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FINITE AXIOMATIZATION OF $\mathbf {Grz}\times \mathbf {S5}$
AND $\mathbf {GL}\times \mathbf {S5}$
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 03 November 2026, pp. 1-30
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- Article
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We develop a selective filtration technique for the monadic Grzegorczyk and Gödel–Löb logics with the monadic Barcan formula, yielding that these logics have the finite model property. As a consequence, we obtain that $\mathbf {GL} \times \mathbf {S5}$
is product matching, and that $\mathbf {Grz} \times \mathbf {S5}$
is axiomatized by adding to the commutator $[\mathbf {Grz},\mathbf {S5}]$
the Gödel translation of the monadic Casari formula. This settles the question of finite axiomatization for two of the product logics listed in [13, p. 137] . As another consequence, we obtain that the one-variable fragments of the predicate Grzegorczyk and Gödel–Löb logics with the Barcan formula are Kripke complete (which contrasts the incompleteness of the full predicate logics).
