[FOM] Understanding universal quantification

[Giuseppina Ronzitti]

It is sometimes claimed that universal quantification  is meaningful
even without reference to the domain of quantification of the variable,
the intended meaning of the quantification being conveyed by one's
understanding of the introduction and elimination rules of a (natural
deduction, sequents) logical system. It seems thus that  universal
quantification would  not commit one to the assertion of the ontological
existence of any totality (omega, power set).

Dag Prawitz remarks ("Proof and the meaning and completeness of the
logical constants") that the conditions for understanding the meaning of
the introduction rules, and thus of the rule of the introduction of
universal quantification, are given by specifying  the forms of the
canonical arguments  ( thus simply applying the inference t --> (x) A(x)
). No problem thus for introduction. But, the conditions for
understanding the meaning of elimination rules, and thus of  the rule of
elimination of universal quantifier, are given by the reduction
procedures (normalization).

The proof-theoretic approach to the understanding of the meaning of
quantification rests thus on not constructive tools and therefore does
not seem  a viable means to eliminate ontological commitments.

G. Ronzitti