User:Logicist/History of Logic

The rise of modern logic
The period which followed the important developments in logic in the thirteenth and early fourteenth century and the beginning of the nineteenth century was largely one of decline and neglect, and is generally regarded as barren by historians of logic. The revival of logic happened in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics. The development of the modern so-called 'symbolic' or 'mathematical' logic during this period is the most significant in the two thousand-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.

A number of features distinguish modern from the old Aristotelian or traditional logic, the most important of which are as follows. Modern logic is fundamentally a calculus. whose rules of operation are determined only by the shape, and not by the meaning of the symbols it employs, as in mathematics. Many logicians were impressed by the 'success' of mathematics, in that there has been no prolonged dispute about any properly mathematical result. C.S. Peirce noted that even though a mistake in the evaluation of a definite integral by Laplace led to an error concerning the moon's orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in metaphysics. He argued that a truly 'exact' logic will depends upon mathematical, i.e. 'diagrammatic' or 'iconic' thought. "Those who follow such methods will ... escape all error except such as will be speedily corrected after it is once suspected". Modern logic is also 'constructive' rather than 'abstractive' i.e. rather than abstracting and formalising theorems derived from ordinary language (or from psychological intuitions about validity) it constructs theorems by formal methods, then looks for an interpretation in ordinary language. It is entirely symbolic, meaning that even the logical constants (which the medevial logicians called 'syncategoremata') as well as the categoric terms are expressed in symbols. Finally, modern logic strictly avoids psychological, epistemological and metaphysical questions.

Periods of modern logic
The development of modern logic falls into roughly five periods, as follows.


 * 1) The pre-historical or embryonic period from Leibniz to 1847, where the notion of a logical calculus is discussed and developed, particularly by Leibniz, but where no schools are formed, and where isolated periodic attempts were abandoned or went unnoticed.
 * 2) The algebraic period from Boole's Analysis to Schroeder's Vorlesungen.  In this period there are more practitioners, and a greater continuity of development.
 * 3) The logicist period from the Begriffsschrift of Frege to the Principia Mathematica of Russell and Whitehead. This was dominated by the 'logicist school', whose aim was to incorporate the logic of all mathematical and scientific discourse in a single unified system, and which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology.  The major logicists were Frege, Russell, and the early Wittgenstein (Companion p.499). It culminates with the Principia, an important work which includes a thorough examination and attempted solution of the antimonies which had been an obstacle to earlier progress.
 * 4) The metalogical period from 1910 to the 1930's which saw the development of metalogic, in the finitist system of Hilbert, and the non-finitist system of Lowenheim and Skolem, the combination of logic and metalogic in the work of Godel and Tarski.  Godel's incompleteness theorem of 1931 was one of the greatest achievements in the history of logic.  Later in the 1930's Godel developed the notion of set-theoretic constructibility.
 * 5) The period after the World war II, and important proof discovered by Paul Cohen of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory, made possible by his invention of the powerful technique called forcing

Embryonic period
The idea that inference could be represented by a purely mechanical process is found as early as Raymond Lull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. Three hundred years later, the English philosopher and logician Thomas Hobbes suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction. The same idea is found in the work of Leibniz, who had read both Lull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like both Lull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was never published until long after his death. Leibniz says that ordinary language are subject to 'countless ambiguities' and are unsuited for a calculus, whose task is to expose mistakes in inference arising from the forms and structures of words. Gergonne (1816) says that reasoning does not have to be about objects which we have perfectly clear ideas about, since algebraic operations can be carried out without our having any idea of the meaning of the symbols involved. Bolzano anticipated a fundamental idea of modern proof theory when he defined logical consequence or 'deducibility' in terms of variables: a set of propositions n, o, p ... are deducible from propositions a, b, c ... in respect of the variables i, j, ... if any substitution for i, j that have the effect of making a, b, c ... true, simultaneously make the propositions n, o, p ... also. This is now known as semantic validity.

Algebraic period
Modern logic begins with the so-called 'algebraic school', originating with Boole and including Peirce, Jevons, Schroeder and Venn. Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions and probabilities.

The school begins with Boole's seminal work Mathematical Analysis of Logic which appeared in 1847, although De Morgan (1847) is its immediate precursor. The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in Lincoln. For example, let x and y stand for classes let the symbol = signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these elective symbols, i.e. symbols which select certain objects for consideration. An expression in which elective symbols are used is called an elective function, and an equation of which the members are elective functions, is an elective equation. The theory of elective functions and their 'development' is essentially the modern idea of truth-functions and their expression in disjunctive normal form.

Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between 'primary propositions' which are the subject of syllogistic theory, and 'secondary propositions', which are the subject of propositional logic, and showed how under different 'interpretations' the same algeberaic system could represent both. An example of a primary proposition is 'All inhabitants are either Europeans or Asiatics'. An example of a secondary proposition is 'Either all inhabitants are Europeans or they are all Asiatics'. These are easily distinguished in modern propositional calculus, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean system.

In his Symbolic Logic (1881), John Venn used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a 'logical machine' which he showed to the Royal Society the following year. In 1885 Allan Marquand proposed an electrical version of the machine that is still extant (picture at the Firestone Library).

The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from Quantity in 1864, where he suggested a symbol to signify exclusive or, which allowed Boole's system to be greatly simplified. This was usefully exploited by Schroeder when he set out theorems in parallel columns in his Vorlesungen (1890-1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive 'either ... or', however, like many of Peirce's innovations, this passed without notice until Sheffer rediscovered it in 1913. Boole's early work also lacks the idea of the logical sum which originates in Peirce (1867), Schroeder (1877) and Jevons (1890), and the concept of inclusion, first suggested by Gergonne (1816) and clearly articulated by Peirce (1870).

The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most ambitious was Schroder's monumental Vorlesungen über die Algebra der Logik (vol iii 1895), although the original idea was again anticipated by Peirce.

Logicist period
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. Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or Begriffschrift is considered the greatest single achievement in the entire history of logic. 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 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'.

Frege's first work, the Begriffschrift 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. The most signification innovation, however, was his explanation of the quantifier in terms of the 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'. 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


 * (x) Ax -> Bx

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. 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 Europeans or Asiatics' is


 * (x) [ I(x) -> (E(x) v A(x)) ]

whereas 'All the inhabitants are Europeans or all the inhabitants are Asiatics' is


 * (x) (I(x) -> E(x)) v (x) (I(x) -> A(x))

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'

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 captures this through the different scope of the quantifiers. Thus


 * (x) [ girl(x) -> E(y) (boy(y) & kissed(x,y) ]

means that to every girl there corresponds some boy (any one will do) who the girl kissed. But


 * (x) [ boy(x) & (y) (girl(y) -> kissed(y,x) ]

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 many-to-one relation, and hence of mathematical induction.

Extra bit
This period overlaps with the work of the so-called 'mathematical school' which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting whose objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory.

Inclusion
The modern symbol for inclusion first appears in Gergonne (1816), who defines it as one idea 'containing' or being 'contained' by another, using the upside-down letter 'C' to express this. Peirce articulated this clearly in 1870, arguing also that inclusion was a wider concept than equality, and hence a logically simpler one. Schroder (also Frege) calls the same concept 'subordination'.