- Locally finite reducts of Heyting algebras and canonical formulas
- Notre Dame Journal of Formal Logic
- Volume | Issue number
- 58 | 1
- Pages (from-to)
- Document type
- Interfacultary Research Institutes
- Institute for Logic, Language and Computation (ILLC)
The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the
→-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics (si-logics for short).
The ∨-free reducts of Heyting algebras give rise to the (∧,→)-canonical formulas that we studied in an earlier work. Here we introduce the (∧,∨)-canonical formulas, which are obtained from the study of the →-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by (∧,∨)-canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas.
One of the main ingredients of these formulas is a designated subset D of pairs of elements of a finite subdirectly irreducible Heyting algebra A. When D=A2, we show that the (∧,∨)-canonical formula of A is equivalent to the Jankov formula of A. On the other hand, when D=∅, the (∧,∨)-canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics.
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