Abstract
For any propositional logic, Sushko’s lemma states that, for any substitution, the preimage of the set of all tautologies of this logic is its theory. The problem of the relationship between the set of all such preimages and the set of all theories for classical propositional logic is considered. It is proved that any consistent theory of classical logic is the preimage of the set of all identically true formulas for some substitution. An algorithm for constructing such a substitution for any consistent finitely axiomatizable theory is presented.
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Notes
Obviously, the preimage of the set of all tautologies under an invertible substitution is the set itself.
It is easy to see that such a substitution does not exist for a contradictory theory of any propositional logic.
Obviously, the number of such distinct classes is at most \(2^n\).
This assertion also holds for any set \(\Omega_\Gamma=\{\nu : \nu(\Gamma)=1\}\) , where \(\Gamma\) is a set of formulas.
References
R. Wojcicki, Lectures on Propositional Calculi (Ossolineum Publ., Wroclaw, 1984).
R. Wojcicki, Lectures on Propositional Calculi, arXiv: http://sl.fr.pl/wojcicki/Wojcicki-Lectures.pdf (1984).
M. Esteban, Duality Theory and Abstract Algebraic Logic, Thesis (Universitat de Barcelona, Barcelona, 2013).
M. Tokarz, “Connections between some notions of completeness of structural propositional calculi,” Studia Logica 32, 77–89 (1973).
Acknowledgments
The author wishes to thank the referee for constructive remarks that helped him to improve the paper.
Funding
This work was supported by the Russian Foundation for Basic Research under grants 17-03- 00818-OGN and 18-011-00869-a.
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Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 856–864 https://doi.org/10.4213/mzm12469.
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Gorbunov, I.A. Theories of the Classical Propositional Logic and Substitutions. Math Notes 110, 887–893 (2021). https://doi.org/10.1134/S0001434621110249
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DOI: https://doi.org/10.1134/S0001434621110249