The first complete English translation of a groundbreaking work. An ambitious account of the relation of mathematics to logic. Includes a foreword by Crispin Wright, translators' Introduction, and an appendix on Frege's logic by Roy T. Cook. The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. (...) It was to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook. Readership: Scholars and advanced students in philosophy of logic, philosophy of mathematics, and early analytic philosophy. (shrink)

By a well-known result of Cook and Reckhow [S.A. Cook, R.A. Reckhow, The relative efficiency of propositional proof systems, Journal of Symbolic Logic 44 36–50; R.A. Reckhow, On the lengths of proofs in the propositional calculus, Ph.D. Thesis, Department of Computer Science, University of Toronto, 1976], all Frege systems for the classical propositional calculus are polynomially equivalent. Mints and Kojevnikov [G. Mints, A. Kojevnikov, Intuitionistic Frege systems are polynomially equivalent, Zapiski Nauchnyh Seminarov POMI 316 129–146] (...) have recently shown p-equivalence of Frege systems for the intuitionistic propositional calculus in the standard language, building on a description of admissible rules of IPC by Iemhoff [R. Iemhoff, On the admissible rules of intuitionistic propositional logic, Journal of Symbolic Logic 66 281–294]. We prove a similar result for an infinite family of normal modal logics, including K4, GL, S4, and. (shrink)

We present a general method for inserting proofs in Frege systems for classical logic that produces systems that can internalize their own proofs.

In this paper we introduce a system AID of bounded arithmetic. The main feature of AID is to allow a form of inductive definitions, which was extracted from Buss’ propositional consistency proof of Frege systems F in Buss 3–29). We show that AID proves the soundness of F , and conversely any Σ 0 b -theorem in AID yields boolean sentences of which F has polysize proofs. Further we define Σ 1 b -faithful interpretations between AID+Σ 0 b (...) -CA and a quantified theory QALV of an equational system ALV in Clote 57–106). Hence ALV also proves the soundness of F. (shrink)

The canonical pair of a proof system P is the pair of disjoint NP sets where one set is the set of all satisfiable CNF formulas and the other is the set of CNF formulas that have P-proofs bounded by some polynomial. We give a combinatorial characterization of the canonical pairs of depth d Frege systems. Our characterization is based on certain games, introduced in this article, that are parametrized by a number k, also called the depth. We (...) show that the canonical pair of a depth d Frege system is polynomially equivalent to the pair (Ad+2,Bd+2) where Ad+2 (respectively, Bd+1) are depth d+1 games in which Player I (Player II) has a positional winning strategy. Although this characterization is stated in terms of games, we will show that these combinatorial structures can be viewed as generalizations of monotone Boolean circuits. In particular, depth 1 games are essentially monotone Boolean circuits. Thus we get a generalization of the monotone feasible interpolation for Resolution, which is a property that enables one to reduce the task of proving lower bounds on the size of refutations to lower bounds on the size of monotone Boolean circuits. However, we do not have a method yet for proving lower bounds on the size of depth d games for d>1. (shrink)

Let L be a first-order language and Φ and ψ two $\Sigma _{1}^{1}$ L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle Chain L,Φ,ψ (n,m) holds: For any chain of finite L-structures C 1 ,...,C m with the universe [n] one of the following conditions must fail: 1. $C_{1}\vDash \Phi $ , 2. C i ≅ C i+1 , for i = 1,...,m — 1, 3. $C_{m}\vDash \Psi $ . For each fixed L and (...) parameters n,m the principle Chain L,Φ,ψ (n,m) can be encoded into a propositional DNF formula of size polynomial in n,m. For any language L containing only constants and unary predicates we show that there is a constant c L such that the following holds: If a constant depth Frege system in DeMorgan language proves Chain L,Φ,ψ (n, c L · n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying ψ by a depth 3 formula of size $2^{{\rm log}(s)^{O(1)}}$ and with bottom fan-in ${\rm log}(s)^{O(1)}$. (shrink)

ABSTRACTIn this paper, we present a generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued Łukasiewicz logics. To this end, we provide proof systems and which augment Avron's Frege system HŁuk with nested and general versions of the disjunction elimination rule, respectively. For these systems, we provide upper bounds on speed-ups w.r.t. both the (...) number of steps in proofs and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus GŁuk for 3-valued Łukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued Łukasiewicz logics. (shrink)

We investigate the substitution Frege () proof system and its relationship to extended Frege () in the context of modal and superintuitionistic propositional logics. We show that is p-equivalent to tree-like , and we develop a “normal form” for -proofs. We establish connections between for a logic L, and for certain bimodal expansions of L.We then turn attention to specific families of modal and si logics. We prove p-equivalence of and for all extensions of , all tabular logics, (...) all logics of finite depth and width, and typical examples of logics of finite width and infinite depth. In most cases, we actually show an equivalence with the usual system for classical logic with respect to a naturally defined translation.On the other hand, we establish exponential speed-up of over for all modal and si logics of infinite branching, extending recent lower bounds by P. Hrubeš. We develop a model-theoretical characterization of maximal logics of infinite branching to prove this result. (shrink)

A Frege proof systemFis any standard system of prepositional calculus, e.g., a Hilbert style system based on finitely many axiom schemes and inference rules. An Extended Frege systemEFis obtained fromFas follows. AnEF-sequence is a sequence of formulas ψ1, …, ψκsuch that eachψiis either an axiom ofF, inferred from previous ψuand ψv by modus ponens or of the formq↔ φ, whereqis an atom occurring neither in φ nor in any of ψ1,…,ψi−1. Suchq↔ φ, is called an extension axiom andqa (...) new extension atom. AnEF-proof is anyEF-sequence whose last formula does not contain any extension atom. In [12], S. A. Cook and R. Reckhow proved that the pigeonhole principlePHPhas a simple polynomial sizeEF-proof and conjectured thatPHPdoes not admit polynomial sizeF-proof. In [5], S. R. Buss refuted this conjecture by furnishing polynomial sizeF-proof forPHP. Since then the important separation problem of polynomial sizeFand polynomial sizeEFhas not shown any progress.In [11], S. A. Cook introduced the systemPV, a free variable equational logic whose provable functional equalities are ‘polynomial time verifiable’ and showed that the metamathematics ofFandEFcan be developed inPVand the soundness ofEFproved inPV. In [3], S. R. Buss introduced the first order systemand showed thatis essentially a conservative extension ofPV. There he also introduced a second order system. (shrink)

A Frege proof systemFis any standard system of prepositional calculus, e.g., a Hilbert style system based on finitely many axiom schemes and inference rules. An Extended Frege systemEFis obtained fromFas follows. AnEF-sequence is a sequence of formulas ψ1, …, ψκsuch that eachψiis either an axiom ofF, inferred from previous ψuand ψv by modus ponens or of the formq↔ φ, whereqis an atom occurring neither in φ nor in any of ψ1,…,ψi−1. Suchq↔ φ, is called an extension axiom andqa (...) new extension atom. AnEF-proof is anyEF-sequence whose last formula does not contain any extension atom. In [12], S. A. Cook and R. Reckhow proved that the pigeonhole principlePHPhas a simple polynomial sizeEF-proof and conjectured thatPHPdoes not admit polynomial sizeF-proof. In [5], S. R. Buss refuted this conjecture by furnishing polynomial sizeF-proof forPHP. Since then the important separation problem of polynomial sizeFand polynomial sizeEFhas not shown any progress.In [11], S. A. Cook introduced the systemPV, a free variable equational logic whose provable functional equalities are ‘polynomial time verifiable’ and showed that the metamathematics ofFandEFcan be developed inPVand the soundness ofEFproved inPV. In [3], S. R. Buss introduced the first order systemand showed thatis essentially a conservative extension ofPV. There he also introduced a second order system. (shrink)

Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable (...) in this extended system. (shrink)

This paper aims at clarifying the nature of Frege's system of logic, as presented in the first volume of the Grundgesetze. We undertake a rational reconstruction of this system, by distinguishing its propositional and predicate fragments. This allows us to emphasise the differences and similarities between this system and a modern system of classical second-order logic.

In the ‘analytic tradition’, Hans Sluga wrote thirty years ago in his book Gottlob Frege, there has been a ‘lack of interest in historical questions — even in the question of its own roots. Anti-historicism has been the baggage of the tradition since Frege’ (Sluga, 1980, p. 2). The state of the discussion of Frege among analytic philosophers, Sluga claimed, illustrated well this indifference. Despite the numbers of pages devoted to Frege, there was still, Sluga claimed, (...) little understanding of the sources of Frege’s ideas, his intellectual debts, and the historical circumstances of his thought. Sluga singled out Michael Dummett’s Frege: Philosophy of Language ‘as a paradigm for the failure of analytic philosophers to come to grips with the actual, historical Frege’. In that book, Dummett famously wrote that the logical system that Frege put forward in his 1879 Begriffsschrift ‘is astonishing because it has no predecessors: it appears to have been born from Frege’s brain unfertilized by external influences’ (Dummett, 1981a, p. xxxv) — and Dummett devoted almost none of its 700 pages to the relationship between Frege and his contemporaries. (shrink)

Frege's logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the "neologicist" approach of Hale and Wright. Less attention has been given to Frege's extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the (...) extension of a concept C, together with extra-logical axioms governing such a predicate, and show that arithmetic can be obtained in such a framework. As a philosophical payoff, we investigate the status of the so-called Hume's Principle and its connections to the root of the contradiction in Frege's system. (shrink)

In this paper I examine the question of logic’s normative status in the light of Carnap’s Principle of Tolerance. I begin by contrasting Carnap’s conception of the normativity of logic with that of his teacher, Frege. I identify two core features of Frege’s position: first, the normative force of the logical laws is grounded in their descriptive adequacy; second, norms implied by logic are constitutive for thinking as such. While Carnap breaks with Frege’s absolutism about logic and (...) hence with the notion that any system of logic should have a privileged claim to correctness, I argue that there is a sense in which Carnap’s framework-relative conception of logical norms has a constitutive role to play: though they are not constitutive for the conceptual activity for thinking, they do nevertheless set the ground rules that make certain forms of scientific inquiry possible in the first place. I conclude that Carnap’s principle of tolerance is tamer than one might have thought and that, despite remaining differences, Frege’s and Carnap’s conceptions of logic have more in common than one might have thought. (shrink)

The great logician Gottlob Frege attempted to provide a purely logical foundation for mathematics. His system collapsed when Bertrand Russell discovered a contradiction in it. Thereafter, mathematicians and logicians, beginning with Russell himself, turned in other directions to look for a framework for modern abstract mathematics. Over the past couple of decades, however, logicians and philosophers have discovered that much more is salvageable from the rubble of Frege's system than had previously been assumed. A variety of repaired (...) class='Hi'>systems have been proposed, each a consistent theory permitting the development of a significant portion of mathematics. This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. John Burgess considers every proposed fix, each with its distinctive philosophical advantages and drawbacks. These systems range from those barely able to reconstruct the rudiments of arithmetic to those that go well beyond the generally accepted axioms of set theory into the speculative realm of large cardinals. For the most part, Burgess finds that attempts to fix Frege do less than advertised to revive his system. This book will be the benchmark against which future analyses of the revival of Frege will be measured. (shrink)

This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in ...

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the (...) question of the independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

Frege claims that the laws of logic are characterized by their “generality,” but it is hard to see how this could identify a special feature of those laws. I argue that we must understand this talk of generality in normative terms, but that what Frege says provides a normative demarcation of the logical laws only once we connect it with his thinking about truth and science. He means to be identifying the laws of logic as those that appear (...) in every one of the scientific systems whose construction is the ultimate aim of science, and in which all truths have a place. Though an account of logic in terms of scientific systems might seem hopelessly antiquated, I argue that it is not: a basically Fregean account of the nature of logic still looks quite promising. (shrink)

Frege's philosophy of language includes detailed views on judgments. His formal logic - the Begriffsschrift - documents some of these views in the introduction and treatment of the judgment stroke. In current logic such an expression is either entirely ignored or, appearing as turnstile, plays an fundamentally different role. In this paper I put forward four claims: (i) Considering Frege's Begriffsschrift, it is methodologically palpable why the judgment stroke was omitted in nearly all logical systems developed after (...) Frege. (ii) The Frege-style inclusion of the judgment stroke in a formal language represents a partial implementation of results from speech act theory. (iii) A more comprehensive implementation, not limited to the judgment stroke, helps to satisfy justified expectations about formal languages and logical calculi. (iv) Such an implementation has not yet been achieved in the current logical mainstream. (shrink)

One of the most striking differences between Frege's Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted , that is, which are being used to express judgements . There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role (...) of the judgement stroke in a way that renders it readily comprehensible why Frege insisted that this symbol was an essential part of his logical system. The key point here is that Frege viewed logic as the study of inference relations amongst acts of judgement , rather than – as in the typical contemporary view – of consequence relations amongst certain objects (propositions or well-formed formulae). The paper also explains why Frege's use of the judgement stroke is not in conflict with his avowed anti-psychologism, and why Wittgenstein's criticism of the judgement stroke as 'logically quite meaningless' is unfounded. The key point here is that while the judgement stroke has no content , its use in logic and mathematics is subject to a very stringent norm of assertion. (shrink)

Two thoughts about the concept of number are incompatible: that any zero or more things have a number, and that any zero or more things have a number only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any things have a number is Frege's; the thought that things have a number (...) only if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous. (shrink)

We show that the untyped λ -calculus can be extended with Frege's interpretation of propositional notions, provided we restrict β -conversion to positive expressions. The system of illative λ -calculus so obtained admits a natural Scott-style semantics.

This paper has three goals: (i) to show that the foundational program begun in the Begriffsschroft, and carried forward in the Grundlagen, represented Frege's attempt to establish the autonomy of arithmetic from geometry and kinematics; the cogency and coherence of 'intuitive' reasoning were not in question. (ii) To place Frege's logicism in the context of the nineteenth century tradition in mathematical analysis, and, in particular, to show how the modern concept of a function made it possible for (...) class='Hi'>Frege to pursue the goal of autonomy within the framework of the system of second-order logic of the Begriffsschrift. (iii) To address certain criticisms of Frege by Parsons and Boolos, and thereby to clarify what was and was not achieved by the development, in Part III of the Begriffsschrift, of a fragment of the theory of relations. (shrink)

What is the number one? How do we know that 2+2=4? These apparently simple questions are in fact notoriously difficult to answer, and in one form or other have occupied philosophers from ancient times to the present. Gottlob Frege's conviction that the truths of arithmetic, and mathematics more generally, are derived from self-evident logical truths formed the basis of a systematic project which revolutionized logic, and founded modern analytic philosophy. In this accessible and stimulating introduction, Joan Weiner traces the (...) development of Frege's thought from his invention of a powerful new logical language in Begriffsschrift, through his explication of his project in the Foundations of Arithmetic and famous papers such as 'On Sense and Reference', to the brilliant, but ultimately doomed, presentation of the system in Basic Laws of Arithmetic. At each stage, she discusses Frege's motivations in a way which enables the modern reader to appreciate the originality, clarity, and profundity of his thought. Past Masters is a series of concise, lucid, authoritative introducitons to the thought of leading intellectual figures of the past whose ideas still influence the way we think today. (shrink)

In this paper I examine the fundamental views on the nature of logical and mathematical truth of both Frege and Carnap. I argue that their positions are much closer than is standardly assumed. I attempt to establish this point on two fronts. First, I argue that Frege is not the metaphysical realist that he is standardly taken to be. Second, I argue that Carnap, where he does differ from Frege, can be seen to do so because of (...) mathematical results proved in the early twentieth century. The differences in their views are, then, not primarily philosophical differences. Also, it might be thought that Frege was interested in analyzing our ordinary mathematical notions, while Carnap was interested in the construction of arbitrary systems. I argue that this is not the case: our ordinary notions play, in a sense, an even more important role in Carnap’s philosophy of mathematics than they do in Frege’s. Finally, I address Tyler Burge’s interpretation of Frege which is in opposition to any Carnapian reading of Frege. (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)

I shall make two main claims. My first main claim is that Frege started out with a view of logic that is closer to Kant’s than is generally recognized, but that he gradually came to reject this Kantian view, or at least totally to transform it. My second main claim concerns Frege’s reasons for distancing himself from the Kantian conception of logic. It is natural to speculate that this change in Frege’s view of logic may have been (...) spurred by a desire to establish the logicality of the axiom system he needed for his logicist reduction, including the infamous Basic Law V. I admit this may have been one of Frege’s motives. But I shall argue that Frege also had a deeper and more interesting reason to reject his early Kantian view of logic, having to do with his increasingly vehement anti-psychologism. (shrink)

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. (shrink)

It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as (...) the first-order completeness is concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete. (shrink)

Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought (1879), by G. Frege.--Some metamathematical results on completeness and consistency; On formally undecidable propositions of Principia mathematica and related systems I; and On completeness and consistency (1930b, 1931, and 1931a), by K. Gödel.--Bibliography (p. [111]-116).

Frege begins his exposition of the symbol system employed in his Begriffsschrift by introducing the sign ⟝, whereby, he says, “[a] judgment is always to be expressed”.[The judgment sign] stands to the left of the sign or complex of signs in which the content of the judgment is given. If we omit the little stroke at the left of the horizontal stroke, then the judgment is to be transformed into a mere complex of ideas; the author is not expressing (...) his recognition or nonrecognition of the truth of this. (shrink)

Gottlob Frege (1848-1925) was a German logician, mathematician and philosopher who played a crucial role in the emergence of modern logic and analytic philosophy. Frege's logical works were revolutionary, and are often taken to represent the fundamental break between contemporary approaches and the older, Aristotelian tradition. He invented modern quantificational logic, and created the first fully axiomatic system for logic, which was complete in its treatment of propositional and first-order logic, and also represented the first treatment of higher-order (...) logic. In the philosophy of mathematics, he was one of the most ardent proponents of logicism, the thesis that mathematical truths are logical truths, and presented influential criticisms of rival views such as psychologism and formalism. His theory of meaning, especially his distinction between the sense and reference of linguistic expressions, was groundbreaking in semantics and the philosophy of language. He had a profound and direct influence on such thinkers as Russell, Carnap and Wittgenstein. Frege is often called the founder of modern logic, and he is sometimes even heralded as the founder of analytic philosophy. (shrink)

In Die Grundgesetze der Arithmetik (I, §48), Frege introduces his rule of the fusion of horizontals, according to which if an occurrence of the horizontal stroke is followed by another occurrence of the same stroke, either in isolation or “contained” in a propositional connective, the two occurrences can be fused with each other. However, the role of this rule, and of the horizontal sign more generally, is controversial; Michael Dummett notoriously claimed, for instance, that the horizontal is “wholly superfluous” (...) in Frege's logical system. In this paper, we challenge Dummett's view by providing a comprehensive analysis of the significance of the horizontal stroke. After some preliminary remarks, we argue that even if Frege's connectives in some sense “contain” the horizontal, yet they are total functions. Then, we take up the question of the sense expressed by the horizontal, and we claim that, unlike other sentential operators, the horizontal is not sense‐compositional. Finally, we consider the semantic and pragmatic aspects of Frege's horizontal in connection to his judgment stroke and the double judgment stroke. Contra Dummett, we argue that the horizontal is a special and indispensable element of Frege's logic. (shrink)

A widely held view about Fregean Sense has it that the determination of a sign's referent by the sign's sense is achieved viasatisfaction: the sense specifies a condition (or set of conditions) and the referent is that entity, if any, which uniquely satisfies that (set of) condition(s). This is usually held in conjunction with the claim that the sense is existentially and qualitatively independent of the referent: if the referent did not exist, or did not uniquely satisfy the sense, the (...) sense would still exist and would still specify the same condition(s) that it actually does (and might determine a different referent than its actual one). Proponents of this view give several reasons for holding it. I describe these reasons and argue that they are not convincing. More generally, I try to show that the notion of satisfaction has no useful application within Frege's system. I then suggest an alternative account of the determination of a referent by a sense that I think is truer to Frege and more illuminating. Compared to the satisfaction view, my account construes determination as a more naturalistic and epistemically real relation between speakers and things in the world. (shrink)

In 1998 a long-lost proposal for an election law by Gottlob Frege (1848–1925) was rediscovered in the _Thüringer Universitäts- und Landesbibliothek_ in Jena, Germany. The method that Frege proposed for the election of representatives of a constituency features a remarkable concern for the representation of minorities. Its core idea is that votes cast for unelected candidates are carried over to the next election, while elected candidates incur a cost of winning. We prove that this sensitivity to past elections (...) guarantees a proportional representation of political opinions in the long run. We find that through a slight modification of Frege’s original method even stronger proportionality guarantees can be achieved. This modified version of Frege’s method also provides a novel solution to the apportionment problem, which is distinct from all of the best-known apportionment methods, while still possessing noteworthy proportionality properties. (shrink)

Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege's; the (...) thought that things have a number only if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC. In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous. (shrink)

Frege believes (1) that his definition of number is (partly) arbitrary; (2) that it "makes" numbers of certain extensions; (3) that without such a definition we cannot even think or understand arithmetical propositions. this position is part of a view according to which mathematics in general involves the free construction of objects, their properties, and the very contents of mathematical propositions. frege tries to avoid excess subjectivism by the kantian device of treating alternative systems of arithmetic (e.g.) (...) as different appearances of a single realm of unstructured truths. (shrink)

Was Frege Liber Wahrheit sagt, lasst sich, mit ein wenig Gewalt, in zwei Schubfacher auiteilen} Das erste Schubfach — es trtigt die Aufschritt ,,Konstrul~:tives" enthalt die Lehre von der Wahrheit als Gegenstand und als Satzbedeutung. Das andere Schubfach —- mit der Aufschrift ,,Destruktives" —e ist reicher gefiilltg es finden sich hier Arguniente gegen die Korrespondenztheorie, gegen die De— iinierbarkeit von Wahrheit, gegen den Nutzen eines Walirheitsprttdikats und insbesondere Diagnosen dafur, welche Irrttimer den von Frege iiir falsch gehaltenen Auffassungen (...) tiber Wahrheit zugrunde liegen. Solche Diagnosen sind in Freges Schriften oft nur angedeutet. Aber sie sind wichtig, wenn man verstehen will, was Frege an Konstwktivem uber Wahrheit zu s-agen hat. Frege halt das Wort ,,wahr" und seine Bedeutung, also den Begriff und damit die Eigenschait der Wahrheit, iiir letztlich witzlos; der Satz ,,Dass Meerwasser salzig ist, ist wahr" sagt seiner Auffassung nach nicht mehr und nicht weniger als der einfache Satz ,,Meeiwasser ist salzig". Aus diesem Grund ist Frege oft als friiher Vertreter der Redundanztheorie der Wahrheit eingeordnet werden.; Im folenden versuche ich zu zeigen, dass Freges Verhtilmis zur Redundanztheorie differenzierter betrachtet werden sollte. Der Gnmd: Frege unterscheidet zwischen verschiedenen sprachlichen Mitteln des Aussagens von Wahrheit, von denen das Wort ,wahr‘ das Unwichtigste ist. Er schreibt: ,,Die Form des Behauptungssatzes ist [...] eigentlich das, womit wir die Wabrheit aussagen, und wir bedtirfen dazu des 'Wortes ,,wahr“ nicht.·}a, wir ktinnen sagen: selbst da, wo wir die Ausdrucksweise ,,es ist wahr, dass ..." anwcnden, ist eigentlich die Form des Behauptungssatzes das Wesentliche." (Frege 1933, S. 140) In der logischen Notation des Systems der Grundgesetze schltigt sich diese Auffassung darin nieder, dass diese Sprache trotz der Redundanzthese zwei nichtredundante wahrheitstliematisierende Ausdrticke enthalt: erstens den sog. Waagerechten ,,—·-— §", der als ,4; ist das .Wahre“ gelesen werden kann, und zweitens den Urteilsstrich ,,|", durch dessen Voransetzung an ,,—— A" ausgedrtickt wird, dass der durch ,,-—— A" bezeichnete Wahrheitswert als wahr be.. (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)

Joan Weiner has recently claimed that Frege neither uses, nor has any need to use, a truth-predicate in his justification of the logical laws. She argues that because of the assimilation of sentences to proper names in his system, Frege does not need to make use of the Quinean device of semantic ascent in order to formulate the logical laws, and that the predicate ‘is the True’, which is used in Frege's justification, is not to be considered (...) as a truth-predicate, because it does not apply to true sentences or true thoughts. The present paper aims to show that Frege needs to use, and does use, a truth-predicate in this context. It is argued, first, that Frege needs to use a truthpredicate in order to show that the truth of the logical laws is evident from the senses of the sentences by means of which they are formulated, and second, that the predicate that he actually uses, ‘is the True’, must be considered as a truth-predicate in the relevant sense, because it can be used and is actually used by Frege to explain the truth-conditions of thoughts. To defend this interpretation, it is discussed whether the explanatory use of ‘is the True’ in Frege's system is compatible with his deflationary analysis of ‘true’. The paper's conclusion is that there is indeed a conflict here; but, from Frege's point of view, this conflict is due merely to the logical imperfection of natural language and does not affect the proper system but only its propaedeutic. (shrink)

I show that Frege's statement (In the Epilogue to his Grundgesetze der Arithmetic v. II) of a way to avoid Russell's paradox is defective, in that he presents two different methods as if they were one. One of these "ways out" is notably more plausible than the other, and is almost surely what Frege really intended. The well-known arguments of Lesniewski, Geach, and Quine that Frege's revision of his system is inadequate to avoid paradox are not affected (...) by the ambiguity of Frede's statement. But a rectnt argument by Linsky and Schumm (Analysis 82 (1971-72), 5-7), intended as a very simple derivation of a contradiction within Frege's revised system, is valid only for the less plausible of the two versions of Frege's way out, and thus is not an effective attack on the revision that Frege intended to make. (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)