Bolesław Sobociński. A contribution to the axiomatization of Lewis' system S5. Notre Dame journal of formal logic, vol. 3 (1962), pp. 51–63. - Bolesław Sobociński. On the generalized Brouwerian axioms.Notre Dame journal of formal logic, vol. 3 (1962), pp. 123–128. - Bolesław Sobociński. A note on modal systems.Notre Dame journal of formal logic, vol. 4 (1963), pp. 155–157. - Ivo Thomas. Solutions of five modal problems of Sobociński.Notre Dame journal of formal logic, vol. 3 (1962), pp. 199–200. - Ivo Thomas. S1° and Brouwerian axioms.Notre Dame journal of formal logic, vol. 4 (1963), pp. 151–152. - Ivo Thomas. S1° and generalized Si-axioms.Notre Dame journal of formal logic, vol. 4 (1963), pp. 153–154. - Ivo Thomas. A final note on S1° and the Brouwerian axioms.Notre Dame journal of formal logic, vol. 4 (1963), pp. 231–232. - Ivo Thomas. Modal systems in the neighbourhood of T.Notre Dame journal of formal logic, vol. 5 no. 1 (1964), pp. 59–61. - Ivo Thomas. Ten modal models. The journal of symbolic logic, vol. 29 no. 3 (1964), pp. 125–128.

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.

Download Full-text

Disarming Stove’s Paradox: In Defence of Formal Logic

The Paideia Archive: Twentieth World Congress of Philosophy ◽

10.5840/wcp20-paideia19988192 ◽

1998 ◽

pp. 127-134

Author(s):

R. Rodrigo Soberano

Keyword(s):

Main Part ◽

Formal Logic ◽

Symbolic Logic ◽

Logical Contradiction ◽

False Conclusion

The argument (d) ("All arguments with true premises and false conclusions are invalid.") is an argument with true premises and false conclusion. Therefore "(d) is invalid" seems to be formally valid. Thus presumably formal logic has to admit it as valid. But then formal logic finds itself in a bind. For the above argument is problematic and even paradoxical since it involves an internal logical contradiction. The paradox, aptly termed "Stove's paradox," is fully realized by demonstrating with the help of symbolic logic the contradiction within the argument. Then as the main part of this essays shows, the paradox is attacked by exposing the paradox's genesis. It is shown that by appeal to some not so obvious logical considerations regarding sound linguistic construction and usage, the above argument could not have been legitimately construction. For its construction must have involved either equivocation or hiatus of meaningfulness in the use of the symbol (d).

Download Full-text