Fifth Example.

A more complex argument, also given by De Morgan,‍[78] contains five terms, and is as stated below, except that the letters are altered.

Every A is one only of the two B or C; D is both B and C, except when B is E, and then it is neither; therefore no A is D.

The meaning of the above premises is difficult to interpret, but seems to be capable of expression in the following symbolic forms—

A = ABc ꖌ AbC,(1)
De = DeBC,(2)
DE = DEbc.(3)

As five terms enter into these premises it is requisite to treat their thirty-two combinations, and it will be found that fourteen of them remain consistent with the premises, namely

ABcdEaBCDeabCdE
ABcdeaBCdEabCde
AbCdEaBCdeabcDE
AbCdeaBcdEabcdE
aBcdeabcde.

If we examine the first four combinations, all of which contain A, we find that they none of them contain D; or again, if we select those which contain D, we have only two, thus—

D = aBCDe ꖌ abcDE.

Hence it is clear that no A is D, and vice versâ no D is A. We might draw many other conclusions from the same premises; for instance—