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This uncertainty as to the status of $\scriptstyle {S}$ and $\scriptstyle {O}$ is not without its effect upon the proofs. Consider, for instance, Th. 3. In the proof, "1°: $\scriptstyle {p}$ true. By Axiom 3, $\scriptstyle {p\triangle p}$ false" will be seen to require $\scriptstyle {pSp}$, concerning the origin of which, and the relation it has to $\scriptstyle {p\supset p}$ (Th. 4), which it indirectly serves to prove, Mr Van Horn says nothing.

(β) In his extensive use of the Principle of Excluded Middle, Mr Van Horn makes no explicit mention of the last steps, that lead from $\scriptstyle {p\supset q,~\sim p\supset q}$, to $\scriptstyle {q}$. These steps would seem to require several propositions: (1) those carrying us from $\scriptstyle {\sim p\vee p}$ to $\scriptstyle {q\vee q}$—"Summation," plus "Permutation," presumably—and (2) "Tautology" $\scriptstyle {q\vee q.\supset .q}$. As Mr Van Horn uses the principle of Excluded Middle in this particular way in the first formal proof given—that of Th. 3—both the principle itself and the propositions required for its use ought, I think, to be deduced immediately from Axiom 3; and I do not see how this is possible.

 

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