Quantifier logic
Propositions analysed into function and argument, 'all' and 'some' expressed by quantifiers; two thousand years of syllogistic were surpassed at a stroke. Inferences with multiple generality could be formalised for the first time.
The Nineteenth Century · 1848 – 1925
Gottlob Frege
弗雷格 · Friedrich Ludwig Gottlob Frege
Inventor of modern logic: quantifiers, function and argument, sense and reference. He set out to reduce arithmetic to logic; a letter from Russell brought the edifice down, but analytic philosophy begins with him.
Analytic PhilosophyBirthplace: Wismar, Mecklenburg
Never ask for the meaning of a word in isolation, but only in the context of a proposition.
Source: The Foundations of Arithmetic, introduction
Frege was born in Wismar on the Baltic to two schoolteachers; his father ran a girls' school. He studied mathematics at Jena and Göttingen, returned to Jena in 1874 as an unsalaried lecturer and taught mathematics there for forty-four years without ever becoming full professor, with almost no dealings with colleagues, and often only one or two students in his lectures, one of them Carnap.
In 1879 he published the Begriffsschrift, subtitled 'a formula language of pure thought modelled on that of arithmetic'. This booklet of eighty-eight pages invented modern logic: function and argument in place of subject and predicate, quantifiers to handle 'all' and 'some', and the first complete axiom system for first-order predicate and propositional calculus. It was barely reviewed, and reviewers complained of the strange notation. It is now regarded as the most important work in logic since Aristotle.
The Foundations of Arithmetic (1884) argued in ordinary language for his logicist programme: arithmetical truths are analytic, numbers are extensions of concepts, and the answer to 'how many Fs' is an object. 'On Sense and Reference' (1892) distinguished an expression's sense (mode of presentation) from its reference (the thing designated): 'the morning star' and 'the evening star' refer to one planet but differ in sense, which is why 'the morning star is the evening star' is informative. The paper is the starting point of the philosophy of language. The first volume of the Basic Laws of Arithmetic (1893) began the formal derivation.
In June 1902, with the second volume in press, he received a letter from Russell pointing out that his Basic Law V allowed the construction of 'the set of all sets that are not members of themselves', which yields a contradiction. He wrote in an appendix: 'Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished.' His attempted repair failed, and late in life he abandoned logicism for the view that arithmetic rests on geometry. His political diary contains antisemitic remarks. Russell, Wittgenstein and Carnap carried his work to the world, and Dummett called him the grandfather of analytic philosophy.
Key Ideas
Sense and reference
An expression has two semantic dimensions, sense and reference. A sentence's reference is its truth value, its sense the thought it expresses. This explains why identity statements can be informative.
Logicism
Arithmetic can be derived wholly from logical laws and definitions; numbers are logical objects. The programme failed, but it gave birth to the modern study of the foundations of mathematics.
Anti-psychologism
The laws of logic are laws of truth, not of thinking; thoughts are objective, not mental images. The first principle of analytic philosophy.
The context principle
Only in the context of a sentence does a word have meaning. The unit of meaning is the sentence, not the word.
Influenced by
- Gottfried Wilhelm Leibniz
His Begriffsschrift realised the 'universal characteristic' Leibniz had dreamed of: a notation in which reasoning is as transparent as calculation.
- Immanuel Kant
He agreed with Kant that geometry is synthetic a priori but set out to show that arithmetic is analytic: number can be derived from logic alone, without intuition.
- Aristotle
- John Stuart Mill
The Foundations of Arithmetic devotes a section to mocking Mill's empiricism about mathematics: if 2+3=5 came from observing pebbles, arithmetic textbooks would be geological reports.
Influenced
- Edmund Husserl
Frege's 1894 attack on the psychologism of his early work pushed him toward the objectivity of logic; the first volume of the Logical Investigations settles accounts with psychologism.
- Bertrand Russell
He was Frege's first important reader and the one whose paradox wrecked the system; the theory of types was his repair of Frege's logicism.
- Ludwig Wittgenstein
He said Frege's work was what he most admired; in 1911 he went to Jena to see him, and Frege sent him to Russell. The Tractatus' context principle and anti-psychologism come straight from Frege.
- Rudolf Carnap
He was one of Frege's very few students at Jena, attending three courses between 1910 and 1914, which he called the source of his whole logical training.
- W. V. O. Quine
- G. E. M. Anscombe
- Saul Kripke
The target of Naming and Necessity is the Frege–Russell description theory: a proper name's reference is not fixed by a sense; names are rigid designators, referring to the same object in every possible world.