 
Summary: Back to the Future: Explicit Logic for Computer
Science
Sergei Artemov #
We will speak about three traditions in Logic:
. Classical, usually associated with Frege, Hilbert, G˜odel, Tarski, and others;
. Intuitionistic, founded by Brouwer, Heyting, Kolmogorov, G˜odel, Kleene,
and others;
. Explicit, which we trace back to Skolem, Curry, G˜odel, Church, and others.
The classical tradition in logic based on quantifiers # and # essentially re
flected the 19th century mathematician's way of representing dependencies be
tween entities. A sentence #x#yA(x, y), though specifying a certain relation
between x and y, did not mean that the latter is a function of the former, let
alone a computable one. The Intuitionistic approach provided a principal shift
toward the e#ective functional reading of the mathematician's quantifiers. A
new, nonTarskian semantics had been suggested by Kleene: realizability that
revealed a computational content of logical derivations. In a decent intuitionstic
system, a proof of #x#yA(x, y) yields a program f that computes y = f(x).
Explicit tradition makes the ultimate step by using representative systems
of functions instead of quantifiers from the very beginning. Since the work of
Skolem, 1920, it has been known that the classical logic can be adequately recast
