Description:
<jats:p>The calculus generated by the addition of the postulate</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481200105079_eqnU1" /></jats:disp-formula></jats:p><jats:p>to S2 will, after Miss Alban, be called “S6”, the calculus generated by the addition of the same postulate to S3, “S7”, and the calculus generated by the addition of the postulate</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481200105079_eqnU2" /></jats:disp-formula></jats:p><jats:p>to S3, “S8”. No interesting interpretation of the calculi S6–8 is known, but they are of some indirect interest because of the connections between them and the five calculi S1–S5. Certain questions concerning them will be treated in the following. In §1 it will be shown that every method of decision for S7 can be turned into a method of decision for S3, and in §2 that the number of complete extensions of S3 is equal to the number of complete extensions of S7 plus one. In §3 it will be shown that McKinsey's method of decision for S2 and S4 can be modified so as to cover S6.</jats:p><jats:p>The letters “<jats:italic>P</jats:italic>”, “<jats:italic>Q</jats:italic>”, “<jats:italic>R</jats:italic>”, and “<jats:italic>S</jats:italic>” will be employed as syntactic variables denoting formulas. Logical expressions will sometimes be used as selfdenotative. “<jats:italic>C</jats:italic>” and “<jats:italic>C</jats:italic>” stand for arbitrary calculi. “<jats:italic>C</jats:italic> + <jats:italic>P</jats:italic>” is the calculus which is the result of adding to <jats:italic>C</jats:italic> as new postulates all formulas which can be derived from <jats:italic>P</jats:italic> by substitution. With “substitution” I mean the operation performed on propositional variables.</jats:p>