  Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html ) is a
research program for redeveloping logic as a formal theory of computability, as
opposed to the formal theory of truth which it has more traditionally been.
Formulas in CL stand for interactive computational problems, seen as games
between a machine and its environment; logical operators represent operations
on such entities; and "truth" is understood as existence of an effective
solution. The formalism of CL is open-ended, and may undergo series of
extensions as the studies of the subject advance. So far three -- parallel,
sequential and choice -- sorts of conjunction and disjunction have been
studied. The present paper adds one more natural kind to this collection,
termed toggling. The toggling operations can be characterized as lenient
versions of choice operations where choices are retractable, being allowed to
be reconsidered any finite number of times. This way, they model
trial-and-error style decision steps in interactive computation. The main
technical result of this paper is constructing a sound and complete
axiomatization for the propositional fragment of computability logic whose
vocabulary, together with negation, includes all four -- parallel, toggling,
sequential and choice -- kinds of conjunction and disjunction. Along with
toggling conjunction and disjunction, the paper also introduces the toggling
versions of quantifiers and recurrence operations.
