  The Sigma formulas of the language of arithmetic express semidecidable
relations on the natural numbers. More generally, whenever a totality of
objects is regarded as incomplete, the Sigma formulas express relations that
are witnessed in a completed portion of that totality when they hold. In this
sense, the Sigma formulas are more concrete semantically than other first-order
formulas.
  We describe a system of deduction that uses only Sigma formulas. Each axiom,
an implication between two Sigma formulas, is implemented as a rewriting rule
for subformulas. We exhibit a complete class of logical axioms for this system,
and we observe that a distributive law distinguishes classical reasoning from
intuitionistic reasoning in this setting.
  Skolem's theory PRA of primitive recursive arithmetic can be formulated in
our deductive system. In Skolem's system, free variables are universally
quantified implicitly, but in our formulation, free variables act as parameters
to the deduction. In this sense, our formulation is more explicitly finitistic.
Furthermore, most of our results are themselves finistic, being theorems of
PRA. In particular, appending our main theorem to a celebrated chain of
reductions from reverse mathematics, we find that an implication of Sigma
formulas is derivable in WKL_0 if and only if there is a deduction from the
antecedent to the consequent in our formulation of PRA.
