  Crispin Wright in his 1982 paper argues for strict finitism, a constructive
standpoint that is more restrictive than intuitionism. In its appendix, he
proposes models of strict finitistic arithmetic. They are tree-like structures,
formed in his strict finitistic metatheory, of equations between numerals on
which concrete arithmetical sentences are evaluated. As a first step towards
classical formalisation of strict finitism, we propose their counterparts in
the classical metatheory with one additional assumption, and then extract the
propositional part of `strict finitistic logic' from it and investigate. We
will provide a sound and complete pair of a Kripke-style semantics and a
sequent calculus, and compare with other logics. The logic lacks the law of
excluded middle and Modus Ponens and is weaker than classical logic, but
stronger than any proper intermediate logics in terms of theoremhood. In fact,
all the other well-known classical theorems are found to be theorems. Finally,
we will make an observation that models of this semantics can be seen as nodes
of an intuitionistic model.
