This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry and (...) the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)

In a ground-breaking essay Nagel contended that the controversy over impossible numbers influenced the development of modern logic. I maintain that Nagel was correct in outline only. He overlooked the fact that the controversy engendered a new account of reasoning, one in which the concept of a well-made language played a decisive role. Focusing on the new account of reasoning changes the story considerably and reveals important but unnoticed similarities between the development of algebraic logic and quantificational logic.

By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s Begriffsschrift, on the other hand, (...) is a visual tool to scrutinize concepts and inferences, and is a calculus only in the thin sense that every possible transition between sentences is fully and unambiguously specified in advance. While Frege’s outlook has dominated much of philosophical thinking about logical symbolism, we believe there is value—particularly in light of recent interest in the role of notations in mathematics and logic—in reviving Boole’s idea of an intrinsic link between, as he put it, a ‘calculus’ and a ‘directive method’ to solve problems. (shrink)

Published in 1908, C.S. Peirce’s ‘A Neglected Argument for the Reality of God’ is one of his most difficult articles. Presenting a peculiar entanglement of scientific method and theology, it sketches a ‘humble’ argument for the reality—and not the existence—of God for Musers, that is, those who pursue the activity he calls ‘Musement’. In Musement, Peirce claims, we can achieve a kind of perception of the intertwinement of the three universes of experience: of feeling, of brute fact, and of reason. (...) He also somehow relates each universe to a distinct phase of inquiry, which is described by the use of induction, deduction, and abduction or retroduction. The way that he develops his claims allows him to outline God as an abductive vague hypothesis to explain how those three universes make up a single whole. The hypothesis being vague means that the principle of contradiction does not hold for it. In this presentation, we aim at throwing some light at these points, focusing on the concept of Musement and what is understood as the vagueness of the hypothesis. (shrink)

In this essay I will consider two theses that are associated with Frege,and will investigate the extent to which Frege really believed them.Much of what I have to say will come as no surprise to scholars of thehistorical Frege. But Frege is not only a historical figure; he alsooccupies a site on the philosophical landscape that has allowed hisdoctrines to seep into the subconscious water table. And scholars in a widevariety of different scholarly establishments then sip from thesedoctrines. I believe (...) that some Frege-interested philosophers at various ofthese establishments might find my conclusions surprising.Some of these philosophical establishments have arisen from an educationalmilieu in which Frege is associated with some specific doctrine at theexpense of not even being aware of other milieux where other specificdoctrines are given sole prominence. The two theses which I will discussillustrate this point. Each of them is called Frege''s Principle, but byphilosophers from different milieux. By calling them milieux I do not want to convey the idea that they are each located at some specificsocio-politico-geographico-temporal location. Rather, it is a matter oftheir each being located at different places on the intellectuallandscape. For this reason one might (and I sometimes will) call them(interpretative) traditions. (shrink)

We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for the origins and development of model theory. (...) This point is illustrated by Carnap’s early semantics and model theory, which he developed within a type theoretical framework and which stand out both for their universalistic treatment and for certain idiosyncratic technicalities by which the construction is supported. One special property is that individuals are context relative in Carnap’s system. This leads to a model theory in which the model domains are more flexible than has been suggested in the literature. (shrink)

It is often argued that by assuming the existence of a universal language, one prohibits oneself from conducting semantical investigations. It could thus be thought that Tarski’s stance towards a universal language in his fruitful Wahrheitsbegriff differs essentially from Carnap’s in the latter’s less successful Untersuchungen zur allgemeinen Axiomatik. Yet this is not the case. Rather, these two works differ in whether or not the studied fragments of the universal language are languages themselves, i.e., whether or not they are closed (...) under derivation rules. In Carnap’s case, axiom systems are not closed under derivation rules, which enables him to adopt a substitutional concept of models. His approach is directly rooted in the tradition of formal axiomatics, we argue, and in this contrary to Tarski’s. In comparing these works by Carnap and Tarski, our aim will be to qualify the connection between Tarski’s approach and the tradition of formal axiomatics, which has been overemphasized in the literature. (shrink)

In this paper I suggest an answer to the question of what Frege means when he says that his logical system, the Begrijfsschrift, is like the language Leibniz sketched, a lingua characteristica, and not merely a logical calculus. According to the nineteenth century studies, Leibniz's lingua characteristica was supposed to be a language with which the truths of science and the constitution of its concepts could be accurately expressed. I argue that this is exactly what the Begriffsschrift is: it is (...) a language, since, unlike calculi, its sentential expressions express truths, and it is a characteristic language, since the meaning of its complex expressions depend only on the meanings of their constituents and on the way they are put together. In fact it is in itself already a science composed in accordance with the Classical Model of Science. What makes the Begrijfsschrift so special is that Frege is able to accomplish these goals with using only grammatical or syncategorematic terms and so has a medium with which he can try to show analyticity of the theorems of arithmetic. (shrink)

In this paper I suggest an answer to the question of what Frege means when he says that his logical system, the Begriffsschrift, is like the language Leibniz sketched, a lingua characteristica, and not merely a logical calculus. According to the nineteenth century studies, Leibniz’s lingua characteristica was supposed to be a language with which the truths of science and the constitution of its concepts could be accurately expressed. I argue that this is exactly what the Begriffsschrift is: it is (...) a language, since, unlike calculi, its sentential expressions express truths, and it is a characteristic language, since the meaning of its complex expressions depend only on the meanings of their constituents and on the way they are put together. In fact it is in itself already a science composed in accordance with the Classical Model of Science. What makes the Begriffsschrift so special is that Frege is able to accomplish these goals with using only grammatical or syncategorematic terms and so has a medium with which he can try to show analyticity of the theorems of arithmetic. (shrink)

It is widely recognized that Frege's systematic conception of science has a major impact on his work. I argue that central to this conception and its impact is Frege's Simplicity Requirement that a scientific system must have as few primitive truths as possible. Frege states this requirement often, justifies it in several ways, and appeals to it to motivate important aspects of his broader views. Acknowledging its central role illuminates several aspects of his work in new ways, including his treatment (...) of truth values, logical concepts, and the dependence relations that induce a ‘natural order’ of truths. (shrink)

Michael Dummett has shown that the fragment ‘17 Kernsätze zur Logik’ is evidence that Frege knew Lotze's Logik Dummett’s dating of this fragment prior to 1879, however, must be rejected.The present paper shows that there are other articles of Frege’s which bear clear traces of Lotze's LogikFirst of all, the expressions Vorstellungsverlauf from ‘Über die wissenschaftliche Berechtigung einer Begriffsschrift’, and veranlassenden Ursachen, from ‘Logik’, certainly are borrowed from Lotze.Second, there are links between ‘Booles rechnende Logik und die Begriffsschrift’ and Lotze's (...) Logik. Furthermore, it is shown that Frege’s ‘Kernsätze’, the ‘Dialog mit Pünjer über Existenz’, and his ‘Logik’ are intimately connected.All of this indicates that these texts were written in roughly the same period, namely the early 1880s.Conclusive evidence for this is that the terms Vorstellung and Vorstellungsverbindung are used indiscriminately in both a psychological and a logical sense in the ‘Begriffsschrift’, a fact which contradicts the ‘Kernsätze’. (shrink)

In a famous passage (A68/B93), Kant writes that “the understanding can make no other use of […] concepts than that of judging by means of them.” Kant's thought is often called the thesis of the priority of judgments over concepts. We find a similar sounding priority thesis in Frege: “it is one of the most important differences between my mode of interpretation and the Boolean mode […] that I do not proceed from concepts, but from judgments.” Many interpreters have thought (...) that Frege's priority principle is close to (or at least derivable from) Kant's. I argue that it is not. Nevertheless, there was a gradual historical development that began with Kant's priority thesis and culminated in Frege's new logic. (shrink)

The Polish logicians' propositional calculi, which consist in a distinct synthesis of the Fregean and Boolean approaches to logic, influenced W. V. Quine's early work in formal logic. This early formal work of Quine's, in turn, can be shown to serve as one of the sources of his holistic conception of natural language.

Frege appears to hold both that thoughts are internally articulated, in a way that mirrors the semantic articulation of the sentences that express them, and that the same thought can be analyzed in different ways, none of which has to be more fundamental than the others. Commentators have often taken these theses to be mutually incompatible and have tended to polarize into two camps, each of which attributes to Frege one of the theses, but maintains that he is only apparently (...) committed to the other. This paper argues that there are good exegetical and philosophical reasons for reconciling the two theses; that this reconciliation can be achieved by rejecting an assumption shared by the two opposite camps of the exegetical debate, i.e., the assumption that essential articulatedness implies unique articulation; and finally, that this crucial assumption can be resisted by appreciating Frege’s anti-atomistic and ‘organic’ conception of the internal complexity of thoughts. (shrink)

This paper identifies a tension in Frege’s philosophy and offers a diagnosis of its origins. Frege’s Context Principle can be used to dissolve the problem of propositional unity. However, Frege’s official response to the problem does not invoke the Context Principle, but the distinction between ‘saturated’ and ‘unsaturated’ propositional constituents. I argue that such a response involves assumptions that clash with the Context Principle. I suggest, however, that this tension is not generated by deep-seated philosophical commitments, but by Frege’s occasional (...) attempt to take a dubious shortcut in the justification of his conception of propositional structure. (shrink)

After the publication of Begriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions of lingua characterica and calculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a mere calculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain (...) Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal. -/- In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a mere calculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system of Begriffsschrift incapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language of Begriffsschrift did not constitute a complete lingua characterica by itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing a lingua characterica was not tied to his later logicist programme. (shrink)

Quine’s philosophy comprises a bewildering set of views whose integrating principle is his "confirmed extensionalism". The paper offers a historical as well as an intellectual reconstruction of extensionalism. Traditional extensionalism (Boole) freed logic from Aristotelian essentialism that had inhibited the development of logic. Quine’s confirmed extensionalism is the acceptance, as a matter of course, of the validity of Frege’s criticism of [Boole’s] extensionalism. His confirmed extensionalism is a generalized version of the philosophy of science known as conventionalism. As such, it (...) places the advancement of science outside the province of science proper. It is, thus, at odds with Quine’s repeated expressions of alliance with the Popperian (hypothetico-deductive) model of science. (shrink)

Quine’s philosophy comprises a bewildering set of views whose integrating principle is his “confirmed extensionalism”. The paper offers a historical as well as an intellectual reconstruction of extensionalism. Traditional extensionalism freed logic from Aristotelian essentialism that had inhibited the development of logic. Quine’s confirmed extensionalism is the acceptance, as a matter of course, of the validity of Frege’s criticism of [Boole’s] extensionalism. His confirmed extensionalism is a generalized version of the philosophy of science known as conventionalism. As such, it places (...) the advancement of science outside the province of science proper. It is, thus, at odds with Quine’s repeated expressions of alliance with the Popperian model of science. (shrink)

I use van Heijenoort’s published writings and manuscript materials to provide a comprehensive overview of his conception of modern logic as a first-order functional calculus and of the historical developments which led to this conception of mathematical logic, its defining characteristics, and in particular to provide an integral account, from his most important publications as well as his unpublished notes and scattered shorter historico-philosophical articles, of how and why the mathematical logic, whose he traced to Frege and the culmination of (...) its formative period in the incompleteness results of Gödel, became modern logic, as distinct from the traditional logic of Aristotle, and why and how the logistic tradition that led from Frege through Russell, rather than the algebraic tradition that led from De Morgan and Boole through Peirce and Schröder, came, in his view, to define modern logic. (shrink)

Van Heijenoort’s account of the historical development of modern logic was composed in 1974 and first published in 1992 with an introduction by his former student. What follows is a new edition with a revised and expanded introduction and additional notes.

In this paper I explore three related topics emerging from Prior's work on the logic of time. First, what is the proper province of logic, if any? Is temporal (modal) logic just logic, on a par with the paradigmatic case of first-order quantification theory or even simple propositional logic? Second, what counts as an interpretation of a formal system? In particular, can formal semantics provide an interpretation? Third, what is the proper role of the meta-theory? In connection with this last (...) question we will see how Prior's attitude towards instants of time may teach us something about the analogous case of possible worlds. (shrink)

This thesis discusses some central aspects of Wittgenstein's conception of language and logic in his Tractatus Logico-Philosophicus and brings them into relation with the philosophies of Frege and Russell. The main contention is that a fruitful way of understanding the Tractatus is to see it as responding to tensions in Frege's conception of logic and Russell's theory of judgement. In the thesis the philosophy of the Tractatus is presented as developing from these two strands of criticism and thus as the (...) culmination of the philosophy of logic and language developed in the early analytic period. Part one examines relevant features of Frege's philosophy of logic. Besides shedding light on Frege's philosophy in its own right, it aims at preparing the ground for a discussion of those aspects of the Tractatus' conception of logic which derive from Wittgenstein's critical response to Frege. Part two first presents Russell's early view on truth and judgement, before considering several variants of the multiple relation theory of judgement, devised in opposition to it. Part three discusses the development of Wittgenstein's conception of language and logic, beginning with Wittgenstein's criticism of the multiple relation theory and his early theory of sense, seen as containing the seeds of the picture theory of propositions presented in the Tractatus. I then consider the relation between Wittgenstein's pictorial conception of language and his conception of logic, arguing that Wittgenstein's understanding of sense in terms of bipolarity grounds his view of logical complexity and of the essence of logic as a whole. This view, I show, is free from the internal tensions that affect Frege's understanding of the nature of logic. (shrink)

When reasons are given for compositionality, the arguments usually purport to establish compositionality in an almost a priori manner. I will rehearse these arguments why one could think that compositionality is a priori true, or almost a priori true, and will find all of them inconclusive. This, in itself, is no reason against compositionality, but a reason to try to establish or defend the principle on other than quasi-a priori grounds.