De Morgan (Formal Logic, p. 14) gives the following proof of this corollary:—“If two propositions P and Q together prove a third R, it is plain that P and the denial of R prove the denial of Q. For P and Q cannot be true together without R. Now, if possible, let P (a particular) and Q (a universal) prove R (a universal). Then P (particular) and the denial of R (particular) prove the denial of Q. But two particulars can prove nothing.”[311]
[311]Further attention will be called in a later chapter to the general principle upon which this proof is based. See section [264].
(iii) From a particular major and a negative minor nothing can be inferred.
Since the minor premiss is negative, the major premiss must by rule 5 be affirmative. But it is also particular, and it therefore follows that the major term cannot be distributed in it. Hence, by rule 4, it must be undistributed in the conclusion, i.e., the conclusion must be affirmative. But also, by rule 6, since we have a negative premiss, it must be negative. This contradiction establishes the corollary that from the given premisses no conclusion can be drawn.
The following mnemonic lines, attributed to Petrus Hispanus, 291 afterwards Pope John XXI., sum up the rules of the syllogism and the first two corollaries:
Distribuas medium: nec quartus terminus adsit:
Utraque nec praemissa negans, nec particularis:
Sectetur partem conclusio deteriorem;
Et non distribuat, nisi cum praemissa, negetve.
201. Restatement of the Rules of the Syllogism.—It has been already pointed out that the first two of the rules given in section [199] are to be regarded as a description of the syllogism rather than as rules for its validity. Again, the part of rule 3 relating to ambiguity may be regarded as contained in the proviso that there shall be only three terms; for, if one of the terms is ambiguous, there are really four terms, and hence no syllogism according to our definition of syllogism. The rules may, therefore, be reduced to four; and they may be restated as follows:—
A. Two rules of distribution:
(1) The middle term must be distributed once at least in the premisses;
(2) No term may be distributed in the conclusion which was not distributed in one of the premisses;
B. Two rules of quality:
(3) From two negative premisses no conclusion follows;
(4) If one premiss is negative, the conclusion must be negative; and to prove a negative conclusion, one of the premisses must be negative.[312]
[312] The rules of quality might also be stated as follows; To prove an affirmative conclusion, both premisses must be affirmative; To prove a negative conclusion, one premiss must be affirmative and the other negative.
202. Dependence of the Rules of the Syllogism upon one another.—The four rules just given are not ultimately independent of one another. It may be shewn that a breach of the second, or of the third, or of the first part of the fourth involves indirectly a breach of the first; or, again, that a breach of the first, or of the third, or of the first part of the fourth involves indirectly a breach of the second.
292 (i) The rule that two negative premisses yield no conclusion may be deduced from the rule that the middle term must be distributed once at least in the premisses.