In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

This work makes available to readers without specialized training in mathematics complete proofs of the fundamental metatheorems of standard (i.e., basically ...

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)

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)

We develop a framework for natural language semantics which handles intensionality via metalogical constructions and deals with degree truth values in an integrated way. We take an axiomatic set theory, ZF, as the foundation for semantic representations, but we make ZF a metalanguage for part of itself by embedding a language ℒ within ZF which is basically a copy of the part of ZF consisting of set expressions. This metalogical set-up is used for handling propositional attitude verbs (limited to believe (...) in this paper). We define a truth function τ which determines the truth value τ(p, T) of an ℒ-proposition p with respect to a theory T. Theories are sets of ℒ-propositions with associated truth values, and can be viewed as a (much more well-defined) replacement for possible worlds. We develop a mechanism for defining belief worlds as theories. We simultaneously develop two different versions of our system - a Boolean version where the set Ω of truth values is {0, 1}, and a degree-truth version where Ω is the interval [0, 1] of real numbers. We use degrees of truth in handling a broad class of semantic predicates that we call base-focus predicates, which include generalized quantifiers as well as many adverb and adjective senses and certain discourse-level predicates. (shrink)

A general meta-logical theory is developed by considering ontological disputes in the systems of metaphysics. The usefulness of this general meta-logical theory is demonstrated by considering the case of the ontological dispute between the metaphysical systems of Lewis’ Modal Realism and Terence Parsons’ Meinongianism. Using Quine’s criterion of ontological commitments and his views on ontological disagreement, three principles of metalogic is formulated. Based on the three principles of metalogic, the notions of independent variable and dependent variable are introduced. (...) Then, the ontological dispute between Lewis’ Modal Realism and Terence Parsons’ Meinongianism are restated in the light of the principles of metalogic. After the restatement, Independent variable and dependent variables are fixed in both Lewis’ Modal Realism and Terence Parsons’ Meinongianism to resolve the dispute. Subsequently, a new variety of quantifiers are introduced which is known as functionally isomorphic quantifiers to provide a formal representation of the resolution of the dispute. The specific functionally isomorphic quantifier which is developed in this work is known as st-quantifier. It is indicated that how st-quantifier which is one of the functionally isomorphic quantifiers can function like existential quantifier. It is also shown that there is some kind of inconsistency which is unavoidable in stating the ontological disagreement and therefore, paraconsistent logic is a requirement in stating the ontological disputes. (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 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)

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)

IT cannot be denied that metaphysics is an unpopular subject among many contemporary professional philosophers. Historically speaking, the reasons for this present-day phenomenon can be traced back to the rise and spread of philosophical Idealism over the past several hundred years. What is common to all Idealistic philosophers is the orientation adopted at the beginnings of their philosophical investigations: they attempt to unravel the mysteries of being by viewing being through ideas. Their prime concern becomes the analyses of terms and (...) propositions. Indeed, this approach is so commonplace that many thinkers cannot imagine doing things in any other way. In our age this approach shows up very clearly in the way many thinkers are perfectly willing to talk about propositions but possess a deep aversion to talking about what a thing is in itself. It is sometimes said that we currently live in a philosophical era dominated by materialism. It would seem, however, that the situation is the other way around; discussions of intramental entities take up most of the time of many philosophers, at least of non-scholastic philosophers. Most scholastic philosophers still want to talk about things i.e. extramental entities. Most of their contemporary philosophers, on the other hand, are perfectly content talking about propositions. If the latter is ever to move closer to the former group, it would seem that what is required is a more materialistic attitude on their part, in the sense of a greater concentration on things as opposed to ideas. (shrink)

With each superintuitionistic propositional logic L with a disjunction property we associate a set of modal logics the assertoric fragment of which is L . Each formula of these modal logics is interdeducible with a formula representing a set of rules admissible in L . The smallest of these logics contains only formulas representing derivable in L rules while the greatest one contains formulas corresponding to all admissible in L rules. The algebraic semantic for these logics is described.

When modeling informal proofs like that of Euclid’s Elements using a sound logical system, we go from proofs seen as somewhat unrigorous – even having gaps to be filled – to rigorous proofs. However, metalogic grounds the soundness of our logical system, and proofs in metalogic are not like formal proofs and look suspiciously like the informal proofs. This brings about what I am calling here the groundedness problem: how can we decide with certainty that our metalogical proofs (...) are rigorous and sustain our logical system? In this paper, I will expose this problem. I will not try to solve it here. (shrink)

The logic of quantum physical propositions can be established by means of dialogs which take account of the general incommensurability of these propositions. Investigated first are meta-propositions which state the formal truth of object-propositions. It turns out that the logic of these meta-propositions is equivalent to ordinary logic. A special class of meta-propositions which state the material truth of object-propositions may be considered as quantum logical modalities. It is found that the logic of these modalities contains all the quantum logical (...) restrictions and thus differs essentially from the modal calculi of ordinary logic. (shrink)

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)

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)

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].

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 π-institution context. Preservation under deductive equivalence of π-institutions is investigated. If a property is known to hold in all algebraic π-institutions and is preserved under deductive equivalence, then it follows that it holds in all algebraizable π-institutions in the sense of [36].

I develop a graph-theoretic model theory for pure and iterative grounding logics.

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.

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)

The paper defends the philosophical method of "regimentation" by example, especially in relation to the theory of mind. The starting point is the Place-Smart after-image argument: A green after-image will not be located outside the skull, but if we cracked open your skull, we won't find anything green in there either. (If we did, you'd have some disturbing medical news.) So the after-image seems not to be in physical space, suggesting that it is non-physical. In response, I argue that the (...) green blob is a fictional object, while assuming a weak sort of realism about fictional objects (where they exist as mind-dependent objects). This view can look like dualism, however, but I try to interpret it not as implying metaphysical dualism, but rather as reflecting a dualism of theory. Roughly, there can be a theory of mind-independent objects, and a theory of mind-dependent objects. Yet there are principled reasons why we cannot integrate the two into a consistent whole, for reasons related to Russell's vicious circle principle. Most of the paper motivates this by an analogy between the physicalist's theory of the world, and drawing a map with a complete representation of the map itself. Regimentation unearths the inconsistencies that arise, when trying to represent as part of a model the very representations used to define the model. (shrink)

In his preface, Hunter explains that this volume is intended to provide for non-mathematicians an introduction to the most important results of modern mathematical logic. The reader will find here the work of Post, Skolem, Gödel, Church, Henkin, and others, presented in a terse and closely-knit style. Though acknowledging the trend toward natural deduction systems, Hunter sticks to more classical axiomatic systems on the grounds that the proofs of metatheorems are simplified by that choice. He begins with a formal system (...) of propositional logic, for which he develops proofs of completeness, consistency, and decidability. Systems of first order predicate logic are then dealt with, and Church's and Gödel's results on undecidability and incompleteness are presented. There is a brief, useful discussion indicating what parts of first order predicate logic are decidable. It is assumed that the reader has some familiarity with elementary formal logic: Hunter develops the set-theoretical results he requires. The compactness of the presentation is facilitated by the use of earlier results for propositional logic to obtain certain results for predicate logic. Especially helpful is Hunter's technique of prefacing each of the longer and more difficult proofs with a section explaining the key ideas and the general method of proof; summaries are also often provided in conclusion. Because of the orientation of the book, a more extensive discussion of the significance of the results obtained and the philosophical issues which have been raised concerning some of them, would have been welcome. Nonetheless, this is a good, concise introduction for those who are not yet prepared to tackle the more technical literature. A bibliography is provided for those interested in pursuing the subject in greater detail.--E. M. F. (shrink)

This paper examines two arguments againstpsychologism advanced by Frege andHusserl. The first argument says that thelaws of logic cannot be justified by thelaws of psychology, because the formerand a priori and certain, but the latterare probable only. The second argumentpoints out that the status of logicallaws as universal principles of thinking isnot intelligible on the psychologisticinterpretation of logic. The author tries toshow how to examine both arguments bymetalogical devices.

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)

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (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)