Logical Constants

Alfred Tarski’s semantic conception of truth is arguably the most influential – certainly, most discussed - modern conception of truth. It has provoked many different interpretations and reactions, some thinkers celebrating it for successfully explicating the notion of truth, whereas others have argued that it is no good as a philosophical account of truth. The aim of this work is to offer a systematic and critical investigation of its nature and significance, based on the thorough explanation of its conceptual, technical as well as historical underpinnings.

The methodological strategy adopted in the thesis reflects the author’s belief that in order to evaluate the import of Tarski’s conception we need to understand what logical, mathematical and philosophical aspects it has, what role they play in his project of theoretical semantics, which of them hang in together, and which should be kept separate. Chapter 2 therefore starts with a detailed exposition of the conceptual and historical background of Tarski’s semantic conception of truth and his method of truth definition for formalized languages, situating it within his project of theoretical semantics, and Chapter 3 explains the formal machinery of Tarski’s truth definitions for increasingly more complex languages. Chapters 4 - 7 form the core of the thesis, all being concerned with the problem of significance of Tarski’s conception. Chapter 4 explains its logico-mathematical import, connecting it to the related works of Gödel and Carnap. Having explained the seminal ideas of the model-theoretic approach to semantics, Chapter 5 tackles the question to what extent Tarski’s ‘The Concept of Truth in Formalized Languages’ (and related articles from the 1930s) anticipates this approach, and what elements might be missing from it. Chapter 6 then deals with the vexed question of its philosophical import and value as a theory of truth, reviewing a number of objections and arguments that purport to show that the method fails as an explanation (explication) of the ordinary notion of truth, and, in particular, that it is a confusion to think that Tarski’s truth definitions have semantic import. Finally, Chapter 7 is devoted to the question whether Tarski’s theory of truth is a robust or rather a deflationary theory of truth.

On the basis of a careful analysis, the thesis aims to substantiate the following view. [A] Tarski’s theory with its associated method of truth definition was primarily designed to serve logico-mathematical purposes. [B] It can be regarded a deflationary theory of a sort, since it completely abstracts from meta- semantical issues concerning the metaphysical or epistemological basis or status of semantic properties. Indeed, [C] this can be interpreted as its laudable feature, since by separating formal (or logico-mathematical) from meta-semantical (or foundational) aspects it usefully divides the theoretical labour to be done in the area of meaning and semantic properties in general. [D] In spite of the fact that Tarski’s conception of truth has this deflationary flavour, the formal structure of its method of truth-definition is quite neutral in that it can be interpreted and employed in several different ways, some of them deflationary, others more robust.

The main of the paper is to discuss the meaning of negation involved in the logic of First Degree Entailment in the context of a dialogical analysis of tonk-like operators The dialogical framework is an approach to meaning that provides an alternative to both the model-theoretical and the proof-theoretical semantics. However, dialogic had and still has a bias towards antirealism. The dialogical approach to logic is not a logic but a semantic rule-based framework where different logics could be developed, combined or compared. But are there any constraints? Can we introduce rules ad libitum to define whatever logical constant? This study should provide the first conceptual moves towards the notion of Dialogical Harmony.

CITE AS: Corcoran, John. 1999.  “Laws of thought”.  Cambridge Dictionary of Philosophy.  R. Audi, Ed.  Cambridge: Cambridge UP. p. 489.

Laws of thought are laws by which or in accordance with which valid thought proceeds, or that justify valid inference, or to which all valid deduction is reducible. Laws of thought are rules that apply without exception to any subject matter of thought, etc.; sometimes they are said to be the object of logic. The expression, rarely used in exactly the same sense by different authors, has long been associated with three equally ambiguous expressions: the law of identity (ID), the law of contradiction (or non-contradiction; NC), and the law of excluded middle (EM).
The so-called “logical truths” of standard first-order logic as discussed in mainstream logic works such as Corcoran-Masoud 2014 and 2015 are uninterpreted sentences and thus not even true.
Nevertheless, they too have been called laws of thought even though they are not laws in any normal sense. Other people unacquainted with traditional usage are inclined to think that a law of thought is a law that might be arrived at by thought, which could make ‘law of thought’ coextensive with ‘law’.
Sometimes, these three expressions—ID, NC, and EM— are taken as propositions of formal ontology having the widest possible subject matter, propositions that apply to entities per se: (ID), every thing is (i.e., is identical to) itself; (NC) no thing having a given quality also has the negative of that quality (e.g., no even number is non-even); (EM) every thing either has a given quality or has the negative of that quality (e.g., every number is either even or non-even).

1981. A note on categoricity and completeness, History and Philosophy of Logic 2, 113–19.

Current study of axiomatic method presupposes concepts and results of string theory and set theory. But axiomatic method was vigorously pursued in the last half of the nineteenth century without explicit reference to the theory of strings, which was not axiomatized until the 1930s (independently by Tarski 1934 and by Hermes
1938). For historical remarks on string theory see Corcoran, Frank and Maloney1974. Moreover, although it is natural to read modern set-theoretic ideas in early methodological work, it is clear that early postulate theorists such as Dedekind, Peano, Hilbert, Huntington and Veblen were not working in any formal set-theoretic framework. Indeed, close correspondence between modern precise explications of methodological concepts and imprecise formulations found in the writings of early postulate theorists suggests that the creators of modern mathematical logic, particularly Carnap, Tarski, and Church, may have been guided by the goal of putting the early ideas on a more mathematically precise basis.

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions, we argue that a proper account of conditional can be obtained by extending the logical notation of Frege's ideography. From a dialogical redefinition of truth-values as moves in a game, it becomes possible to characterize the logical meaning of " If " , and only " If ". That is: by getting rid of the paradoxes of material implication, whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections. Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed in this algebraic and game-theoretical light.

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions, we argue that a proper account of conditional can be obtained by extending the logical notation of Frege’s Begriffsschrift. After discarding his Equivalence Thesis, according to which the denial of a sentence equates with the affirmation of its negation, it becomes possible to characterize the logical meaning of “if”, and only “if”. That is: without irremediably leading from entailment into formal implication, whilst sticking to a bivalent definition of conditional as a speech-act commitment. Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed in this new light.

Este texto presenta, y en cierta medida analiza, ambigüedades existentes en textos de lógica y filosofía de la lógica (como la interpretación de los llamados principios, postulados, leyes o verdades lógicas, la coexistencia de la tesis de que toda relación presupone  la existencia de al menos dos relata y la de que una cosa puede relacionarse consigo misma, o la llamada "paradoja del mentiroso") bajo el supuesto de que, dado  que la lógica no es anterior a la semántica, un análisis semántico puede mostrar no solo que tal polisemia sugiere que ni la disciplina ni las teorías lógicas son tan rigurosas como se ha afirmado, sino también que varias  afirmaciones de esas disciplinas carecen de sentido.

The well-known necessary-consequence/necessary-consequent ambiguity—a species of segmentation, scope, and structural ambiguities—is found in conditional sentences such as:

(A) If zero is oblong, then necessarily some square is oblong.

A can read as expressing a proposition to the effect that “some square is oblong” is a necessary consequence of “zero is oblong”. This necessary-consequence segmentation parses A with the adverb as part of the discontinuous modal connective ‘if … then necessarily’. The consequent ‘some square is oblong’ is non-modal. Alternatively, the adverb can be taken instead as part of the consequent yielding the necessary-consequent segmentation: A can read as expressing roughly that the modal proposition “necessarily some square is oblong” is a material consequence of “zero is oblong”. Brackets indicate segmentation:

Necessary-consequence segmentation
[If] zero is oblong, [then necessarily] some square is oblong.

Necessary-consequent segmentation
If zero is oblong, then [necessarily some square is oblong].

Similar confusing ambiguities arise in logic texts where ‘valid’, ‘logical’, ‘necessary’, ‘sound’, ‘derivable’, and the like occur before ‘conclusion’, ‘inference’, ‘deduction’ and the like.

This paper, supplementing [1], compares corresponding segmentations of various conditional sentences. E.g., taking both “necessities” as logical, the following have different truth-values—first true, second false. The second implies the first.

[If] Twain is Clemens, [then necessarily] Clemens is Twain.
If Twain is Clemens, then [necessarily Clemens is Twain].

The first says that “Clemens is Twain” follows logically from “Twain is Clemens”; the second that “Clemens is Twain” is logically necessary if Twain is Clemens.

Theory of Knowledge - International Baccalaureate Essay
Subject: “A model is a simplified representation of some aspect of the world. In what ways may models help or hinder the search for knowledge?”

Author: Angel Versetti
Word-count: 1599 Words

Pretendemos, en esta parte, primeramente establecer una caracterización de la Lógica Matemática por comparación con la Lógica Clásica. Así hablaremos de la Lógica Matemática como una ciencia positiva, de su hiperformalismo, como de su pandeductivismo y a-metafisicismo. Luego, una vez caracterizada la Lógica Matemática, estableceremos finalmente, qué tipo de relación existe entre ambas lógicas y el fundamento de esa relación. Indagamos la relación comparando a ambas lógicas desde tres perspectivas. De esta comparación surgen, según nuestra apreciación, la siguiente caracterización, que no pretende ser ni excluyente ni exclusiva. Pero sí pretendemos que sea lo suficiente manifestativa de la realidad lógico-matemática.
La Lógica matemática es:
1. Es una ciencia positiva;
2. Es hiperformalista o lógico-formalista;
3. Es pandeductivista;
4. Es un automatismo irreflexivo;
5. Es a-metafisicista.

This article is focused on answering the question to what extent one can be a sceptic. Sextus Empiricus’s Outlines of Scepticism serves as a guide. In section 1, it is investigated whether three logical laws have a certain foundation or are subject to doubt. In section 2, Sextus’s way to deal with these laws is examined; the question arises how dogmatic his approach is. After that, a possible ‘reply’ by Sextus to the criticism receives attention. Section 3 is concentrated on a possible alternative to Sextus’s approach. Besides logical laws, some important methods are concerned.

In the 1970s the Chilean biologist Francisco Varela developed a systems theory of life as autopoietic, or self-creating, on the formal basis of the mathematician George Spencer-Brown's 1969 book Laws of Form. Although Varela saw the great power and simplicity of Spencer-Brown' s calculus, he appropriated Spencer-Brown's calculus by adding a third law to the original two laws of form. This law makes self-reference explicit, and I'll argue that it is a redundant addition because self-reference is already implied by indication. The addition of this redundancy has the effect of separating the transcendental realm of constitutive operations from the natural realm of effects. Anna-Teresa Tymieniecka's phenomenology of life as ontopoiesis, on the other hand, succeeds in developing a theory of life on the formal ontological basis prepared by Husserl's phenomenology, transcendental logic, and the living tradition of a mathesis universalis. Tymieniecka's phenomenological theory of life is a breakthrough to life's inner workings, and rests on an ontologically robust foundation. By drawing attention to the calculus behind the calculus, I hope to initiate the interpretation of Spencer-Brown's calculus of indications as an important manifestation of the mathesis universalis, and apply the interpretation to the place prepared by Tymieniecka's phenomenology of ontopoiesis.

This essay discusses Wittgenstein's conception of logic, early and late, and some of the types of logical system that he constructed. The essay shows that the common view according to which Wittgenstein had stopped engaging in logic as a philosophical discipline by the time of writing Philosophical Investigations is mistaken. It is argued that, on the contrary, logic continued to figure at the very heart of later Wittgenstein's philosophy; and that Wittgenstein's mature philosophy of logic contains many interesting thoughts which have gone widely unnoticed.

What is a proof? What are the formal relations between proof and truth? What kind of paradoxes and possible errors do arise from an unsound use of such notions?
The First Lecce Workshop in Philosophy of Logic (Filog), named "Truth, Proof and Paradox", will try to provide possible answers to these questions. Filog aims at formulating an explication of the concepts of proof and truth, in the light of the recent philosophical and logical methodology, in order to evaluate their formal and material adequacy.
A special emphasis will be given to different varieties of semantic realism and antirealism in philosophy of logic and to their relations with the systems of formal pragmatics.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ must always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. MSC: 11-XX (Number Theory)

I argue against inferentialism about logic. First, I argue against an analogy between logic and chess, before considering a more basic objection to stipulating inference rules as a way of establishing the meaning of logical constants. The objection-the Mushroom Omelette Objection-is that stipulative acts are partly constituted by logical notions, and therefore cannot be used to explain logical thought. I then argue that the same problem also attaches to following existing conventional rules, since either those rules have logical contents, or following those conventional rules is done for logical reasons. Lastly, I compare this argument with other arguments found in Quine's early work, and consider two attempts to reply to Quine.

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications.

Three classic distinctions specify that truths can be necessary versus contingent, analytic versus synthetic, and a priori versus a posteriori. The philosopher reading this article knows very well both how useful and ordinary such distinctions are in our conceptual work and that they have been subject to many and detailed debates, especially the last two. In the following pages, I do not wish to discuss how far they may be tenable. I shall assume that, if they are reasonable and non problematic in some ordinary cases, then they can be used in order to understand what kind of knowledge the maker's knowledge is. By this I mean the sort of knowledge that Alice enjoys when she holds the information (true content) that Bob's coffee is sweetened because she just put two spoons of sugar in it herself. The maker's knowledge tradition is quite important but it is not mainstream in modern and analytic epistemology and lacks grounding in terms of exactly what sort of knowledge one is talking about. My suggestion is that this grounding can be provided by a minimalist approach, based on an information-theoretical analysis. In the article, I argue that (a) we need to decouple a fourth distinction, namely informative versus uninformative, from the previous three and, in particular, from its implicit association with analytic versus synthetic and a priori versus a posteriori; (b) such a decoupling facilitates, and is facilitated by, moving from a monoagent to a multiagent approach: the distinctions qualify a proposition, a message, or some information not just in themselves but relationally, with respect to an informational agent; (c) the decoupling and the multiagent approach enable a re-mapping of currently available positions in epistemology (Classic, Innatist, Kant's and Kripke's) on these four dichotomies; (d) within such a re-mapping, two positions, capturing the nature of a witness' knowledge and of a maker's knowledge, can best be described as contingent, synthetic, a posteriori, and uninformative and as
contingent, synthetic, weakly a priori (ab anteriori), and uninformative respectively. In the conclusion, I indicate why the analysis of the maker’s knowledge has important consequences in all those cases in which the poietic (constructive) intervention on a system determines the truth of the model of that system.

Contemporary accounts of the possession conditions for basic logical concepts rely on either the subject being primitively compelled by transitions that are isomorphic to the introduction and elimination rules for the logical constant or the rules are an innate property of the subject’s language of thought. Drawing from Heidegger’s work in the philosophy of logic and Boghossian’s work on inference, I argue that any attempt to explain logical inference as a form of rule following will lead to arguments for the impossibility of logic or an infinite regress. An under-explored alternative to defining all inferences as instances of following a rule is to take some inferences as basic – inferences that do not follow rules and provide a starting point for our inferential practices. I propose that neural networks can provide a useful model for understanding basic inferences as a primitive ability since they show how one can infer in accordance with a logical principle without necessarily following a rule. Determining whether a subject possesses a basic logical concept is arrived at then by interpreting their inferential behaviour.

In: Wittgenstein and Hegel. Reevaluation of Difference. Jakub Mácha and Alexander Berg (ed.), Boston/Berlin, De Gruyter (On Wittgenstein), 2019, pp. 291-309. https://www.degruyter.com/view/product/496759

Wittgenstein opens his preface to the TLP with the following words: “Perhaps this book will be understood only by someone who has himself already had the thoughts that are expressed in it—or at least similar thoughts. […] Its purpose would be achieved if it gave pleasure to one person who read and understood it.” In this chapter, I shall take this invitation literally and try to interpret the TLP by reference to the reader’s thoughts and understanding, imagining this reader to be Hegel. To this end, I give an anachronistic “Hegelian reading of the TLP” in which I, firstly, focus on the thoughts—and not mere propositions, as Wittgenstein puts it in his preface—expressed in the Tractatus and, secondly, examine how Hegel’s SL could be seen as a response to the TLP. To cut a long story short, the most important Tractarian thought from a Hegelian point of view would be “the sole logical constant” (TLP: 5.47), which the SL refers to as the Concept.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]

JOHN CORCORAN’S JUNE 2015 ANNOTATIONS OF TARSKI’S 1994 INTRODUCTION TO LOGIC.
The immediate purpose of this posting is to provide information about Tarski’s use of ‘equals’, ‘is’, is identical to’, ‘=’, and the like to participants in the session on FORMALIZING EUCLID’S FIRST AXIOM Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný).

Here is a tidbit from page 56.
In this book we consider the notion of equality among numbers always as a special case of the general concept of logical identity. One should add, however, that there have been (and perhaps still are) mathematicians who—as opposed to the standpoint adopted here—did not identify the symbol "=" occurring in arithmetic with the symbol of logical identity; they did not consider equal numbers to be necessarily identical, and therefore looked upon the notion of equality among numbers as a specifically arithmetical concept. 1994/61

This book is alleged to be a comprehensive but largely elementary description of mathematical logic including its historical development, its most important achievements and its implications for philosophy. Although the intended audience includes both mathematicians
and philosophers, 'the book presupposes no knowledge of logic and only high school mathematics". The author avows an aim "to stimulate long-lived interest by communicating some of the excitement and beauty of the subject". Details and techniques are eschewed in favor of emphasis on the development of an overview.
The book is also intended by the author to serve as an introduction to logic. His opinion of the shortcomings of most introductions to logic is perceptive and insightful.

This is a survey paper in German covering a number of central themes in the philosophy of logic, including the debates surrounding the notion of logical consequence, the problem of demarcating logic, logical pluralism vs. logical monism and the question of the normativity of logic.