- 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.
- Accepted author manuscript
Final publisher version
If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library, or send a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.