popper-logic-scientific-discovery

appendix *iv 347
if we consider that B2 is needed to prove ‘p(ba, c) � p(a, c)’, that is to
say, the dual of B1. This suggests that we may use the second example
for B1, changing only the value of 1.0 from 0 to 1, and that of 0.1 from
1 to 0. Everything else may be left unchanged. B2 fails for a = 1, b = 0,
and c = 2.)
Ultimately, for showing that C is independent, we take again the
same S, but assume that ā = a. If we now take p (0, 1) = 0 and in all
other cases p(a, b) = 1, then C fails, because p(0¯, 1) ≠ p(1, 1). The other
axioms are satisfied.
This concludes the proofs of the independence of the operational
axioms.
As to the non-operational parts of the postulates, a proof of the
independence of postulate 1 has been given above (when I commented
upon this postulate).
Postulate 2 requires (in its non-operational part) that whenever a
and b are in S, p(a, b) is a real number. In order to show the independence
of this requirement—which we may briefly refer to as ‘postulate
2’—we first consider a non-numerical Boolean interpretation of S. To this end,
we interpret S as an at most denumerable and non-numerical Boolean
algebra (such as a set of statements, so that ‘a’, ‘b’, etc. are variable names
of statements). And we stipulate that ‘x¯’ is to denote, if x is a number, the
same as ‘ − x’; and if x is a Boolean element (say, a statement) then ‘x¯’ is
to denote the Boolean complement (negation) of x. Similarly, we stipulate
that ‘xy’; ‘x + y’; ‘x = y’; ‘x ≠ y’; and ‘x � y’, have their usual arithmetical
meaning if x and y are numbers, and their well-known Boolean
meanings whenever x and y are Boolean elements. (If x and y are
statements, ‘x � y’ should be interpreted as ‘x entails y’.) In order to
prove the independence of postulate 2, we now merely add one more
stipulation: we interpret ‘p(a, b)’ as synonymous with ‘a + b¯’ in the
Boolean sense. Then postulate 2 breaks down while A1, A2, A3 and all
the other axioms and postulates turn into well-known theorems of
Boolean algebra. 11
The proofs of the independence of the existential parts of postulates
3 and 4 are almost trivial. We first introduce an auxiliary system
11 A slight variant of this interpretation transforms all the axioms into tautologies of the
propositional calculus, satisfying all the postulates except postulate 2.