In this paper I will develop a lambda-term calculus, lambda-2Int, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry-Howard correspondence, which has been well-established between the simply typed lambda-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system (...) of Wansing's bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed lambda-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view. (shrink)

What, in Frege’s view, makes definitions fruitful? In Grundlagen §70, Frege offers an answer: Unfruitful definitions are definitions that “could just as well be omitted and leave no link missing in the chain of our proofs”. The §70 passage, however, poses an interpretive puzzle as its characterization of fruitfulness appears to conflict with other conditions that Frege imposes on definitions, namely, eliminability and conservativeness. It appears that the only way to resolve this conflict is to attribute to Frege a notion (...) of fruitfulness that is trivially satisfied and, hence, poorly motivated. I argue that this worry is misplaced. This is because Frege distinguishes between two roles of definitions, namely, between definitions qua explanations of concepts, and definitions qua resources of a proof system. I use this distinction to argue that a fruitful definition, for Frege, is a definition that plays both roles, and that to play both roles, the definition has to be used in the proof of sentences containing the term so defined. Starting from §70, I develop and defend this reading of Frege’s notion of fruitful definition. (shrink)

This paper describes both an exegetical puzzle that lies at the heart of Frege’s writings—how to reconcile his logicism with his definitions and claims about his definitions—and two interpretations that try to resolve that puzzle, what I call the “explicative interpretation” and the “analysis interpretation.” This paper defends the explicative interpretation primarily by criticizing the most careful and sophisticated defenses of the analysis interpretation, those given my Michael Dummett and Patricia Blanchette. Specifically, I argue that Frege’s text either are inconsistent (...) with the analysis interpretation or do not support it. I also defend the explicative interpretation from the recent charge that it cannot make sense of Frege’s logicism. While I do not provide the explicative interpretation’s full solution to the puzzle, I show that its main competitor is seriously problematic. (shrink)

What makes certain definitions fruitful? And how can definitions play an explanatory role? The purpose of this paper is to examine these questions via an investigation of Frege’s treatment of definitions. Specifically, I pursue this issue via an examination of Frege’s views about the scientific unification of logic and arithmetic. In my view, what interpreters have failed to appreciate is that logicism is a project of unification, not reduction. For Frege, unification involves two separate steps: (1) an account of the (...) content expressed by arithmetical claims and (2) the justification of that content. The distinction between these steps allows us to see that there are two notions of definition at play in Frege’s logicist work, viz., one concerned with conceptual analysis, the other concerned with the construction of gap-free proof. I then use this discussion to explain how Frege employs his definitions to defend an epistemological thesis about arithmetic, and to clarify Grundlagen’s fruitfulness condition of definitions, and thereby address two interpretive puzzles from the recent literature. (shrink)

El artículo no pretende ser estudio detallado de la semántica de Gottlob Frege, más bien, se caracteriza por ser un acercamiento a la distinción entre signo (Zeichen), representaciones (Vortellungen), sentido (Sinn) y referencia (Bedeutung) expuesta por el autor. La argumentación fregeana discute la manera desde la cual se puede referir, a través de signos, y como, de éstos, emerge un mundo (tercer mundo), de sentido, que puede estar o no ligado a una referencia. El presente estudio recurre a dos obras (...) fregeanas, especíicamente: ueber sinn und Bedeutung (1892) y Der Gedanke (1918-1919), como fuentes para establecer una distinción que permita un acercamiento preliminar a la semántica fregeana. (shrink)

In the philosophy of mathematical practice, the aim is to understand the various aspects of this practice. Even though definitions are a central element of mathematical practice, the study of this aspect of mathematical practice is still in its infancy. In particular, there is little empirical evidence to substantiate claims about definitions in practice. In this article, we address this gap by reporting on an empirical investigation on how mathematicians create definitions and which roles and properties they attribute to them. (...) On the basis of interviews with thirteen research mathematicians, we provide a broad range of relevant aspects of definitions. In particular, we address various roles of definitions and show that definitions are not just a product of mathematical factors, but also of social and contingent factors. Furthermore, we provide concrete examples of how mathematicians interact and think about definition. This broad empirical basis with a variety of examples provides an optimal starting point for future investigations into definitions in mathematical practice. (shrink)

Why did Frege offer only proper names as examples of presupposition triggers? Some scholars claim that Frege simply did not care about the full range of presuppositional phenomena. This paper argues, in contrast, that he had good reasons for employing an extremely narrow notion of ‘Voraussetzung’. On Frege’s view, many devices that are now construed as presupposition triggers either express several thoughts at once or merely ‘illuminate’ a thought in a particular way. Fregean presuppositions, in contrast, are essentially tied to (...) names. (shrink)

In his Logical Investigations Edmund Husserl criticizes John Stuart Mill’s account of meaning as connotation, especially Mill’s failure to separate the distinction between connotative and non-connotative names from the distinction between the meaningful and the meaningless. According to Husserl, both connotative and non-connotative names have meaning or “signification”, that is, what Gottlob Frege calls the sense (“Sinn”) of an expression. The distinction between connotative and non-connotative names is a distinction between two kinds of meaning (or sense), attributive and non-attributive meaning (...) (“attributive und nicht-attributive Bedeutung”). Attributive (connotative) names denote (refer to) objects through their attributes, whereas a non-attributive name means a thing directly (“direkt”). In this paper I examine the concepts of attributive and non-attributive meaning by means of the semiotic theory of Charles S. Peirce, and compare Peirce’s account with the views of Frege, Husserl, Alexius Meinong, and David Kaplan and Gareth Evans. (shrink)

Chomsky’s (1995, 2000a) Minimalist Program (MP) invites a perspective on semantics that is distinctive and attractive. In section one, I discuss a general idea that many theorists should find congenial: the spoken or signed languages that human children naturally acquire and use— henceforth, human languages—are biologically implemented procedures that generate expressions, whose meanings are recursively combinable instructions to build concepts that reflect a minimal interface between the Human Faculty of Language (HFL) and other cognitive systems. In sections two and three, (...) I develop this picture in the spirit of MP, in part by asking how much of the standard Frege-Tarski apparatus is needed in order to provide adequate and illuminating descriptions of the “concept assembly instructions” that human languages can generate. I’ll suggest that we can make do with relatively little, by treating all phrasal meanings as instructions to assemble number-neutral concepts that are monadic and conjunctive. But the goal is not to legislate what counts as minimal in semantics. Rather, by pursuing one line of Minimalist thought, I hope to show how such thinking can be fruitful. (shrink)

Over the past decades, Patricia Blanchette has developed a sophisticated account of Frege's conception of logic and his views on logical consequence. One of the central components of her interpretation is the idea that Frege's conception of logical consequence is ‘semantically laden’ and not purely formal. The aim of the present paper is to provide precise explications of this as well as related ideas that inform her account, and to discuss their significance for the philosophy of logic in general and (...) Frege's philosophy in particular. (shrink)