popper-logic-scientific-discovery

(1,1)
N(α.β.γ)
N(α)
N(α.β) N(α.β.γ)
= ·
N(α) N(α.β)
which proves to be an identity after cancellation of ‘N(α.β)’. (Contrast
with this proof, and with the proof of (2 s), by Reichenbach in
Mathematische Zeitschrift 34, p. 593.)
If we assume independence (cf. section 53), i.e.
(1 s )
α.βF″(γ) = αF″(γ)
we obtain from (1) the special multiplication theorem
(1 s)
αF″(β.γ) = αF″(β). αF″(γ)
With the help of the equivalence of (1) and (1′), the symmetry of the
relation of independence can be proved (cf. note 4 to section 53).
The addition theorems deal with the frequency of those elements which
belong either to β or to γ. If we denote the disjunctive combination
bination of these classes by the symbol ‘β + γ’, where the sign ‘ + ’, if
placed between class designations, signifies not arithmetical addition but the
non-exclusive ‘or’, then the general addition theorem is:
(2)
αF″(β + γ) = αF″(β) + αF″(γ) − αF″(β.γ)
This statement follows from the definition in section 52 if we use
the universally valid formula of the calculus of classes
(2,2)
α.(β + γ) = (α.β) + (α.γ),
and the formula (which is also universally valid)
(2,1)
N(β + γ) = N(β) + N(γ) − N(β.γ)
Under the assumption that, within α, β and γ have no member in
common—a condition which can be symbolized by the formula
(2 s )
N(α.β.γ) = 0
appendix ii 287