Freges Logic

2005.11.07

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  • Frege's Logic, Theorem, and Foundations for Arithmetic
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    • Frege and Carnap on the Normativity of Logic.

    An encyclopedia of philosophy articles written by professional philosophers.

    Florian Steinberger - - Synthese 1: Frege on the Relations Between Logic and Thought. Simon Evnine - manuscript. Meaning and Ontology in Frege's Philosophy, Vol. Sense and Reference in Frege's Philosophy. Frege's Logic, Theorem, and Foundations for Arithmetic.

    Zalta - - Stanford Encyclopedia of Philosophy. Frege on Truth, Beauty and Goodness. Simon Evnine - - Manuscrito 26 2: From Arithmetic to Analytic Philosophy. Joan Weiner - - Open Court. Truth, Assertion, and the Horizontal: Frege on "the Essence of Logic". Taschek - - Mind Frege on Indirect Proof. Ivan Welty - - History and Philosophy of Logic 32 3: Frege's Conception of Logic: Frege's goal was to show that mathematics grows out of logic , and in so doing, he devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down to him in the logical tradition.

    In effect, Frege invented axiomatic predicate logic , in large part thanks to his invention of quantified variables , which eventually became ubiquitous in mathematics and logic, and which solved the problem of multiple generality. Previous logic had dealt with the logical constants and , or , if A frequently noted example is that Aristotle's logic is unable to represent mathematical statements like Euclid's theorem , a fundamental statement of number theory that there are an infinite number of prime numbers.

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    The most dramatic difference is that Frege's logic allows us to define concepts using nested. The axioms for propositional logic. . Though Frege essentially had a second- order logic in Gg, his rules of inference don't look as.

    Frege's "conceptual notation" however can represent such inferences. One of Frege's stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to "intuition". If there was an intuitive element, it was to be isolated and represented separately as an axiom: Having exhibited this possibility, Frege's larger purpose was to defend the view that arithmetic is a branch of logic, a view known as logicism: Already in the Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy , were derived within what Frege understood to be pure logic.

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    This idea was formulated in non-symbolic terms in his The Foundations of Arithmetic Later, in his Basic Laws of Arithmetic vol. Most of these axioms were carried over from his Begriffsschrift , though not without some significant changes.

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    The one truly new principle was one he called the Basic Law V: The crucial case of the law may be formulated in modern notation as follows. The case is special because what is here being called the extension of a predicate, or a set, is only one type of "value-range" of a function. In a famous episode, Bertrand Russell wrote to Frege, just as Vol.

    It is easy to define the relation of membership of a set or extension in Frege's system; Russell then drew attention to "the set of things x that are such that x is not a member of x ". The system of the Grundgesetze entails that the set thus characterised both is and is not a member of itself, and is thus inconsistent. Frege wrote a hasty, last-minute Appendix to Vol.

    Frege opened the Appendix with the exceptionally honest comment: This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion. Frege's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourse , and hence is worthless indeed, this would make for a contradiction in Frege's system if he had axiomatized the idea, fundamental to his discussion, that the True and the False are distinct objects; see, for example, Dummett , but recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways:.

    Frege's work in logic had little international attention until when Russell wrote an appendix to The Principles of Mathematics stating his differences with Frege. The diagrammatic notation that Frege used had no antecedents and has had no imitators since.

    Frege's Logic, Theorem, and Foundations for Arithmetic

    Moreover, until Russell and Whitehead's Principia Mathematica 3 vols. Frege's logical ideas nevertheless spread through the writings of his student Rudolf Carnap — and other admirers, particularly Bertrand Russell and Ludwig Wittgenstein — Frege is one of the founders of analytic philosophy , whose work on logic and language gave rise to the linguistic turn in philosophy.

    His contributions to the philosophy of language include:.

    • Il messico insorge (Libri da premio) (Italian Edition);
    • Similar books and articles.
    • Frege's Logic — Danielle Macbeth | Harvard University Press.
    • Elaboraciones básicas y platos elementales con pescados (Spanish Edition).

    As a philosopher of mathematics, Frege attacked the psychologistic appeal to mental explanations of the content of judgment of the meaning of sentences. His original purpose was very far from answering general questions about meaning; instead, he devised his logic to explore the foundations of arithmetic, undertaking to answer questions such as "What is a number? It should be kept in mind that Frege was employed as a mathematician, not a philosopher, and he published his philosophical papers in scholarly journals that often were hard to access outside of the German-speaking world.

    He never published a philosophical monograph other than The Foundations of Arithmetic , much of which was mathematical in content, and the first collections of his writings appeared only after World War II. A volume of English translations of Frege's philosophical essays first appeared in , edited by students of Wittgenstein, Peter Geach and Max Black —88 , with the bibliographic assistance of Wittgenstein see Geach, ed.

    Despite the generous praise of Russell and Wittgenstein, Frege was little known as a philosopher during his lifetime. His ideas spread chiefly through those he influenced, such as Russell, Wittgenstein, and Carnap, and through work on logic and semantics by Polish logicians. While conventional accounts of meaning took expressions to have just one feature reference , Frege introduced the view that expressions have two different aspects of significance: Reference , or, "Bedeutung" applied to proper names , where a given expression say the expression "Tom" simply refers to the entity bearing the name the person named Tom.

    Frege also held that propositions had a referential relationship with their truth-value in other words, a statement "refers" to the truth-value it takes. By contrast, the sense or "Sinn" associated with a complete sentence is the thought it expresses.

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    Let us call the new, defined symbol introduced in a definition the definiendum , and the term that is used to define the new term the definiens. In Begr , Frege invented the predicate calculus. Thus, the incomplete expression "the square root of " contains a blank spot, which, when completed by an expression referring to a number, yields a complex expression also referring to a number, e. Thus, Frege analyzed the above inferences in the following general way:. The traditional view is that one must either restrict Basic Law V or restrict the Comprehension Principle for Concepts. This notation can be extended for relational concepts.

    The sense of an expression is said to be the "mode of presentation" of the item referred to, and there can be multiple modes of representation for the same referent. The distinction can be illustrated thus: But the sense of the word "Wales" is a part of the sense of the latter expression, but no part of the sense of the "full name" of Prince Charles. These distinctions were disputed by Bertrand Russell, especially in his paper " On Denoting "; the controversy has continued into the present, fueled especially by Saul Kripke 's famous lectures " Naming and Necessity ".

    Frege's published philosophical writings were of a very technical nature and divorced from practical issues, so much so that Frege scholar Dummett expresses his "shock to discover, while reading Frege's diary, that his hero was an anti-Semite. In the last year of his life, at the age of 76, his diary contains extreme right-wing political opinions, opposing the parliamentary system, democrats, liberals, Catholics, the French and Jews, who he thought ought to be deprived of political rights and, preferably, expelled from Germany. Some interpretations have been written about that time.

    Frege had friendly relations with Jews in real life: Frege was described by his students as a highly introverted person, seldom entering into dialogue, mostly facing the blackboard while lecturing though being witty and sometimes bitterly sarcastic. Die Grundlagen der Arithmetik: Verlag Hermann Pohle online version. Frege intended that the following three papers be published together in a book titled Logische Untersuchungen Logical Investigations. Though the German book never appeared, the papers were published together in Logische Untersuchungen , ed.