popper-logic-scientific-discovery

338
new appendices
The following comments may be made upon this system of
postulates:
The six axioms—A1, A2, A3, B1, B2, and B3—are explicitly used
in the actual operations of deriving the theorems. The remaining
(existential) parts of the postulates can be taken for granted, as in the
paper in which I first published this system of axioms. (Cf. note 1,
above.)
At the price of introducing a fourth variable, ‘d’, in Postulates 3 and
4, these six axioms may be replaced by a system of only four axioms,
consisting of A1, A2, and the following two:
B If p(a, bc)p(b, c) ≠ p(d, c) provided that p(a, c) � p(d, c), then
+
p(ab, c) ≠ p(d, c)
C If p(a, b) + p(ā, b) ≠ p(c, c) then p(c, c) = p(d, b)
+
In this system, B + is equivalent to the conjunction of B1 and B2, and
This system looks very much like some of the customary systems (except for the omission
of antecedents in the axioms other than C′, and the form of the antecedent of C′);
and it is remarkable that it yields for the elements a, b, . . . , as does the system in the text,
the theorems of Boolean algebra which ordinarily are separately assumed. Nevertheless it
is unnecessarily strong; not only because it introduces the numbers 1 and 0 (thus hiding
the fact that these need not be mentioned in the axioms) but also because A3, B1, and
C follow immediately from A3′, A4′, and C′, while for the opposite derivations, all
the axioms of the system given in the text except A2 are indispensable. (For these
derivations, see appendix *v.)
Within the system of axioms here described, and also within the system given in
the text, the conjunction of the axioms A4′ and B1′ is replaceable by B1, and vice
versa. My independence proofs (given below) are applicable to the system here
described.
The derivation of B1 from A4′ and B1′, in the presence of the axioms A3 or A3′, C or
C′, and B2, is as follows:
(1) 0 � p(a, b) � p(a, a) A4′; C or C′; A3 or A3′
(2) p(a, a) � p((aa)a, a) = p(aa, aa)p(a, a) = p(a, a) 2 1, A3′; B2; A3 or A3′
(3) 0 � p(a, b) � p(a, a) � 1 1, 2
(4) p(ba, c) � p(a, c) B2, 3
Now we apply B1′
(5) p(ab, c) � p(a, c) 4. B1′
For the derivation of A4′ and B1′ from B1, see appendix *v.