as well as shows how to understand induction in a semantic way. A closer analysis of deductivism in the theory of justification shows that it is a hidden inductivism. (shrink)

A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. (...) We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic. (shrink)

This paper briefly discusses the relations between logic and metalogic in history. Metalogic is understood as a reflection on logic in its various senses, particularly sensu stricto (formal, mathematical) and sensu largo (formal logic plus semantic plus methodology of science). It is shown that metalogic in its contemporary understanding arose after mathematical logic had become a mature discipline. Special passage is devoted to metalogic in Poland. The last part of the paper discussed so-called logocentric predicament.

Since McCall (1966), the heterodox principle of propositional logic that it is impossible for a proposition to be entailed by its own negation—in symbols, ¬(¬φ→φ)—has gone by the name of Aristotle’s thesis, since Aristotle apparently endorses it in Prior Analytics 2.4, 57b3–14. Scholars have contested whether Aristotle did endorse his eponymous thesis, whether he could do so consistently, and for what purpose he endorsed it if he did. In this article, I reconstruct Aristotle’s argument from this passage and show that (...) he accepts this thesis. Further, I show that the argument he gives is, making plausible assumptions, a correct proof in a consistent fragmentary nonclassical metalogic for a metatheorem he previously states concerning his assertoric syllogistic. In this way, Aristotle’s argument emerges as a fascinating case study in the use of a nonclassical metalogic to prove a result about a nonclassical object system. (shrink)

Metalogical properties that have traditionally been studied in the deductive system context (see, e.g., [21]) and transferred later to the institution context [33], are here formulated in the -institution context. Preservation under deductive equivalence of -institutions is investigated. If a property is known to hold in all algebraic -institutions and is preserved under deductive equivalence, then it follows that it holds in all algebraizable -institutions in the sense of [36].

This article describes a specific pedagogical context for an advanced logic course and presents a strategy that might facilitate students’ transition from the object-theoretical to the metatheoretical perspective on logic. The pedagogical context consists of philosophy students who in general have had little training in logic, except for a thorough introduction to syllogistics. The teaching strategy tries to exploit this knowledge of syllogistics, by emphasizing the analogies between ideas from metalogic and ideas from syllogistics, such as existential import, the (...) distinction between contradictories and contraries, and the square of opposition. This strategy helps to improve students’ understanding of metalogic, because it allows the students to integrate these new ideas with their previously acquired knowledge of syllogistics. (shrink)

These remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre’s theory of pure reflection, the linchpin of the works of Sartre’s early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Gödel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception through (...) the discovery of the diagonal theorems can be recognized as a particularly clear instance of the drama of reflection according to Sartre, especially in the positing and overcoming of its proper value-ideal, viz. the synthesis of consistency and completeness. Conversely, this translation solves a number of systematic problems about pure reflection’s relations to accessory reflection, phenomenological reflection, pre-reflective self-consciousness, conversion, and value. Negative foundations, the metaphysical position emerging from this translation between existential philosophy and metalogic, concurs by different paths with Badiou’s Being and Event in rejecting both ontotheological foundationalisms and constructivist antifoundationalisms. (shrink)

The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's.First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. In this family one (...) finds systems that correspond to the associative Lambek calculus, linear logic, relevant logics, BCK logic and intuitionistic logic. Above these basic systems, sequent systems parallel to the basic systems are constructed, which formalize various notions of derived rules for the basic systems. The deduction theorem is provable for the basic systems if, and only if, they are at least as strong as systems corresponding to linear logic, or BCK logic, depending on the language, and their deductive metalogic is not stronger than they are. (shrink)

Leśniewski’s metalogic is often considered to be difficult to understand because it differs greatly from its standard formulation. In this paper I try to explain the reasons of these idiosyncrasies. I claim that they have mainly two sources. First of all there is Leśniewski’s conviction that a formal system should be conceived as a set of concrete marks that can always physically and syntactically be expanded by the addition of new theses. Secondly there is Leśniewski’s conviction that definitions should (...) neither be formulas belonging to the metalanguage, nor deduction rules, but formulas belonging to the object-language and expressed with the help of the biconditional functor. The realisation of the first point is linked to the second one in so far as the metalinguistic rule for the writing out of definitions has to be formulated in a way that makes it possible to build the formal system in agreement with Leśniewski’s conception. While explaining these points I give an overview of the main peculiarities of Leśniewski’s metalogic. (shrink)

_An Introduction to Metalogic_ is a uniquely accessible introduction to the metatheory of first-order predicate logic. No background knowledge of logic is presupposed, as the book is entirely self-contained and clearly defines all of the technical terms it employs. Yaqub begins with an introduction to predicate logic and ends with detailed outlines of the proofs of the incompleteness, undecidability, and indefinability theorems, covering many related topics in between.

This volume contains a selection of nine essays with an interdisciplinary perspective. They were originally presented at the Sixth International Conference on Word and Music Studies, which was held at Edinburgh University in June 2007 and was organized by the International Association for Word and Music Studies (WMA). The contributions to this volume focus on self-reference in various systematic, historical and intermedial ways. Self-reference - including, as a special case, metareference (the self-conscious reflection on music, literature and other (...) medial concerns) - is explored, among others, in instrumental music by Mozart, Mahler and Satie, in the structure and performance of (meta-)operas, in operatic adaptations of drama and filmic adaptations of opera, as well as in intermedial novelistic references to music. The essays cover a historical range from the 18th century to the present and are of interest to literary and opera scholars and students, musicologists as well as all readers generally interested in medial self-reference and intermediality studies. (shrink)

This volume puts forward a distinct new theory of direct reference, blending insights from both the Fregean and the Russellian traditions, and fitting the general theory of language understanding used by those working on the pragmatics of natural language.

John Campbell investigates how consciousness of the world explains our ability to think about the world; how our ability to think about objects we can see depends on our capacity for conscious visual attention to those things. He illuminates classical problems about thought, reference, and experience by looking at the underlying psychological mechanisms on which conscious attention depends.