Kit Fine. The logics containing S4.3. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 371–376. - Kit Fine. An ascending chain of S4 logics. Theoria, vol. 40 (1974), pp. 110–116. - Kit Fine. Logics containing K4. Part I. The journal of symbolic logic, vol. 39 (1974), pp. 31–42. - L. Esakia and V. Meskhi. Five critical modal systems. Theoria, vol. 43 (1977), pp. 52–60.

1985 ◽  
Vol 50 (1) ◽  
pp. 231-234
Author(s):  
R. A. Bull
Keyword(s):  
Symbolic Logic ◽  
Kit Fine ◽  
Modal Systems

On finite and infinite modal systems

1938 ◽  
Vol 3 (2) ◽  
pp. 77-82 ◽  
Author(s):  
C. West Churchman
Keyword(s):  
Modal Logic ◽  
Symbolic Logic ◽  
Modal Operator ◽  
Image Position ◽  
Modal Systems ◽  
Complex Modal

In Oskar Becker's Zur Logik der Modalitäten four systems of modal logic are considered. Two of these are mentioned in Appendix II of Lewis and Langford's Symbolic logic. The first system is based on A1–8 plus the postulate,From A7: ∼◊p⊰∼p we can prove the converse of C11 by writing ∼◊p for p, and hence deriveThe addition of this postulate to A1–8, as Becker points out, allows us to “reduce” all complex modal functions to six, and these six are precisely those which Lewis mentions in his postulates and theorems: p, ∼p, ◊p, ∼◊p, ∼◊∼p, and ◊∼p This reduction is accomplished by showingwhere ◊n means that the modal operator ◊ is repeated n times; e.g., ◊3p = ◊◊◊p. Then it is shown thatBy means of (1), (2), and (3) any complex modal function whatsoever may be reduced to one of the six “simple” modals mentioned above.It might be asked whether this reduction could be carried out still further, i.e., whether two of the six “irreducible” modals could not be equated. But such a reduction would have to be based on the fact that ◊p = p which is inconsistent with the set B1–9 of Lewis and Langford's Symbolic logic and independent of the set A1–8. Hence for neither set would such a reduction be possible.

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