Two negative premisses are really tantamount to a declaration that there is no connexion whatever between the Major and Minor (as quantified in the premisses) and the term common to both premisses; in short, that this is not a Middle term—that the condition of a valid Syllogism does not exist.
There is an apparent exception to this when the real Middle in an argument is a contrapositive term, not-M. Thus:—
Nobody who is not thirsty is suffering from fever.
This person is not thirsty.
... He is not suffering from fever.
But in such cases it is really the absence of a quality or rather the presence of an opposite quality on which we reason; and the Minor Premiss is really Affirmative of the form S is in not-M.
Canon V. If one premiss is negative, the conclusion must be negative.
If one premiss is negative, one of the Extremes must be excluded in whole or in part from the Middle term. The other must therefore (under Canon IV.) declare some coincidence between the Middle term and the other extreme; and the conclusion can only affirm exclusion in whole or in part from the area of this coincidence.
Canon VI. No conclusion can be drawn from two particular premisses.