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A Reduction in the number of the Primitive Propositions of Logic. By J. G. P. Nicod, Trinity College. (Communicated by Mr G. H. Hardy.)

[Received and read 30 October 1916.]

Of the four elementary truth functions needed in logic, only two are taken as indefinables in Principia Mathematica. These two have now been defined by Mr Sheffer^[1] in terms of a single new function $\scriptstyle {p|q}$, "p stroke q." I propose to make use of Mr Sheffer's discovery in order to reduce the number of primitive propositions needed for the logical calculus.

There are two slightly different forms of the new indefinable, for we may treat $\scriptstyle {p|q}$ as meaning the same thing as either $\scriptstyle {\sim p.\sim q}$, or $\scriptstyle {\sim p\vee \sim q}$^[2]. The definition of $\scriptstyle {\sim p}$ is the same in both cases, namely $\scriptstyle {p|p}$, while that of $\scriptstyle {p\vee q}$ simply changes from $\scriptstyle {p/q|p/q}$ with the AND-form into $\scriptstyle {p/p|q/q}$ with the OR-form.

However, the best course is for us to define all the four truth-functions directly in terms of the new one. In so doing, we find that, while the definition of $\scriptstyle {\sim p}$ remains the same, and those of $\scriptstyle {p\vee q}$, $\scriptstyle {p.q}$ simply permute, as we pass from the AND-form to the OR-form, the definition of $\scriptstyle {p\supset q}$ is simpler in the latter form. It is $\scriptstyle {p{\mathsf {I}}q/q}$, as against $\scriptstyle {p/p|q{\mathsf {I}}p/p|q}$.

The OR-form is therefore to be preferred^[3].

Definitions.

${\begin{aligned}\scriptstyle {\sim p}&\scriptstyle {.=.p|p}\quad &\scriptstyle {\text{Df.}}\qquad &&\scriptstyle {p\vee q}&\scriptstyle {.=.p/p|q/q}\quad &\scriptstyle {\text{Df.}}\\\scriptstyle {p\supset q}&\scriptstyle {.=.p|q/q}\quad &\scriptstyle {\text{Df.}}\qquad &&\scriptstyle {p.q}&\scriptstyle {.=.p/q|p/q}\quad &\scriptstyle {\text{Df.}}\end{aligned}}$

Remarks on these Definitions.

One ought not to aim at retaining before one's mind the complex translation into the usual system, "$\scriptstyle {\sim p\vee \sim q}$," as the "real meaning" of the stroke. For the stroke, in the stroke-system, is simpler than either $\scriptstyle {\sim }$ or $\scriptstyle {\vee }$, and from it both of them arise. We may not be able to think otherwise than in terms of the four usual functions; it will then be more in accordance with the nature of the new system to think of the $\scriptstyle {|}$, not as some fixed compound of $\scriptstyle {\sim }$ and $\scriptstyle {\vee }$, but as a bare structure, out of which, in various ways, $\scriptstyle {\sim }$ and $\scriptstyle {\vee }$ will grow.

1. Sheffer, Trans. Amer. Math. Soc. Vol. xiv. pp. 481–488.
2. Sheffer, loc. cit., footnote^[1], p. 488.</ref>
3. $\scriptstyle {p|q}$ thus corresponds to what is termed the Disjunctive relation in Mr W. E. Johnson's writings.