structural:scholar_edition
History

Term logic, also called Aristotelian logic, treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Term logic treated All, Some and No in the 4th century BC, in an account also touching on the alethic modalities.

In 1827, George Bentham published his Outline of a New System of Logic: With a Critical Examination of Dr. Whately's Elements of Logic, describing the principle of the quantifier, but the book was not widely circulated.

William Hamilton claimed to have coined the terms "quantify" and "quantification", most likely in his Edinburgh lectures c. 1840. Augustus De Morgan confirmed this in 1847, but modern usage began with De Morgan in 1862 where he makes statements such as "We are to take in both all and some-not-all as quantifiers".

Gottlob Frege, in his 1879 Begriffsschrift, was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates. He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas. Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀ x ~, or contraposition. Frege's treatment of quantification went largely unremarked until Bertrand Russell 's 1903 Principles of Mathematics.

In work that culminated in Peirce (1885), Charles Sanders Peirce and his student Oscar Howard Mitchell independently invented universal and existential quantifiers, and bound variables. Peirce and Mitchell wrote Π and Σ where we now write ∀ x and ∃ x. Peirce's notation can be found in the writings of Ernst Schröder, Leopold Loewenheim, Thoralf Skolem, and Polish logicians into the 1950s. Most notably, it is the notation of Kurt Gödel 's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic.

Peirce's approach to quantification also influenced William Ernest Johnson and Giuseppe Peano, who invented yet another notation, namely (x) for the universal quantification of x and (in 1897) ∃ x for the existential quantification of x. Hence for decades, the canonical notation in philosophy and mathematical logic was (x) P to express "all individuals in the domain of discourse have the property P," and "(∃ x) P " for "there exists at least one individual in the domain of discourse having the property P." Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the Principia Mathematica of Whitehead and Russell, Quine, and Alonzo Church. In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1960s.

Around 1895, Peirce began developing his existential graphs, whose variables can be seen as tacitly quantified. Whether the shallowest instance of a variable is even or odd determines whether that variable's quantification is universal or existential. (Shallowness is the contrary of depth, which is determined by the nesting of negations.) Peirce's graphical logic has attracted some attention in recent years by those researching heterogeneous reasoning and diagrammatic inference.