The theory of constructive types (principles of logic and mathematics). By Leon Chwistek.

25
sio-ns: E(X), G(Ç,u), H( J,. v), I(X v. ) are propositional
expressions 1).
0'15 If E(') is any propositional expression containing the individual (or primitive, or determined) real variable k, then
(E)J (X) is a propositional expression. Here we have tu r n e d
a real variable in to ani apparent variable.
0 151 If the expressions:
are propositional expressions, they denote the same logical
propositions, i. e they have the same meaning2).
0~152 Any functional expression containing no fundamental letters
and no apparent variables is a primitive functional
expression.,
016 Suppose that X, y". v, p are any individual. (or primitive. or
determined real variables) and F(X). or G(.),. or H(,,.v)
or I(Ç,.J v, ) are any propositional expressions containing
the variables f. or ): u. or i, v. or ), U, v; p. Suppose
that, ', or.u.,.', or v. v', or ' are at the same time
two different. individual (or primitive, or fundamental) real
variables, or that À, (,u? v, or p) is a determined real variable
and ' (,.' v'. or ') is any functional expression having
no letters in common with À. (y., v, or?). Then. if PC(V),
or G(',.'), or H(\'^?, V'), or l(a',., /. ) denotes a logical proposition. ) [o( ) I ()'} or i uy. (j(. U.)] {' y.' } or
x v A^A ^,.,,.. o i,a A A,.,A
À y. IV 1 (I i,, )] 'o, J. '. '} or l',) ',
denotes the same logical proposition 3). Here k,: V' and y., I':
and v, v' anfd. s are c on n e x ed one with another.
Note that here the alphabetical order and the order of variables in expressions F(I), G (Av.), H(0..,.: V) lIJ(;, vy, p) are
irrelevant.
0'161 If À is a determined variable: aund if is an argument connexed with ÀA then. is a determined variable.
') Directions 0'13'131 correspond to X 9315 of Principia.
2) This direction correspond to:* 11'07 of Principia.
s) Without a direction of this kind we could not write e. g,
Ex[c{p }] == y{a}.
dt
The need of a particular direction concerning this matter was tirst pointed out to
mie by Prof. Lesniewski,