there is no direct predication concerning Joe Smith, but
only a predication of one of the alternatives conditionally on the other
being denied, as, _If Joe Smith was not a prophet he was an impostor_;
or, _If he was not an impostor, he was a prophet_. Symbolically,
Disjunctives may be represented thus:

A is either B or C,
Either A is B or C is D.

Formally, every Conditional may be expressed as a Categorical. For our
last example shows how a Disjunctive may be reduced to two Hypotheticals
(of which one is redundant, being the contrapositive of the other; see
chap. vii. Sec. 10). And a Hypothetical is reducible to a Categorical thus:
_If the sky is clear, the night is cold_ may be read--_The case of the
sky being clear is a case of the night being cold_; and this, though a
clumsy plan, is sometimes convenient. It would be better to say _The sky
being clear is a sign of the night being cold_, or a condition of it.
For, as Mill says, the essence of a Hypothetical is to state that one
clause of it (the indicative) may be inferred from the other (the
conditional). Similarly, we might write: _Proof of Joe Smith's not being
a prophet is a proof of his being an impostor_.

This turning of Conditionals into Categoricals is called a Change of
Relation; and the process may be reversed: _All the wise are virtuous_
may be written, _If any man is wise he is virtuous_; or, again, _Either
a man is not-wise or he is virtuous_. But the categorical form is
usually the simplest.

If, then, as substitutes for the corresponding conditionals,
categoricals are formally adequate, though sometimes inelegant, it may
be urged that Logic has nothing to do with elegance; or that, at any
rate, the chief elegance of science is economy, and that therefore, for
scientific purposes, whatever we may write further about conditionals
must be an ugly excrescence. The scientific purpose of Logic is to
assign the conditions of proof. Can we, then, in the conditional form
prove anything that cannot be proved in the categorical? Or does a
conditional require to be itself proved by any method not applicable to
the Categorical? If not, why go on with the discussion of Conditionals?
For all laws of Nature, however stated, are essentially categorical. 'If
a straight line falls on another straight line, the adjacent angles are
together equal to two right angles'; 'If a body is unsupported, it
falls'; 'If population increases, rents tend to rise': here '