B—­is before—­C;
      A—­is before—­B: 
    .’.  A—­is before—­C.

And in like manner A—­is simultaneous with—­B; etc. Such arguments (as well as the mathematical) are intuitively sound and verifiable, and might be generalised in axioms if it were worth while:  but it is not, because no method could be founded on such axioms.

The customary use of relative terms justifies some Mediate Inferences, as, The father of a father is a grand-father.

Some cases, however, that at first seem obvious, are really delusive unless further data be supplied.  Thus A co-exists with B, B with C; .’.  A with C—­is not sound unless B is an instantaneous event; for where B is perdurable, A may co-exist with it at one time and C at another.

Again:  A is to the left of B, B of C; .’.  A of C.  This may pass; but it is not a parallel argument that if A is north of B and B west of C, then A is north-west of C:  for suppose that A is a mile to the north of B, and B a yard to the west of C, then A is practically north of C; at least, its westward position cannot be expressed in terms of the mariner’s compass.  In such a case we require to know not only the directions but the distances of A and C from B; and then the exact direction of A from C is an affair of mathematical calculation.

Qualitative reasoning concerning position is only applicable to things in one dimension of space, or in time considered as having one dimension.  Under these conditions we may frame the following generalisation concerning all Mediate Inferences:  Two terms definitely related to a third, and one of them positively, are related to one another as the other term is related to the third (that is, positively or negatively); provided that the relations given are of the same kind (that is, of Time, or Coinherence, or Likeness, or Equality).

Thus, to illustrate by relations of Time—­

      B is simultaneous with C;
      A is not simultaneous with B: 
    .’.  A is not simultaneous with C.

Here the relations are of the same kind but of different logical quality, and (as in the syllogism) a negative copula in the premises leads to a negative conclusion.

An examination in detail of particular cases would show that the above generalisation concerning all Mediate Inferences is subject to too many qualifications to be called an Axiom; it stands to the real Axioms (the Dictum, etc.) as the notion of the Uniformity of Nature does to the definite principles of natural order (cf. chap. xiii.  Sec. 9).

CHAPTER X

CATEGORICAL SYLLOGISMS

Sec. 1.  The type of logical, deductive, mediate, categorical Inference is a Syllogism directly conformable with the Dictum:  as—­

      All carnivores (M) are excitable (P);
      Cats (S) are carnivores (M): 
    .’.  Cats (S) are excitable (P).