## Notre Dame Journal of Formal Logic

### A Closer Look at Some Subintuitionistic Logics

#### Abstract

In the present paper we study systematically several consequence relations on the usual language of propositional intuitionistic logic that can be defined semantically by using Kripke frames and the same defining truth conditions for the connectives as in intuitionistic logic but without imposing some of the conditions on the Kripke frames that are required in the intuitionistic case. The logics so obtained are called subintuitionistic logics in the literature. We depart from the perspective of considering a logic just as a set of theorems and also depart from the perspective taken by Restall in that we consider standard Kripke models instead of models with a base point. We study the relations between subintuitionistic logics and modal logics given by the translation considered by Došen. Moreover, we classify the logics obtained according to the hierarchy considered in Abstract Algebraic Logic.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 42, Number 4 (2001), 225-255.

Dates
First available in Project Euclid: 12 September 2003

https://projecteuclid.org/euclid.ndjfl/1063372244

Digital Object Identifier
doi:10.1305/ndjfl/1063372244

Mathematical Reviews number (MathSciNet)
MR2010183

Zentralblatt MATH identifier
1034.03007

#### Citation

Celani, Sergio; Jansana, Ramon. A Closer Look at Some Subintuitionistic Logics. Notre Dame J. Formal Logic 42 (2001), no. 4, 225--255. doi:10.1305/ndjfl/1063372244. https://projecteuclid.org/euclid.ndjfl/1063372244

#### References

• [1] Ardeshir, M., and W. Ruitenburg, Basic propositional calculus I,'' Mathematical Logic Quarterly, vol. 44 (1998), pp. 317--43.
• [2] Ardeshir, M., and W. Ruitenburg, \mboxBasic propositional calculus II. Interpolation,'' Archive for Mathematical Logic, vol. 40 (2001), pp. 349--64.
• [3] Blok, W. J., and D. Pigozzi, Algebraizable Logics, vol. 77, Memoirs of the American Mathematical Society, American Mathematical Society, 1989.
• [4] Corsi, G., "Weak logics with strict implication", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), pp. 389--406.
• [5] Czelakowski, J., Protoalgebraic Logics, Kluwer Academic Publishers, Dordrecht, 2001.
• [6] Došen, K., "Modal translations in $\bf K$" and $\bf D$, pp. 103--27 in Diamonds and Defaults, edited by M. de Rijke, Kluwer Academic Publishers, Dordrecht, 1993.
• [7] Epstein, G., and A. Horn, "Logics which are characterized by subresiduated lattices", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 199--210.
• [8] Hacking, I., What is strict implication?'' The Journal of Symbolic Logic, vol. 28 (1963), pp. 51--71.
• [9] Rasiowa, H., An Algebraic Approach to Non-Classical Logics, North-Holland Publishing Company, Amsterdam, 1974.
• [10] Restall, G., "Subintuitionistic logics", Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 116--29.
• [11] Ruitenburg, W., "Constructive logic and the paradoxes", Modern Logic, vol. 1 (1991), pp. 271--301.
• [12] Ruitenburg, W., "Basic logic, K4, and persistence", Studia Logica, vol. 63 (1999), pp. 343--52.
• [13] Sazaki, K., "Formalization for the consequence relation of Visser's propositional logic", Reports on Mathematical Logic, vol. 33 (1999), pp. 65--78.
• [14] Suzuki, Y., Non-normal propositional languages on transitive frames and their embeddings, Ph.D. thesis, JAIST, Ishikawa, 1999.
• [15] Suzuki, Y., F. Wolter, and M. Zakharyaschev, "Speaking about transitive frames in propositional languages", Journal of Logic, Language and Information, vol. 7 (1998), pp. 317--39.
• [16] Visser, A., Aspects of Diagonalization and Provability, Ph.D. thesis, University of Utrecht, Utrecht, 1981.
• [17] Visser, A., "A propositional logic with explicit fixed points", Studia Logica, vol. 40 (1981), pp. 155--75.
• [18] Wansing, H., "Displaying as temporalizing: Sequent systems for subintuitionistic logics", pp. 159--78 in Logic, Language and Computation, Kluwer Academic Publishers, Dordrecht, 1997.