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40 Dem.: Taut. Syll.; then, Perm., Taut., and Syll., or $$\scriptstyle{S'}$$.

Reciprocal theorem by Add.$$\scriptstyle{\frac{p/p}{p,s}}$$ instead of Taut.

$$\scriptstyle{p.q.\supset.\sim(\sim p\vee\sim q)}$$ and reciprocal theorem.

Dem.: Id., Def. of $$\scriptstyle{\sim}$$, preceding theorem, and Syll.

Reciprocal theorem in the same manner.

Appendix. After the substance of this paper had been written, I was given the opportunity of seeing Mr Van Horn's very interesting and original paper dealing with what is practically the same subject. Mr Van Horn recognises clearly the superiority of what has been called above the OR-form over the AND-form chosen in Sheffer's text. This deserves the more notice, as Mr Van Horn, I understand, had not Sheffer's article at hand in the time he was writing his own paper. His $$\scriptstyle{\triangle}$$, as will be seen from the definition he gives, is indistinguishable from $$\scriptstyle{\mathsf{I}}$$. I was much attracted by the harmonious character of Mr Van Horn's third Axiom. It seems to me therefore all the more desirable that certain objections, which Mr Van Horn's proofs in their present form naturally suggest to the reader, should be dealt with.

(α) It is not quite plain to me whether "of the same truth-value" (say $$\scriptstyle{S}$$ for short), "of opposite truth-values" (say $$\scriptstyle{O}$$), are used as indefinables, or as abbreviations. If the former, we have no right to go, e.g., from $$\scriptstyle{pOq}$$ and $$\scriptstyle{\sim p}$$, to $$\scriptstyle{q}$$, etc., without some axiom to that effect, connecting $$\scriptstyle{O}$$ and $$\scriptstyle{S}$$ with $$\scriptstyle{\triangle}$$. If, on the other hand, $$\scriptstyle{S}$$ and $$\scriptstyle{O}$$ are abbreviations&mdash;as it seems to me they are&mdash;the two parts of Axiom 3 stand for not less than four propositions: We cannot assert the first two, or the last two, or all four, propositions together, because we should then need $$\scriptstyle{p.q.\supset.p}$$, $$\scriptstyle{p.q.\supset.q}$$, before we could make any use of such a synthetic Axiom.