In fact (as observed in chap. ii. § 2) the sign of logical relation (_is_ or _is not_), whilst usually adequate for class-reasoning (coinherence) and sometimes extensible to causation (because a cause implies a class of events), should never be stretched to include other relations in such a way as to sacrifice intelligence to formalism. And, besides mathematical or quantitative relations, there are others (usually considered qualitative because indefinite) which cannot be justly expressed by the logical copula. We ought to read propositions expressing time-relations (and inferences drawn accordingly) thus: 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.