Four Notions of Independence

by

BOGUS E AW WO L N I E W IC Z (University of Warsaw)

T h e notion of mutually independent propositions looms large in Wittgenstein’s Tractatus. In 5.152 we have the definition:

(D) “Satze, welche keine Wahrheitsargumente mit einander gemein haben, nennen wir von einander unabhiingig.

In the original edition of 1922 this definition is followed by another criterion of mutual independence:

(C) ‘Von einander unabhiingige Satze (z.B. irgend zwei Elementarsatze) geben einander die Wahr- scheinlichkeit 112.”

But in the second impression of 1933 this criterion has been dropped, for in its place we have there the obviously truncated passage:

“Zwei Elementarsatze geben einander die Wahr- scheinlichkeit 1 /2.”

This is, by the way, the only substantial change introduced by Wittgenstein into his text, and oddly enough it is not even mentioned in Black‘s Companion. What was the reason for that change?

Apparently Wittgenstein thought first that C is entailed by D, or even that they are equivalent, but neither turns out to be the case. For consider three elementary propositions “el”, “e2”, 11 - Theoria, 2 1970

162 BOGUSLAW WOLNIEWICZ

“es”, and a proposition ”p; such that p i el v es. The matrix of their truth-possibilities has the form:

el e2 e3 I P i

T F T T T F

. T F

According to D, propositions “es” and “pf” having no common truth-arguments are certainly independent of each other. But what about C? Let’s write ”I@,q)” for “p and q are mutually inde- pendent”. Using the notion of conditional probability we may state C on its weaker interpretation thus:

(C) I(p,q) -+ P(p /q ) = m / P ) = 1/2

where ”P@/q)” is read as usual “the probability of p given 4”. Computing from our matrix the relevant conditional probabili- ties by Wittgenstein’s method given in 5.15- 5.151, we get

p l e h ) = 1 /2

which is all right, but

Plp ilea) = 3 /4

which obviously violates condition C. Rejecting C altogether, however, Wittgenstein went too far,

obliterating thereby the important connexion between his notion of independence and that of probability. It will do to replace C by the weaker

IC’) Ilp,q) -+ Plplq) = P(P/ - 4)

FOUR NOTIONS OF INDEPENDENCE 163

i.e. by the familiar criterion of statistical independence. It is easily checked that C’ is satisfied in our example.

It should be stressed that the consequent of C’ gives only a necessary condition of independence as defined by D, not a sufficient one. For there are truth-functions satisfying it which do not satisfy D. To put “el-e,” for ”p”, and “es-er” for “q“, would be a case in point. But as a condition of statistical indepen- dence the consequent of C’ is both necessary and sufficient. So we have two different notions of independence here: the sta- tistical (I8) and the Wittgensteinian one ( Iw) . Their relation is simple:

but not conversely.

which might be called the moda2 one (Im). This we define: In turn I, is related to yet another notion of independence,

z m b , d - + d f ObAdA ObA “ 4 ) ” o ( - p A d A 0 c “P A “ 4).

Modal independence follows from statistical independence. For if I&,q), then p is compatible with q. Upon the plausible inter- pretation of compatibility as compossibility we have thus:

But it is well know from the theory of probability that:

I.@,q) = I,@, - q) - I.( -p ,q) = I,( -p,q)

which with our definition of I , yields immediately the result:

12) UP,q) + I m ( P , q )

but not conversely. The last notion to be brought into this context is that of

deductive independence (L). As it is usually understood, two propositions are deductively independent if they are compa- tible, and neither is entailed by the other. Interpreting here entailment as strict implication we get the definition:

164 BOGUSLAV WOLNIEWICZ

I d P , 4 ) -+dl 0 0 A 4) A 0 (P A - 4 ) A 0 ( “ P A 4) Clearly we have then:

(3) Im(P,q) -+ I d ( P d

but once again not conversely. Taking (l), (2) and (3) together we come finally to the con-

clusion that the four different notions of independence discussed here are related in the following way:

(4) I, +I, +I, + I d

and that Wittgenstein’s notion of independence is the strongest of them all.

It should be clear that this investigation is meant to apply in a straightforward way to the case of the independence of two propositions only. In the general case of the mutual independence of n propositions things get very complicated formally; e.g. then to define I , we need 2” - n - 1 equations, as pointed out by A. N. Kolmogorov. But however important these complications might be mathematically, philosophically they don’t seem to bring in anything essentially new.

Received on November 26,1969.