Where is ‘There is’ in ‘∃’?

Abstract

ABSTRACT The paper offers a survey of four key moments in which symbolisms for quantification were first introduced: §§11–2 of Frege’s Begriffsschrift (1879); Peirce’s ‘Algebra of Logic’ (1885); Peano’s ‘Studii di Logica matematica’ (1897); and *9 (‘replaced’ by *8 in the second edition) of Whitehead and Russell’s Principia Mathematica (1910). Despite their divergent aims, these authors present substantially equivalent visions of what their differing symbolisms express. In each case, some passage suggests that one (but not the only) way to render one of the symbols into ordinary-language words (German, English and Italian) is to say that there is something or some thing(s) exist(s). Exactly how this comes out varies from language to language, but the point remains the same. As a result, in almost all recent logic manuals that introduce ‘∃’, some passage says that the symbol means ‘there is’ or ‘there exists’, and the tradition has grown up of labelling ‘∃’ ‘the existential quantifier’. Some considerations are offered for deprecating this reading of ‘∃’ and the label adopted for it, and for preferring to read it as ‘for some (at least one)’ and for speaking of the ‘particular quantifier’.