Lewis has made an attempt to construct a logistic system containing an implication-relation ⊰ in such a way that p⊰q shall have the meaning: q is deducible from p. Lewis and Langford admit that “the one serious doubt which can arise concerning the equivalence of p⊰q to the relation of deducibility … arises from the fact that strict implication has its corresponding paradoxes:

19.74˜⋄ p ⊰ · p ⊰ q,

‘If p is impossible, then p strictly implies any proposition q’; and

19.75 ˜⋄˜p ⊰ · q ⊰ p,

‘If p is necessary, then any proposition q strictly implies p.’”

Indeed, it is not obvious that 19.74 and 19.75 should hold. It is true that in many cases 19.75 (as well as 19.74) is valid; for in the system of Lewis and Langford

p · ˜p · ⊰ q

is provable, and the conformity of this law with real deduction is shown by them. In all cases in which a proposition of the form p·∼p can be derived from an impossible proposition, 19.75 holds. But it is not obvious that there are no other kinds of impossible propositions.

Fortunately it is easy to see what is the origin of the paradoxes. They are introduced into the system by the definition

11.02 p ⊰ q · = ˜ ⋄(p · ˜q).

Assume this definition. Then if q is necessary, p·∼q is impossible, and so, in accordance with 11.02, p⊰q is valid.

It is known that the usual definition of a dense series without extreme elements is complete with respect to first-order functions, in the sense that any first-order function on the base of a set of postulates defining such a series either is implied by the postulates or is inconsistent with them. It is here understood, in accordance with the usual convention, that when we speak of a function on the base , the function shall be such as to place restrictions only upon elements belonging to the class determined by f; or, more exactly, every variable with a universal prefix shall occur under the hypothesis that its values satisfy f, while every variable with an existential prefix shall have this condition categorically imposed upon it.

Consider a set of postulates defining a dense series without extreme elements, and add to this set the condition of Dedekind section, to be formulated as follows. Let the conjunction of the three functions,

be written H(ϕ), where the free variables f and g, being parameters throughout, are suppressed. This is the hypothesis of Dedekind's condition, and the conclusion is

which may be written C(ϕ).

In Quine's New foundations the axiom of infinity does not appear to be provable. In a certain stronger system, very closely related to Quine's New foundations, the axiom of infinity is provable. One of the peculiarities of this latter system is that even unstratified propositions can be proved by induction (this is used in the proof of the axiom of infinity). It would seem that definition by induction should also be possible quite irrespective of any conditions of stratification in this latter system. In this paper it is shown that such is the case.

Theorem. If α, α1, α2, …, αs are any classes, and if there is no confusion of bound variables in {Sxp}(v) or {Sxq}(v) and neither z nor v occurs free in p or q, and if

⊦ (y1, y1, … ys):. y1ϵα1 . y2ϵα2 . … . ysϵαs : ⊃ : (Ez) : zϵα : (v) : v . ≡ . {Sxp}(v),

⊦ (y, n, y1, y2, …, ys) :. yϵα . n ϵ Fin . y1ϵα1 . y2ϵα2. … . ysϵαs : ⊃ : (Ex) :zxα : (v) : v=z . ≡ . {Sxq}(v),

then there is an r such thai there is no confusion of bound variables in {Sxr}(v) and neither z nor v occurs free in r, and

⊦ (n, y1, y2, …, ys) :. nϵ Fin . y1ϵα1 . y2ϵα2 . … . ysϵαs : ⊃ : (Ez) : zϵα : (v) : v=z . ≡ . {Sxr}(v),

⊦ (y1, y2, …, ys) : y1ϵα1 . y2ϵα2 . … . ysϵαs . ⊃ . ιx{Snr}(0) = ιxp,

⊦ (n, y1, y2, …, ys) : nϵ Fin . y1ϵα1 . y2ϵα2 . … . ysϵαs . ⊃ . ιx{Snr}(n+1) = ιx{Syq}(ιxr).