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

2O
It is easy to see that my system must be much more eco iplicated than that of Whitehead and Russell. It might be thought
that any further complication must be useless to clear up the ideas
on which Mathematies is to be based. But it may be erroneous to
think that clear ideas are never complicated: while, we must agree
that many simple ideas are, as a matter of fact, very obscure.
The system of Whitehead and Russeli: being the most perfect and most ingeniously constructed system of Logic I know,
I hardly conceive that any other method in working on these matters
ean be used. The knowledge of Principia is therefore quite sufficient to understand what is said in this paper. All the propositions
used as corollaries being statedQ there is as a matter of fact no
essential difficulty in uTnderstanding my proofs without the knowledge
of Principia.
To sum up my system is basèd on a most consistent appli
cation of the Russellian theory of types. Mathematical ideas are
developed step by step, with the help of special hypotheses, if necessary, which affords a base for co-structing the hierarchy of différent stages of Mathematicso This method seems to prove that there
is no ole unique system, but on the contrary.ma'ny exclusive sy"
stems of MathemraticsO
The anme,,constructive types is based on the theoretical
pOssibility of (construction of all filnctions belonging to a given type
io my system,
iL DBireetionis feteer ing tihI t ieInî-al ad à the ltSi
of synibtolso
It is to be remarked that we can hardlu imagine a system
of symbolic Logic without some directions concerning the meaniiig
of symbols. Take e. g. the proposition '{p p}. We having an operation consisting in talking q for p in?p, we might think. that q was
one of the possible values of }{p}. Now it is easy to see that
such an interpretation of our symbols implies a ontradiction. Let
us write:
(where the Leibnizian idea of identity is assumed).