Philosophy of Logic

Classical recapture

by Andrew Aberdein

In Prospettive della logica e della filosofia della scienza, V. Fano, M. Stanzione, & G. Tarozzi, edd. (Catanzaro: Rubettino, 2001), pp. 11–18.

The recapture relationship is an important element to any understanding of the connexion between different systems of... more The recapture relationship is an important element to any understanding of the connexion between different systems of logic. Loosely speaking, one system of logic recaptures another if it is possible to specify a subsystem of the former system which exhibits the same patterns of inference as the latter system. In particular if a relationship of this kind can be shown to exist between a non-classical logic and classical logic, the non-classical system is said to exhibit classical recapture. This has been invoked by several proponents of non-classical logics to argue that their system retains classical logic as a limit case, and is therefore a methodologically progressive successor to classical logic. In this paper I advance and defend a new and more precise account of recapture and the character of its reception by the proponents of the recapturing system. I then indicate some of the applications of classical recapture which this account makes possible.

The Philosophy of Alternative Logics

by Andrew Aberdein

Co-authored with Stephen Read. In Development of modern logic, L. Haaparanta, ed. (Oxford: Oxford University Press, 2009), pp. 613-723.

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that... more This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals.

Keywords: classical logic, logical theory, intuitionistic logic, quantum logic, relevance logic, paraconsistent logic

How Fundamental is the Fundamental Assumption?

by Nils Kürbis

published in Teorema XXXI/2 (2012), pp.5-19.

The fundamental assumption of Dummett’s and Prawitz’ proof-theoretic justification of deduction is that ‘if we have a... more The fundamental assumption of Dummett’s and Prawitz’ proof-theoretic justification of deduction is that ‘if we have a valid argument for a complex statement, we can construct a valid argument for it which finishes with an application of one of the introduction rules governing its principal operator’. I argue that the assumption is flawed in this general version, but should be restricted, not to apply to arguments in general, but only to proofs. I also argue that Dummett’s and Prawitz’ project of providing a logical basis for metaphysics only relies on the restricted assumption.

The Stoic Anomaly: An Inquiry into Some Possible Semitic Components in Stoic Logic and Physics (Spanish)

by Carlos Segovia

"La anomalía estoica: En torno a los posibles componentes semíticos de la lógica y la física estoicas," Paideia 89 (2010) 295-307.

1. Introducción
2. La anomalía lingüístico-temporal (sobre la relativa indistinción del presente y el futuro en... more 1. Introducción
2. La anomalía lingüístico-temporal (sobre la relativa indistinción del presente y el futuro en el estoicismo)
3. La anomalía ontológica (sobre la supresión del verbo "ser" en la física estoica)
4. La anomalía lógica (sobre la supresión de la cópula verbal en la lógica estoica)
5. A modo de conclusión

Misyurov D.A. Dialectical formulas based on the binary notation as the development formulas // Credo New. 2012. №2

by Dmitry Misyurov

The article suggests dialectical formulas based on the binary notation as the development formulas: formula with... more The article suggests dialectical formulas based on the binary notation as the development formulas: formula with dominant and the non-dominant elements; universal formula; formula with symbolic weight of elements; tautological formula. For example, it suggests an opportunity to use the dialectical formulas for modeling and artificial intelligence creation, etc.

Die dreifache Stellung der Reflexion zu ihren Gründen in Hegels Logik

by Stefan Schick

Semantic Conception of Truth. What It Is and What It Is Not

by Ladislav Koren

This is a slightly corrected and revised version of the thesis for PhD degree: "Truth and Meaning: the Dialectics of Theory and Practice".

Alfred Tarski’s semantic conception of truth is arguably the most influential – certainly, most discussed - modern... more 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.

Il n’y a pas de rapport sexuel: The Irresolvability of the Gadamer-Habermas Debate

by William J. Urban

class paper written Good Friday, April 6, 2012

Der phänomenologische Ursprung des Logischen: E. Krit. Analyse d. Phänomenolog. Grundlegung d. Logik in Edmund Husserls" Logischen …

by Bernward Grünewald

Identity, Possibility, Rigid Designators / Identità, possibilità, designatori rigidi [English and Italian Version]

by antonio caronia

Published in: NAME Readymade, Moderna galerija, Ljubljuana 2008.

Considerations about the "JJJ project" (three Slovenian artists changing their names in Janez Janša - the... more Considerations about the "JJJ project" (three Slovenian artists changing their names in Janez Janša - the Slovenian Prime Minister at the time - in 2007) from a philosophic, linguistic and logical standpoint, published in the book related to the exhbition "NAME Readymade", Ljubljana 2008. The subtitle ("On Formally Undecidable Propositions of Janez Janša and Concerning Systems") is an adaptation of the one of Kurt Goedel's paper about his first theorem demonstrating the incompleteness of arithmetic.

Considerazioni sul "Progetto Janez Janša" (tre artisti sloveni che nel 2007 cambiarono il proprio nome assumendo quello dell'allora primo ministro) dal punto di vista filosofico, logico e linguistico, contenuto nel libro/catalogo della mostra "NAME Readymade" tenuta a Ljubljana nel 2008. Il sottotitolo del saggio ("Sulle proposizioni formalmente indecidibili di Janez Janša e dei sistemi relativi") è un calco di quello del famoso saggio di Kurt Goedel del 1931 in cui si dimostrava il primo teorema di incompletezza dei sistemi formali.

Schroyens, W. (2010). Reasoning from or reasoning about beliefs: Truth-based or possibility-based compatibility judgments and Handley et al.’s (2006) litmus test of the suppositional conditional. Current Psychology letters [Online], 25 (3).

by Walter Schroyens

LÓGICA FORMAL E TRANSCENDENTAL. Kant e a questão das relações entre intuição e conceito no juízo

by Luciano Codato

According to Kant, logic has a transcendental nature insofar as the content of representations is concerned, and not... more According to Kant, logic has a transcendental nature insofar as the content of representations is concerned, and not only the form of the relation between them. Despite this apparent feature, there has been some misunderstanding concerning the fact that transcendentality and formality do not correspond to two distinct logics, but are both attributes of the same science of the "mere form of thought", the basic operation of which is judgment. The consequence of this misunderstanding is the complete omission of the conditions that make the transcendentalization of logic possible according to Kant's interpretation of the form of judgment. Contrary to this dominant approach, if it is possible to elicit not only the mode whereby concepts are related in judgment, but also the sense of the form of the concept, it may be possible to understand under what conditions reference to a very particular kind of object is necessary to logic, notwithstanding its strictly formal nature.

Extensão e forma lógica na Crítica da razão pura

by Luciano Codato

Since the logical form of the judgment is interpreted by Kant as a subordination of extensions, how to understand its... more Since the logical form of the judgment is interpreted by Kant as a subordination of extensions, how to understand its relation to an unknown = x? Against traditional interpretations, elaborated from the background of analytical philosophy or of Port-Royal Logic, one intends to recover the specificity of Kant’s notion of extension, so that one can comprehend the judicative relation between universal and singular. Within this context, two relations performed in the judgment are distinguished: the predicative relation between the superior concept P and the inferior concept S, and the non-predicative relation between the intuition of something individual = x and the universals S and P. As a result, besides a reflective activity on the grounds of the Transcendental Analytic, one tries to identify the conception of reason which Kant presupposes as a given.

Judgment, extension, logical form

by Luciano Codato

http://www.degruyter.com/view/supplement/9783110210347_Inhaltsverzeichnis_Vol5.pdf

In Kant’s logical texts the reference of the form of the judgment to an “unknown = x” is well known, but its... more In Kant’s logical texts the reference of the form of the judgment to an “unknown = x” is well known, but its understanding remains far from consensual. Due to the universality of all concepts, the subject as much as the predicate, in the form S is P, is regarded as predicate of the x, which, in turn, is regarded as the subject of the judgment. In the CPR, particularly in the text on the “logical use of the understanding”, this Kantian interpretation of the subject-predicate relation leads to the question about the relations that must hold between intuition and concept in the judgment. In contrast to intuition, if no concept, due to its universal character, refers immediately to an object, how should we understand the relations of subject and predicate to one another, as well as their relations to intuition, which corresponds to the very special individuality of that object in general = x? In the Kant-Literatur, the relations between intuition and concept in the judgment have been considered in diverse theoretical backgrounds, mainly in Fregean logic and in the logic of Port-Royal. Although so markedly different, these two solutions to the problem above seem to share a common thesis, in so far as they claim, though in different ways, a predicative character to those relations. If the analytic tradition recognizes in the relation between x and the concept S the marks of a propositional function Sx, in turn, the interpretation elaborated from the background of Port-Royal recognizes in this relation the minor premise x is S implicit in the judgment every S is P. This being the case, if it were possible to prove, on the contrary, that the relations between intuition and concept in the judgment could only be of a non-predicative character, then a third solution would be open to us, a solution that could enable us to track down the sense of the conceptions of judgment and logical form in the CPR. In applying this argumentative strategy, it is of the utmost importance to insist on the specificity of Kant’s notion of extension, in order to prove its irreducibility to the Port-Royal notion of extension as well as to the modern one.

The structure of logical consequence: proof-theoretic conceptions

by Ole Hjortland

PhD Thesis, University of St Andrews, 2009

The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple... more The model-theoretic analysis of the concept of logical consequence has come under heavy criticism in the last couple of decades. The present work looks at an alternative approach to logical consequence where the notion of inference takes center stage. Formally, the model-theoretic framework is exchanged for a proof-theoretic framework. It is argued that contrary to the traditional view, proof-theoretic semantics is not revisionary, and should rather be seen as a formal semantics that can supplement model-theory. Specifically, there are formal resources to provide a proof-theoretic semantics for both intuitionistic and classical logic.

We develop a new perspective on proof-theoretic harmony for logical constants which incorporates elements from the substructural era of proof-theory. We show that there is a semantic lacuna in the traditional accounts of harmony. A new theory of how inference rules determine the semantic content of logical constants is developed. The theory weds proof-theoretic and model-theoretic semantics by showing how proof-theoretic rules can induce truth-conditional clauses in Boolean and many-valued settings. It is argued that such a new approach to how rules determine meaning will ultimately assist our understanding of the apriori nature of logic.

Harmony, Normality and Stability

by Nils Kürbis

As the title says: my account of proof-theoretic harmony, normality and stability!

Gentzen mentions that it should be possible to specify a function that maps introduction rules onto elimination rules... more Gentzen mentions that it should be possible to specify a function that maps introduction rules onto elimination rules in systems of natural deduction. This paper specifies such a function. I specify two kinds of rules, one in which it is more natural to assume an introduction rule to be given and elimination rules are determined from it, and another kind in which it is an elimination rule which is given and the introduction rules are determined from it. The process also works the other way round, so that it doesn't really matter which rules are supposed to be given first. The process is very general and applies to a large class of logics. The paper begins with a discussion of the philosophical importance of this in connection with the notion of harmony. I discuss Dummett's ideas on harmony and stability, which is supposed to be stronger than harmony. Dummett suggests that normalisability is a formal criterion of harmony. However, he seems to aim at something else, and this criterion does not give an independent formally precise notion of stability. I propose formally precise definitions of harmony and stability, which are distinct from normalisability. My aim is not exegetical, and according to my definitions, classical as well as intuitionist logics count as governed by stable (and hence harmonious) rules of inference.

Stable Harmony

by Nils Kürbis

A talk I gave at Logica 2007. Not sure where the title came from ...

Negation: A Problem for the Proof-Theoretic Justification of Deduction

by Nils Kürbis

I won the Jacobsen Essay Price of the University of London for this essay!

What is Wrong with Classical Negation?

by Nils Kürbis

The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on... more The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on consideration about the meanings of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett’s outlook on the theory of meaning. In particular, I shall grant Dummett his general bias towards verificationism or justificationism, encapsulated in the slogan ‘meaning is use’. The second general assumption I see no need to question is Dummett’s particular breed of molecularism. Some of Dummett’s assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inferences in the Dummettian framework.

Remarks on some general features of abduction

by Luis Estrada-González

If you are interested in a pre-print please e-mail me.

Overemphasizing the features of abduction as it occurs in scientifi c practices and daily life scenarios has led to... more Overemphasizing the features of abduction as it occurs in scientific practices and daily life scenarios has led to overlook some features that abduction in those circumstances shares with other phenomena in which some given outputs fail to stand in a certain relation with some given inputs, and thus a modification on those inputs is in order. We propose here a top-down, conceptual and taxonomic investigation on what the most general purely logical features of abduction could be, as well as a research program to investigate to what extent it is a pervasive notion in logic. We start by motivating some broadenings in the most common approaches to abduction, then we characterize inferential problems and finally give general characterizations of the notions of abductive problem, abductive solution and abductive inference.