Are Logical Languages Compositional?

In this paper I argue that in contrast to natural languages, logical languages typically are not compositional. This does not mean that the meaning of expressions cannot be determined at all using some well-defined set of rules. It only means that the meaning of an expression cannot be determined without looking at its form. If one is serious about the compositionality of a logic, the only possibility I see is to define it via abstraction from a variable free language.     

Hintikka’s Logical Revolution

Hintikka thinks that second-order logic is not pure logic, and because of Gödel’s incompleteness theorems, he suggests that we should liberate ourselves from the mistaken idea that first-order logic is the foundational logic of mathematics. With this background he introduces his independence friendly logic (IFL). In this paper, I argue that approaches taking Hintikka’s IFL as a foundational logic of mathematics face serious challenges. First, the quantifiers in Hintikka’s IFL are not distinguishable from Linström’s general quantifiers, which means that the quantifiers in IFL involve higher order entities. Second, if we take Wright’s interpretation of quantifiers or if we take Hale’s criterion for the identity of concepts, Quine’s thesis that second-order logic is set theory will be rejected. Third, Hintikka’s definition of truth itself cannot be expressed in the extension of language of IFL. Since second-order logic can do what IFL does, the significance of IFL for the foundations of mathematics is weakened.     

Logical structuralism and Benacerraf’s problem

There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by a corresponding account of how it is that we acquire knowledge of those objects. The connection between these problems results in what is often called “Benacerraf’s Problem”, which is a dilemma that many philosophical views about mathematical objects face. It will be my goal here to present a view, attributed to Richard Dedekind, which approaches the initial questions in a different way than many other philosophical views do, and in doing so, avoids the dilemma given by Benacerraf’s problem.     

Walter Pitts and “A Logical Calculus”

Many years after the publication of “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Warren McCulloch gave Walter Pitts credit for contributing his knowledge of modular mathematics to their joint project.

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A New–old Characterisation of Logical Knowledge

We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ‘sortal terms’, two theories that will feature prominently. Second, we propose that logic comprises four ‘momental sectors’: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ‘external’ negation not-R; and the assertion of R in the pair of propositions ‘it is (un)true that R’ belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic among other foundational studies: set theory and other theories of collections, metamathematics, axiomatisation, definitions, model theory, and abstract and operator algebras. Fifth, we test this characterisation in two important contexts: the formulation of some logical paradoxes, especially the propositional ones; and indirect proof-methods, especially that by contradiction. The outcomes differ for asserted propositions from those for unasserted ones. Finally, we reflect upon self-referring self-reference, and on the relationships between logical and mathematical knowledge. A subject index is appended.     

Rayo’s The Construction of Logical Space

Abstract

I wonder which one in a series of characters Agustín Rayo really is, with an emphasis on objective correctness and semantics.     

A New Interpretation of Carnap’s Logical Pluralism

Topoi - Rudolf Carnap’s logical pluralism is often held to be one in which corresponding connectives in different logics have different meanings. This paper presents an alternative view of...     

The Logical Style of Confucius’ Analects

Considered from a logical point of view, Confucius’ Analects contain many implicit forms of reasoning and argumentation. This is shown first by analyzing the phenomenon of parallelism: direct parallelism is often a way of hinting at a general assertion, whereas anti-parallelism hides dilemmas, generalizations and modal notions of “moral preference.” The Analects also have various types of conditionals, ranging from material implications, to modalized implications, and counterfactual conditionals, which are the germs of implicit reasoning, concluding with a moral recommendation. Analogies are particularly abundant and a presentation of three examples suggests that, beyond their explicative role, they also involve moral recommendations. The implicit logic of The Analects requires an active, albeit unconscious participation of the reader, which could be an important element in explaining the enduring influence of the text.     

Can First-Order Logical Truth be Defined in Purely Extensional Terms?

W. V. Quine thinks logical truth can be defined in purely extensional terms, as follows: a logical truth is a true sentence that exemplifies a logical form all of whose instances are true. P. F. Strawson objects that one cannot say what it is for a particular use of a sentence to exemplify a logical form without appealing to intensional notions, and hence that Quine's efforts to define logical truth in purely extensional terms cannot succeed. Quine's reply to this criticism is confused in ways that have not yet been noticed in the literature. This may seem to favour Strawson's side of the debate. In fact, however, a proper analysis of the difficulties that Quine's reply faces suggests a new way to clarify and defend the view that logical truth can be defined in purely extensional terms.     

On Kripke’s Dogmatism Paradox: A Logical Dynamical Analysis

As a byproduct of solving the surprise-exam paradox, Saul Kripke formulates a “dogmatism paradox” which seems to show that knowledge entails dogmatism. In this paper, the author analyzes the nature of the dogmatism paradox from a logical dynamical perspective. The author suggests that the dogmatism paradox is better understood as a paradox of knowledge attribution rather than of knowledge. Therefore, the dogmatism paradox could be solved without sacrificing the principle of epistemic closure. Based on a famous version of relevant alternatives theory, the author formalizes a logic of knowledge attribution in the style of logical dynamics, namely, public retraction logic, and analyzes how knowledge attributions are retracted with the expansion of relevant alternatives.     

Tūsī on Avicenna’s Logical Connectives ∗

Tūsī, a thirteenth century logician writing in Arabic, uses two logical connectives to build up molecular propositions: ‘if-then’, and ‘either-or’. By referring to a dichotomous Tree, Tūsī shows how to choose the proper disjunction relative to the terms in the disjuncts. He also discusses the disjunctive propositions which follow from a conditional proposition     

Logical and psychological partitioning of mind: Depicting the same map?

The aim of this paper is to demonstrate that empirically delimited structures of mind are also differentiable by means of systematic logical analysis. For the sake of this aim, the paper first summarizes Demetriou’s theory of cognitive organization and growth. This theory assumes that the mind is a multistructural entity that develops across three fronts: the processing system that constrains processing potentials, a set of specialized structural systems (SSSs) that guide processing within different reality and knowledge domains, and a hypercognitive system that monitors and controls the functioning of all other systems. The second part of the paper focuses on the SSSs, which are the target of our logical analysis, and it summarizes a series of empirical studies demonstrating their autonomous operation. The third part develops the logical proof showing that each SSS involves a kernel element that cannot be reduced to standard logic or to any other SSS. The implications of this analysis for the general theory of knowledge and cognitive development are discussed.     

Roger Swyneshed’s Obligationes: A Logical Game of Inference Recognition?

In [Dutilh Novaes, Medieval-obligations as logical Games of Consistency maintenance, synthese, (2004)], I proposed a reconstruction of Walter Burley’s theory of obligationes, based on the idea that Burley’s theory of obligationes could be seen as a logical game of consistency maintenance. In the present paper, I intend to test the game hypothesis on another important theory of obligationes, namely Roger Swyneshed’s theory. In his treatise on obligationes [edited by P.V. Spade, cf. Spade History and philosophy of Logic 3(1982) 1-32], Swyneshed introduced significant modifications to the general framework of obligationes. To compare the two theories, I apply the same formal apparatus used in the previous paper. It will become patent that Swyneshed’s theory is considerably different from Burley’s, among other reasons because the dynamic aspects that play a major role in the latter are simply not present in the former. My conclusion is that Swyneshed’s version of obligationes is not directed towards consistency maintenance, but rather towards inference recognition, and that it is, from a game-theoretical perspective, less interesting a theory than Burley’s.     

Why Ask Why? Logical Fallacies in the Diagnosis of Fetal Alcohol Spectrum Disorder

A diagnosis of Fetal Alcohol Spectrum Disorder (FASD) ascribes cause to developmental disability; however, there are logical issues in causation with ethical implications. This article focuses on the use of fallacious logic (affirming the consequent) in FASD, focusing on the Canadian Guidelines for diagnosis, and knowledge translation issues from science to practice. The clinician’s logical fallacy is an ethical issue of veracity in the clinician–patient relationship; this then leads to issues of nonmaleficence, because the diagnosis in turn blames the mother for her child’s difficulties. Suggestions for revised diagnostic practices that avoid allusions to causation and responsibility are discussed.     

On the Lack of Direction in Rayo’s The Construction of Logical Space

Abstract

I argue that Agustín Rayo’s symmetric ‘just is’ statements cannot be defined in terms of notions like essence, grounding or metaphysical truth-conditions. I go on to argue that one of these latter notions, which allow us to express an asymmetric relationship between facts, is needed to do some of the work that Rayo intends ‘just is’ statements to do, such as stating reductionist claims.     

On the Proximity of the Logical and ‘Objective Bayesian’ Interpretations of Probability

In his Bayesian Nets and Causality, Jon Williamson presents an ‘Objective Bayesian’ interpretation of probability, which he endeavours to distance from the logical interpretation yet associate with the subjective interpretation. In doing so, he suggests that the logical interpretation suffers from severe epistemological problems that do not affect his alternative. In this paper, I present a challenge to his analysis. First, I closely examine the relationship between the logical and ‘Objective Bayesian’ views, and show how, and why, they are highly similar. Second, I argue that the logical interpretation is not manifestly inferior, at least for the reasons that Williamson offers. I suggest that the key difference between the logical and ‘Objective Bayesian’ views is in the domain of the philosophy of logic; and that the genuine disagreement appears to be over Platonism versus nominalism (within weak psychologism).

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