An (m + n)-valued propositional calculus2
may happen to be a subsystem of an m-valued propositional calculus
, though the converse is never true. This fact may give us the impression that, as m grows, the content of
becomes meagre. The present treatment is intended to remove this impression by constructing a complete, m-valued sub-system of any (m + n)-valued propositional calculus.
In the following we adopt the customary, autonymous mode of speech according to which symbols belonging to the object calculi or languages are used in the syntactic language as names for themselves, and juxtaposition serves to denote juxtaposition.
2.1 = df stands for definational identity in the syntactic language.
2.11 ≡ stands for definational identity in the object calculi.
2.2 ∊, ⊂, ∩, {x1, …, xn} are used in their meanings as customarily employed in the theory of sets—∊ for class membership, ⊂ for proper inclusion, ∩ for the product operation of classes, {x1, …, xn} for the class with x1, …, xn as its only elements.
2.3 x, y, z are used as unspecified natural numbers including 0. m, n, i, j are used as unspecified natural numbers other than 0.
2.401 Definition. δ = df as the function of two variables defined for any x, y such that
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200105717/resource/name/S0022481200105717_eqnU1.gif?pub-status=live)
2.41 Definition. For m ≧ 2, ιm = df the function of two variables denned on the set {0, …, m − 1} such that ιm (x, y) = y − x for x ≦ y and ιn(x, y) = 0 for x > y.