Editing History of logic (section)

===Logicist period===
[[File:Young frege.jpg|thumb|160px|Gottlob Frege.]]
After Boole, the next great advances were made by the German mathematician [[Gottlob Frege]]. Frege's objective was the program of [[Logicism]], i.e. demonstrating that arithmetic is identical with logic.<ref name="k435">Kneale p. 435</ref> Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or [[Begriffsschrift]] is important.<ref name="k435"/> Frege also tried to show that the concept of [[number]] can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first writer to suggest this. In his pioneering work {{Lang|de|Die Grundlagen der Arithmetik}} (The Foundations of Arithmetic), sections 15–17, he acknowledges the efforts of Leibniz, [[J. S. Mill]] as well as Jevons, citing the latter's claim that "algebra is a highly developed logic, and number but logical discrimination."<ref>Jevons, ''The Principles of Science'', London 1879, p. 156, quoted in ''Grundlagen'' 15</ref>

Frege's first work, the ''Begriffsschrift'' ("concept script") is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of inference (''modus ponens'' and substitution), and six axioms. Frege referred to the "completeness" of this system, but was unable to prove this.<ref>Beaney p. 10 – the completeness of Frege's system was eventually proved by [[Jan Łukasiewicz]] in 1934</ref> The most significant innovation, however, was his explanation of the [[Quantifier (logic)|quantifier]] in terms of mathematical functions. Traditional logic regards the sentence "Caesar is a man" as of fundamentally the same form as "all men are mortal." Sentences with a proper name subject were regarded as universal in character, interpretable as "every Caesar is a man".<ref>See for example the argument by the medieval logician [[William of Ockham]] that singular propositions are universal, in [[Summa Logicae]] III. 8 (??)</ref> At the outset Frege abandons the traditional "concepts ''subject'' and ''predicate''", replacing them with ''argument'' and ''function'' respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words ''if, and, not, or, there is, some, all,'' and so forth, deserves attention".<ref>{{harvnb |Frege |1879}} in {{harvnb |van Heijenoort |1967 |p=7}}</ref> Frege argued that the quantifier expression "all men" does not have the same logical or semantic form as "all men", and that the universal proposition "every A is B" is a complex proposition involving two ''functions'', namely ' – is A' and ' – is B' such that whatever satisfies the first, also satisfies the second. In modern notation, this would be expressed as

: <math>\forall \; x \big( A(x) \rightarrow B (x) \big)</math>

In English, "for all x, if Ax then Bx". Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. not reducible to a general proposition. Universal and particular propositions, by contrast, are not of simple subject-predicate form at all. If "all mammals" were the logical subject of the sentence "all mammals are land-dwellers", then to negate the whole sentence we would have to negate the predicate to give "all mammals are ''not'' land-dwellers". But this is not the case.<ref>"On concept and object" p. 198; Geach p. 48</ref> This functional analysis of ordinary-language sentences later had a great impact on philosophy and [[linguistics]].

This means that in Frege's calculus, Boole's "primary" propositions can be represented in a different way from "secondary" propositions. "All inhabitants are either men or women" is
[[File:BS-13-Begriffsschrift Quantifier2-svg.svg|130px|alt=Straight line with bend; text "x" over bend; text "F(x)" to the right of the line.|thumb|[[Frege]]'s "Concept Script"]]
: <math>\forall \; x \Big( I(x) \rightarrow \big( M(x) \lor W(x) \big) \Big) </math>

whereas "All the inhabitants are men or all the inhabitants are women" is

: <math>\forall \; x \big( I(x) \rightarrow M(x) \big) \lor \forall \;x \big( I(x) \rightarrow W(x) \big)</math>

As Frege remarked in a critique of Boole's calculus:

: "The real difference is that I avoid [the Boolean] division into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it."<ref>BLC p. 14, quoted in Beaney p. 12</ref>

As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient [[problem of multiple generality]]. The ambiguity of "every girl kissed a boy" is difficult to express in traditional logic, but Frege's logic resolves this through the different scope of the quantifiers. Thus

:<math>\forall \; x \Big( G(x) \rightarrow  \exists \; y \big( B(y) \land K(x,y) \big) \Big)</math>
[[File:Giuseppe_Peano.jpg|thumb|120px|Peano]]
means that to every girl there corresponds some boy (any one will do) who the girl kissed. But

:<math>\exists \;x \Big( B(x) \land \forall \;y \big( G(y) \rightarrow K(y, x) \big) \Big)</math>

means that there is some particular boy whom every girl kissed. Without this device, the project of logicism would have been doubtful or impossible. Using it, Frege provided a definition of the [[ancestral relation]], of the [[Injective function|many-to-one relation]], and of [[mathematical induction]].<ref>See e.g. [http://www.utm.edu/research/iep/f/frege.htm The Internet Encyclopedia of Philosophy], article "Frege"</ref>
[[File:Ernst Zermelo 1900s.jpg|thumb|left|130px|Ernst Zermelo]]
This period overlaps with the work of what is known as the "mathematical school", which included [[Richard Dedekind|Dedekind]], [[Moritz Pasch|Pasch]], [[Giuseppe Peano|Peano]], [[David Hilbert|Hilbert]], [[Ernst Zermelo|Zermelo]], [[Edward Vermilye Huntington|Huntington]], [[Oswald Veblen|Veblen]] and [[Arend Heyting|Heyting]]. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was [[Hilbert's Program]], which sought to ground all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. The standard [[axiomatization]] of the [[natural number]]s is named the [[Peano axioms]] eponymously. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of Frege's work, he independently recreated his logical apparatus based on the work of Boole and Schröder.<ref>Van Heijenoort 1967, p. 83</ref>

The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by [[Bertrand Russell]]. This proved Frege's [[naive set theory]] led to a contradiction. Frege's theory contained the axiom that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not).<ref>See e.g. Potter 2004</ref> This contradiction is now known as [[Russell's paradox]]. One important method of resolving this paradox was proposed by [[Ernst Zermelo]].<ref>Zermelo 1908</ref> [[Zermelo set theory]] was the first [[axiomatic set theory]]. It was developed into the now-canonical [[Zermelo–Fraenkel set theory]] (ZF). Russell's paradox symbolically is as follows:

:<math>\text{Let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R</math>
The monumental [[Principia Mathematica]], a three-volume work on the [[foundations of mathematics]], written by Russell and [[Alfred North Whitehead]] and published 1910–1913 also included an attempt to resolve the paradox, by means of an elaborate [[system of types]]: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "[[set of all sets]]". The ''Principia'' was an attempt to derive all mathematical truths from a well-defined set of [[axiom]]s and [[inference rule]]s in [[Mathematical logic|symbolic logic]].