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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| History of logic | 8/13 | https://en.wikipedia.org/wiki/History_of_logic | reference | science, encyclopedia | 2026-05-05T03:59:53.898187+00:00 | kb-cron |
The idea that inference could be represented by a purely mechanical process is found as early as Raymond Llull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the Oxford Calculators led to a method of using letters instead of writing out logical calculations (calculationes) in words, a method used, for instance, in the Logica magna by Paul of Venice. Three hundred years after Llull, 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 Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death. Leibniz says that ordinary languages 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; hence, he proposed to identify an alphabet of human thought comprising fundamental concepts which could be composed to express complex ideas, and create a calculus ratiocinator that would make all arguments "as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate."
Gergonne (1816) said that reasoning does not have to be about objects about which one has perfectly clear ideas, because algebraic operations can be carried out without 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:Hence I say that propositions
M
{\displaystyle M}
,
N
{\displaystyle N}
,
O
{\displaystyle O}
,... are deducible from propositions
A
{\displaystyle A}
,
B
{\displaystyle B}
,
C
{\displaystyle C}
,
D
{\displaystyle D}
,... with respect to variable parts
i
{\displaystyle i}
,
j
{\displaystyle j}
,..., if every class of ideas whose substitution for
i
{\displaystyle i}
,
j
{\displaystyle j}
,... makes all of
A
{\displaystyle A}
,
B
{\displaystyle B}
,
C
{\displaystyle C}
,
D
{\displaystyle D}
,... true, also makes all of
M
{\displaystyle M}
,
N
{\displaystyle N}
,
O
{\displaystyle O}
,... true. Occasionally, since it is customary, I shall say that propositions
M
{\displaystyle M}
,
N
{\displaystyle N}
,
O
{\displaystyle O}
,... follow, or can be inferred or derived, from
A
{\displaystyle A}
,
B
{\displaystyle B}
,
C
{\displaystyle C}
,
D
{\displaystyle D}
,.... Propositions
A
{\displaystyle A}
,
B
{\displaystyle B}
,
C
{\displaystyle C}
,
D
{\displaystyle D}
,... I shall call the premises,
M
{\displaystyle M}
,
N
{\displaystyle N}
,
O
{\displaystyle O}
,... the conclusions.This is now known as semantic validity.
=== Algebraic period ===
Modern logic begins with what is known as the "algebraic school", originating with Boole and including Peirce, Jevons, Schröder, 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, Lincolnshire. 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 algebraic 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 predicate logic, 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).