[515] Thus the rule for elimination from particular negatives is practically identical with the rule for elimination from universal negatives. The same rule may be deduced by obversion from the result obtained in the preceding section. Something is not either PX or Qx or R ; therefore, Something is either prX or qrx or pqr ; therefore, Something is either pr or qr ; therefore, Something is not either PQ or R.
494. Order of procedure in the process of elimination.—Schröder (Der Operationskreis des Logikkalkuls, p. 23) points out that first to eliminate and then combine is not the same thing as first to combine and then eliminate. For, as a rule, if a term X is eliminated from several isolated propositions the combined results give less information than is afforded by first combining the given propositions and then effecting the required elimination.
There are indeed many cases in which we cannot eliminate at all unless we first combine the given propositions. This is of course obvious in syllogisms; and we have a similar case if we take the premisses Everything is A or X, Everything is B or x. We cannot eliminate X from either of these propositions taken by itself, since in each of them X (or x) appears as an isolated alternant. But by 512 combination we have Everything is Ax or BX ; and this by the elimination of X becomes Everything is A or B.[516]
[516] Working with negatives we get the same result. Taking the propositions Nothing is ax, Nothing is bX, separately, we cannot eliminate X from either of them. But combining them in the proposition Nothing is ax or bX, we are able to infer Nothing is ab.
There are other cases in which elimination from the separate propositions is possible, but where this order of procedure leads to a weakened conclusion. Take the propositions Everything is AX or Bx, Everything is CX or Dx. By first eliminating X and then combining, we have Everything is AC or AD or BC or BD. But by first combining and then eliminating X our conclusion becomes Everything is AC or BD, which gives more information than is afforded by the previous conclusion.
EXERCISES.
495. Suppose that an analysis of the properties of a particular class of substances has led to the following general conclusions, namely:
1st, That wherever the properties A and B are combined, either the property C, or the property D, is present also; but they are not jointly present;
2nd, That wherever the properties B and C are combined, the properties A and D are either both present with them, or both absent;
3rd, That wherever the properties A and B are both absent, the properties C and D are both absent also; and vice versâ, where the properties C and D are both absent, A and B are both absent also.
Shew that wherever the property A is present, the properties B and C are not both present; also that wherever B is absent while C is present, A is present.
[Boole, Laws of Thought, pp. 118 to 120; compare also Venn, Symbolic Logic, pp. 276 to 278.]
A solution of this problem has already been given in section [488]. We may also proceed as follows. The premisses are:
All AB is Cd or cD, (i)