The intellectual innovations of two men were integral to the creation of FOL. FOL is a synthesis of these two men’s achievements; Frege’s in the field of Logic, Cantor’s in Mathematics. The essential role that continues to be played by the 4 symbols, which together make up the concrete, active residue of their contribution, in the construction of arguments in FOL and achievement of results using the FOL system represents an undeniable testimony to their considerable influence. Neither lived to see the fruits their isolated and independent intellectual projects would bring forth when carefully synthesized. Neither took any steps to achieve this synthesis, that was the work of others, those anonymous logician hordes whose own isolated, piecemeal projects unconsciously harkened to the dictates of an imperative that would only reveal itself as coursing through their efforts, with the stern, invisible inevitability of a gravitational force pulling space-lumps together into a novel constellational equipoise, retrospectively.
Cantor’s ‘Set Theory’ was an original, and massively influential, conceptualization of mathematical phenomena. It was a new way of thinking about relations between mathematical objects: in Cantor’s picture a mathematical object is a category, a definition, a set, into which other objects either did or did not fit, like a Russian Doll. When subjected to Set Theoretic exploration a mathematical object’s identity is rendered highly sensitive to its basis in multiplicity, to its existentially precarious status as the outcome of a certain combination of other identities that can also be combined to produce an entirely different identity – undergoing a Set-Theoretic exploration entails being subjected to the various enticements of such recombination. Instrumental to the functioning of the Set-Theoretic conceptualization are the axiom of Belonging and the axiom of Inclusion. For an object to Belong to a set every component into which it can be broken must also belong to the set, it must be able to resist recombination as a different set no matter how it is broken apart. This fidelity isn’t required of components that are included in a set, Inclusion is a more relaxed affair, come and go as you please. The sign for Belonging is ∈, the sign for Inclusion is ⊆. These signs were lifted for use in FOL, but purely as signs, not axioms – logical systems don’t run on axioms but on rules governing inference, there’s a difference.
For items in logical arguments to have relations of Belonging and Inclusion with one another, for logical arguments to be made about such relations, the items themselves must be different from those used in pre-FOL systems. After all, systems of logic aren’t just differentiated by the ways in which they argue truthfully but also by the subjects they make arguments about, the kinds of things they can be truthful about. In the case of FOL the necessary change had already been furnished – in the 1860’s Frege had created two quantifying symbols, the existential qualifier and the upside-down A – it just needed to be incorporated. Frege had developed these symbols in order to conscript new subjects for the arguments he hoped would be made in his own system, the Begriffschift. Before Frege logical propositions were only concerned with, and built up out of, singular items, atoms, ‘a’’s and ‘b’’s and ‘c’’s. But with the arch distinction between two different types of multiplicity, ‘some of’ and ‘all of’ (as in the distinction between ‘some of x have this property’ and ‘all of x have this property’), encoded in Frege’s existential qualifier (whose symbol is ∃ ) and his upside-down A (whose symbol is ∀ ), a wedge is driven into the conceptual field named ‘plurality’, sending potential lines of logical incursion through it like hairline fractures. This conceptual field has another name – ‘mathematics.’ Frege’s blunt binary, the distinction between ‘some of’ and ‘all of’, mathematizes logic. The incorporation of Cantorian set theory, which pertains to relations between multiples not singular objects, contemporizes this mathematization, allowing logic and maths to share a common language, not just a common subject (which is what Frege’s achievement would have amounted to on its own, regardless of whatever he may argue to the contrary).
The incorporation of these 4 new signs (whose succinct prettiness would have amounted to an argument for their incorporation that even the most pragmatic logician would have been loathe to disregard entirely) and the conceptual fields they grant access to (like magic keys) is what made FOL a step beyond the logical systems that preceded it.
The intellectual innovations of two men, Frege’s in the field of Logic and Cantor’s in the field of Mathematics, were especially important to the establishment of the FOL system. The Set-Theoretic conception of mathematical objects and the axioms which express and formulate this conception, which include the axiom of Belonging and the axiom of Inclusion among others, had a significant impact on the establishment of the FOL system, the Set-Theoretic conception does have bearing upon the way arguments are constructed in FOL. The inclusion of Frege’s existential qualifier and his ∀ in FOL allow the distinction between ‘some of’ and ‘all of’, as in the distinction between ‘some of x have this property’ and ‘all of x have this property’, to be expressed in logical form. FOL is nevertheless a different system from Frege’s Begriffschift, which he hoped would provide a new basis for Logic on its own. The construction of the FOL system is associated with the ‘mathematisation of logic’, the intellectual innovations of Frege and Cantor influenced the construction of the FOL system. The construction of the FOL system represents a ‘step beyond’ the types of logic that preceded it, the intellectual innovations of Frege and Cantor influenced the construction of the FOL system.
The intellectual innovations of Frege, in the field of Logic, and Cantor, in the field of Mathematics, influenced the construction of the FOL system. The ∃ and the ∀ symbols are different and are used in FOL.