There are two Propositions, A and B.
Let it be granted that
If A is true, B is true. (i)
Let there be another Proposition C, such that
If C is true, then if A is true B is not true. (ii)
Nemo and Outis differ about the truth of C.
Nemo says C cannot be true: Outis says it may be.
Nemo’s Argument
Number (ii) amounts to this:—
If C is true, then (i) is not true.
But, ex hypothesi, (i) is true.
∴ C cannot be true; for the assumption of C involves an absurdity.
Outis’s Reply
Nemo’s two assertions, “if C is true, then (i) is not true” and “the assumption of C involves an absurdity”, are erroneous.
The assumption of C does not involve any absurdity; since the two propositions, “if A is true, B is true” and “if A is true, B is not true”, are compatible.
But the assumption of C and A together does involve an absurdity; since the two propositions, “B is true” and “B is not true”, are incompatible.
Hence it follows, not that C is untrue, but that C and A cannot be true together.
Nemo’s Rejoinder
Outis has wrongly divided protasis and apodosis in (ii).
The absurdity is not the last clause of (ii), “B is not true”, but all that follows the word “then”, i. e. the Hypothetical “If A is true B is not true”; and, by (ii), it is the assumption of C only which causes this absurdity.
In fact, Outis has made (ii) equivalent to “If C is true [and if A is true] then if A is true B is not true”. 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 (ii) originally given.
[April, 1894.]