Referentiality in Frege's Grundgesetze

In §§28-31 of his Grundgesetze der Arithmetik, Frege forwards a demonstration that every correctly formed name of his formal language has a reference. Examination of this demonstration, it is here argued, reveals an incompleteness in a procedure of contextual definition. At the heart of this incompleteness is a difference between Frege’s criteria of referentiality and the possession of reference as it is ordinarily conceived. This difference relates to the distinction between objectual and substitutional quantification and Frege’s vacillation between the two

Minimal Logicism

PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the existence of individuals rather than on the existence of concepts.PLV (Plural Basic Law V) est un système de second ordre cohérent qui vise à dériver l'arithmétique de Peano du second ordre. Il emploie la notion de quantification plurielle et une formulation du premier ordre de la tristement célèbre Loi Fondamentale V de Frege. La sémantique plurielle de George Boolos est remplacée par la Acts of Choice Semantics (ACS) de Enrico Martino, qui est développée à partir de la notion de référence arbitraire en raisonnement mathématique. ACS fournit une forme de logicisme qui est radicalement alternative à celle de Frege et qui est fondée sur l'existence des individus plutôt que sur l'existence des concepts

Frege's Basic Law V and Cantor's Theorem

The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom

The Semantics of Value-Range Names and Frege's Proof of Referentiality

In this article, I try to shed some new light on Grundgeselze 10, 29-31 with special emphasis on Frege's criteria and proof of referentiality and his treatment of the semantics of canonical value-range names. I begin by arguing against the claim, recently defended by several Frege scholars, that the first-order domain in Grundgesetze is restricted to value-ranges (including the truth-values), but conclude that there is an irresolvable tension in Frege's view. The tension has a direct impact on the semantics of the concept-script, not least on the semantics of value-range names. I further argue that despite first appearances truth-value names (sentences) play a distinguished role as semantic "target names" for "test names" in the criteria of referentiality (29) and do not figure themselves as "test names" regarding referentiality. Accordingly. I show in detail that Frege's attempt to demonstrate that by virtue of his stipulations "regular" value-range names have indeed been endowed with a unique reference, can plausibly be regarded as a direct application of the context principle. In a subsequent section, I turn to some special issues involved in 10. 10 is closely intertwined with 31 and in my and Richard Heck's view would have been better positioned between 30 and 31. In a first step, I discuss the piecemeal strategy which Frege applies when he attempts to bestow a unique reference on value-range names in 3, 10-12. In a second step, I critically analyze his tentative, but predictably unsuccessful proposal (in a long footnote to 10) to identify all objects whatsoever, including those already clad in the garb of value-ranges, with their unit classes. In conclusion, I present two arguments for my claim that Frege's identification of the True and the False with their unit classes in 10 is illicit even if both the permutation argument and the identifiability thesis that he states in 10 are regarded as formally sound. The first argument is set out from the point of view of the syntax of his formal language. It suggests though that a reorganization of the exposition of the concept-script would have solved at least one of the problems to which the twin stipulations in 10 give rise. The second argument rests on semantic considerations. If it is sound, it may call into question, if not undermine the legitimacy of the twin stipulations

The Logical Problem of Identity

Abstract Keith A. Coleman Department of Philosophy, February 2008 University of Kansas A traditional problem concerning the meaning or logical content of statements of identity received its modern formulation in Gottlob Frege's "On Sense and Reference." Identity is taken either as a relation between objects or a relation between terms. If identity is interpreted as a relation between objects, then identity statements seem to be of little value since everything is clearly identical to itself. Assertions of identity are thought to convey significant information, but it is hard to see how they can on this interpretation. If identity is instead interpreted as a relation between terms, then identity statements still seem to be of little value since apparently they only convey a linguistic pronouncement to use certain terms interchangeably. Assertions of identity do not appear to be about the use of language, but, on this understanding of identity, they evidently are. I examine the nature of the problem (and what it would take to solve it) and the advantages and disadvantages of each one of the two approaches to interpreting the content of identity statements. I then investigate two approaches for solving the problem from the perspective of identity as a relation between objects. The first of these represents the account provided by Gottlob Frege, and the second represents the account provided by Saul Kripke. I conclude that neither one of these accounts finally solves the problem of identity in its entirety. I then examine Michael Lockwood's approach to resolving the problem of identity based on the idea of identity as a relation between terms. I discuss and critically evaluate Lockwood's account together with a modified version of that account. After arguing for the inadequacy of the views examined as ultimate solutions to the problem of identity, I end by suggesting a strategy prompted by treating identity as indiscernibility

Remarks on the problem of Sense in a pragmatic reading

The division of the problems about Meaning in an extensional and an intensional theory was accused of unnecessarily duplicating the theoretical problems involved in the knowledge of Meaning. The result was either 1. an aggressive rejection of the intensional part of the problem or 2. the adoption of non-classical semantics to account for intensions. This paper suggests a reading of Frege's theory of Meaning that preserves the contribution of the intensional aspect of the question without sacrificing the benefits of the classical insights. For this, however, we need to expand his theory to cases of semantic values which are not directly assertive, that is, what Micheal Dummett called ingredient value. This solution preserves the direct and intuitive insights of classical semantics, expanding it to cases where the designation of truth or falsehood does not have a unified interpretation: speculative, modal, ambiguous, relative cases or cases in which theories disputing Meaning is not straightforwardly mapped to the truth or the not-truth. The paper is a contribution to support the pragmatic interpretation of Frege's legacy by Micheal Dummett

Intersubjectivity and contemporary social theory: the everyday as critique

Traditional analyses of popular or everyday culture have been couched in terms of, on the one hand, variants of the Frankfurt School view that it distorts or offers a false consciousness of an underlying reality, whilst on the other, the response from discourse theorists and more traditional forms of social constructionism has been philosophically conventionalist in arguing that discourses, definitions and labelling activities directly construct meaning. The argument of this thesis is that it is possible, via Dummett's reading of Frege, to construct a realist account of meaning or, more strictly, sense, which preserves an element of rationality in everyday cultural reception without thereby effecting a radical relativisation of the notion of what constitutes rational processes and practices. This is achieved via Dummett's Context Principle, through which it is argued that the actual meaning of an utterance is not given directly by the conventional meaning of its constituent terms and phrases, but rather that the object picked out, its reference, depends on the context in which the utterance occurs. Hence the terms and phrases offer no more than a clue, a route to the reference, not the actuality of the reference. Consequently, whilst the sedimented meanings of the elements from which an utterance is composed provide a structuring of meaning, the actual constitution of meaning will depend on the context. The thesis explores the view that meaning or sense has an open-ended, but ontological quality by examining a variety of issues and themes including reflexivity, forms of conventionalism, conceptions of the everyday, perspectives in phenomenological social theory and philosophy, rationality, semiotics, reference, discursivity, spatial and temporal locations of sense. It counterposes the emphasis on the contextual structuring of meaning as an effect of the subject's everyday appropriation of background routines, to passive constructions of subjectivity as offered by the Frankfurt School, and what it sees as overly direct constructions of meaning via classification systems in social constructionist approaches, which again have a reductive effect on the subject's role in the production of meaning