Sec. 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.  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.