Frege on Identity as a Relation of Names

This essay offers a detailed philosophical criticism of Frege’s popular thesis that identity is a relation of names. I consider Frege’s position as articulated both in ‘On Sense and Reference’, and in the Grundgesetze, where he appears to take an objectual view of identity, arguing that in both cases Frege is clearly committed to the proposition that identity is a relation holding between names, on the grounds that two different things can never be identical. A counterexample to Frege’s thesis is considered, and a positive thesis is developed according to which, in contradistinction to the Fregean position, identity is a reflexive, symmetric, and transitive relation holding only between a thing and itself which can be expressed as a relation between names.     

From Plato to Frege: Paradigms of Predication in the History of Ideas

One of the perennial questions of philosophy concerns the simple statements which say that an object is so and so or that such and such objects are so and so related: simple predicative statements. Do such statements have an ontological basis, and if so, what is that basis? The answer to this question determines—or in any case, is expressive of—a specific fundamental outlook on the world. In the course of the history of Western philosophy, various philosophers have given various answers to the question of predication. This essay presents the main, crucial answers: the paradigms and theories of predication of the Sophists (and of all later radical relativists), of Plato, of Aristotle, of the Aristotelian-minded non-nominalists, of Leibniz, and of Frege. In addition, the essay follows (to some extent) the most influential—the Aristotelian or mereological—paradigm of predication in its continuity and modification through the many centuries of its reign. However, the essay is not content to adopt the merely historical point of view; it also poses the question of adequacy. Prior to Frege, there was no philosophically adequate theory of predication, and the essay points out the shortcomings (besides aspects that can be viewed as advantages) of each pre-Fregean predication theory considered in it. Frege, in the nineteenth century, brought the philosophy of predication on the right track, but his own theory of predication has its own deficits. The essay ends with the presentation of a theory of predication that the author himself considers adequate.     

Logic as a Science and Logic as a Theory: Remarks on Frege, Russell and the Logocentric Predicament

Since its publication in 1967, van Heijenoort??s paper, ??Logic as Calculus and Logic as Language?? has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege??s Begriffsschrift (1879) to the work of Herbrand, G?del and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on some aspects of van Heijenoort??s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called ??Logocentric Predicament?? (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed.     

Frege,Dedekind, and the Modern Epistemology of Arithmetic

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.     

An Informational Theory of Counterfactuals

Backtracking counterfactuals are problem cases for the standard, similarity based, theories of counterfactuals e.g., Lewis (Noûs13.4, 455–476, 1979). These theories usually need to employ extra-assumptions to deal with those cases (e.g., Lewis’ “standard resolution of vagueness”). Hiddleston (Noûs 39(4), 632–657, 2005) proposes a causal theory of counterfactuals that, supposedly, deals well with backtracking. The main advantage of the causal theory is that it provides a unified account for backtracking and non-backtracking counterfactuals (no extra-assumption is needed). In this paper, I present a backtracking counterfactual that is a problem case for Hiddleston’s account. Then I propose an informational theory of counterfactuals, which deals well with this problem case while maintaining the main advantage of Hiddleston’s account (the unified account for backtracking and non-backtracking counterfactuals). In addition, the informational theory offers a general theory of backtracking that provides clues for the semantics and epistemology of counterfactuals. I propose that backtracking is reasonable when the (possibly non-actual) state of affairs expressed in the antecedent of a counterfactual transmits less information about an event in the past than the actual state of affairs.     

Reform of the construction of the number system with reference to Gottlob Frege

Due to missing ontological commitments Frege rejected Hilbert’s Fundamentals of Geometry as well as the construction of the system of real numbers by Dedekind and Cantor. Almost all of school mathematics is ontologically committed. Therefore, H.-G. Steiner considered Frege’s viewpoint of mathematics fundamentals, refined by Tarski’s semantics, as suitable for math education. Frege committed numbers ontologically by using measurement to define numbers. He invented the concept of quantitative domain (Größengebiet), which – it is now known by reconstruction of that concept by the New-Fregean Movement – agrees with the concept of quantity domain (Größenbereich) as established in the German reform of the application-oriented construction of the system of real numbers. Concepts of quantity (ratio-scale) and interval-scale in comparative measurement theory – going beyond Frege – show the way how the negative numbers can be ontologically committed and the operations of addition and multiplication can be included. In this work it is shown how Frege’s viewpoint of mathematics fundamentals, as propagated by H.-G. Steiner, can be better implemented in the current construction of the system of real numbers in school.     

Peirce the logician

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schröder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Löwenheim wrote famous papers in Peirce-Schröder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Löwenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.     

Horwich on Natural and Non-Natural Meaning

Paul Horwich’s Use Theory of Meaning (UTM) depends on his rejection of Paul Grice’s distinction between natural and non-natural meaning and his Univocality of Meaning Thesis, as he wishes to deflate the meaning-relation to usage. Horwich’s programme of deflating the meaning-relation (i.e. how words, sentences, etc., acquire meaning) to some basic regularity of usage cannot be carried through if the meaning-relation depends on the minds of users. Here, I first give a somewhat detailed account of the distinction between natural and non-natural meaning in order to set the stage for Horwich’s critique of it. I then present Horwich’s critique of the distinction and show how that rejection accords with his overall view of meaning as use. Horwich’s rejection of the distinction between natural and non-natural meaning, I argue in the last section, is ill founded, and because UTM depends on this rejection, UTM is stillborn.     

Intuitionistic Frege systems are polynomially equivalent

In a paper by Cook and Reckhow (1979), it is shown that any two classical Frege systems polynomially simulate each other. The same proof does not work for intuitionistic Frege systems, since they can have nonderivable admissible rules. (The rule A/B is derivable if the formula A → B is derivable. The rule A/B is admissible if for all substitutions σ, if σ(A) is derivable, then σ(B) is derivable.) In this paper, we polynomially simulate a single admissible rule. Therefore any two intuitionistic Frege systems polynomially simulate each other. Bibliography: 20 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 129–146.     

The Grammar of Platonism

In this paper, based on a critical analysis of ideas of Frege, Quine and Prior, we show how Lambda Calculus and Hilbert’s Epsilon Calculus are useful to give us a good understanding of Platonic objects.     

Weak theories of linear algebra

We investigate the theories of linear algebra, which were originally defined to study the question of whether commutativity of matrix inverses has polysize Frege proofs. We give sentences separating quantified versions of these theories, and define a fragment in which we can interpret a weak theory V¹ of bounded arithmetic and carry out polynomial time reasoning about matrices - for example, we can formalize the Gaussian elimination algorithm. We show that, even if we restrict our language, proves the commutativity of inverses.This work was done while a postdoctoral research fellow at the Department of Computer Science, University of Toronto, Canada.     

Matrix identities and the pigeonhole principle

We show that short bounded-depth Frege proofs of matrix identities, such as PQ=IQP=I (over the field of two elements), imply short bounded-depth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential-size bounded-depth Frege proofs, it follows that the propositional version of the matrix principle also requires bounded-depth Frege proofs of exponential size.     

Frege’s Ancestral and Its Circularities

After presenting the ordinary and the Fregean formulations of the ancestral, I raise the question of what is their relationship, the natural candidate being that the Fregean version is an analysans intended to improve upon, and replace, the common notion of ancestral (the analysandum). Next, two types of circles that arise in connection with the Fregean ancestral are presented, and it is claimed that one of the circles makes it impossible to maintain the just described (??replacement??) interpretation. A reference is made to Kerry, who was the first to point out a circularity in Frege??s ancestral. Some of Frege??s remarks are examined in order to tentatively sketch, an answer to the issue of the relationship between ordinary and Fregean ancestral; the latter, if not as an analysans replacing the common notion, can still be seen as a profound enrichment of the former.     

Towards a More General Concept of Inference

The main objective of this paper is to sketch unifying conceptual and formal framework for inference that is able to explain various proof techniques without implicitly changing the underlying notion of inference rules. We base this framework upon the so-called two-dimensional, i.e., deduction to deduction, account of inference introduced by Tichý in his seminal work The Foundation’s of Frege’s Logic (1988). Consequently, it will be argued that sequent calculus provides suitable basis for such general concept of inference and therefore should not be seen just as technical tool, but philosophically well-founded system that can rival natural deduction in terms of its “naturalness”.     

The slingshot argument and the correspondence theory of truth

The correspondence theory of truth holds that each true sentence corresponds to a discrete fact. Donald Davidson and others have argued (using an argument that has come to be known as the slingshot) that this theory is mistaken, since all true sentences correspond to the same “Great Fact.” The argument is designed to show that by substituting logically equivalent sentences and coreferring terms for each other in the context of sentences of the form ‘P corresponds to the fact that P’ every true sentence can be shown to correspond to the same facts as every other true sentence. The claim is that all substitution of logically equivalent sentences and coreferring terms takes place salva veritate. I argue that the substitution of coreferring terms in this context need not preserve truth. The slingshot fails to refute the correspondence theory.     

Logical form,mathematical practice,and Frege's Begriffsschrift

For over a century we have been reading Frege's Begriffsschrift notation as a variant of standard notation. But Frege's notation can also be read differently, in a way enabling us to understand how reasoning in Begriffsschrift is at once continuous with and a significant advance beyond earlier mathematical practices of reasoning within systems of signs. It is this second reading that I outline here, beginning with two preliminary claims. First, I show that one does not reason in specially devised systems of signs of mathematics as one reasons in natural language; the signs are not abbreviations of words. Then I argue that even given a system of signs within which to reason in mathematics, there are two ways one can read expressions involving those signs, either mathematically or mechanically. These two lessons are then applied to a reading of Frege's proof of Theorem 133 in Part III of his 1879 logic, a proof that Frege claims is at once strictly deductive and ampliative, a real extension of our knowledge. In closing, I clarify what this might mean, and how it might be possible.     

Behavior strategies,mixed strategies and perfect recall

When perfect recall is not satisfied, the informational contents of mixed and behavior strategies differ and are more than what the information partition describes. First, we consider two kinds of additional information strategies may carry, and show that such information leads to theperfect recall refinement of a given information partition. This does not, however, imply that the strategies compensate fully for the lack of perfect recall. We give a necessary and sufficient condition on an information partition, calledA-loss, for the informational content of mixed strategies to fully compensate for the lack of perfect recall. The informational content of behavior strategies never fully compensates.     

Some arguments against intentionalism

Recently, many have argued that phenomenal content supervenes on representational content; i.e. that the phenomenal character of an experience is wholly determined (metaphysically, not causally) by the representational content of that experience. This paper it identifies many counter-examples to intentionalism. Further, this paper shows that, if intentionalism were correct, that would require that an untenable form of representational atomism also be correct. Our argument works both against the idea that phenomenal content supervenes on “conceptual” content and also against the idea that it supervenes on “non-conceptual” content. It is also shown that the distinction between conceptual and non-conceptual content has been wrongly conceived as distinction between different kinds of information: in fact, it is a distinction between ways of packaging information that is, in itself, neither conceptual or non-conceptual.