Here A. I. E. O. correspond with those similarly symbolised in the usual list, merely designating in the predicates the quantity which was formerly treated as implicit.
§ 4. As to Relation, propositions are either Categorical or Conditional. A Categorical Proposition is one in which the predicate is directly affirmed or denied of the subject without any limitation of time, place, or circumstance, extraneous to the subject, as _All men in England are secure of justice_; in which proposition, though there is a limitation of place ('in England'), it is included in the subject. Of this kind are nearly all the examples that have yet been given, according to the form _S is P_.
A Conditional Proposition is so called because the predication is made under some limitation or condition not included in the subject, as _If a man live in England, he is secure of justice_. Here the limitation 'living in England' is put into a conditional sentence extraneous to the subject, 'he,' representing any man.
Conditional propositions, again, are of two kinds--Hypothetical and Disjunctive. Hypothetical propositions are those that are limited by an explicit conditional sentence, as above, or thus: _If Joe Smith was a prophet, his followers have been unjustly persecuted_. Or in symbols thus:
If A is, B is; If A is B, A is C; If A is B, C is D.
Disjunctive propositions are those in which the condition under which predication is made is not explicit but only implied under the disguise of an alternative proposition, as _Joe Smith was either a prophet or an impostor_. Here 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. § 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_.