There are three men in a house, Allen, Brown, and Carr, who may go in and out, provided that (1) they never go out all at once, and that (2) Allen never goes out without Brown.
Can Carr ever go out?
Nemo and Outis differ on this point.
Nemo says he cannot: Outis says he can.
The rules, by which the men are bound, may be expressed thus:—
(1) If Carr goes out, then if Allen goes out Brown does not go out.
(2) If Allen goes out, Brown goes out.
Nemo’s Argument
Number (1) amounts to this:—
If Carr goes out, then (2) is not true.
But, ex hypothesi, (2) is true.
∴ Carr cannot go out; for the assumption that he goes out involves an absurdity.
Outis’s Reply
Nemo’s two assertions, “if Carr goes out, then (2) is not true” and “the assumption that Carr goes out involves an absurdity”, are erroneous.
The assumption, that Carr goes out, does not involve any absurdity; since the two propositions, “if Allen goes out Brown does not go out” and “if Allen goes out Brown goes out”, are compatible.
But the assumption, that Carr and Allen go out both at once, does involve an absurdity; since the two propositions, “Brown does not go out” and “Brown goes out”, are incompatible.
Hence it follows, not that Carr cannot go out, but that Carr and Allan cannot go out both at once.
Nemo’s Rejoinder
Outis has wrongly divided protasis and apodosis in (1).
The absurdity is not the last clause of (1), “Brown does not go out”, but all that follows the word “then”, i. e. the Hypothetical “If Allen goes out Brown does not go out”; and, by (1), it is the assumption, that Carr goes out, which causes this absurdity.
In fact, Outis has made (1) equivalent to “If Carr goes out [and if Allen goes out] then if Allen goes out Brown does not go out”. This is erroneous: the words in the brackets in the compound protasis are superfluous, and the remainder is the true protasis which conditions the absurd apodosis, as is evident from the form of (1) originally given.
[May 1, 1894.]