Of course syllogistic reasoning, like all other reasoning, presupposes the laws of thought, and in the process of indirect reduction, which occupies a not unimportant place in the doctrine of the syllogism, these laws come in explicitly.
It is not necessary to consider in detail formal inferences belonging to the logic of relatives, e.g., B is greater than C, A is greater than B, therefore, A is greater than C. Here we require the principle that whatever is greater than anything that is greater than a third thing is itself greater than the third thing; and it would be still more difficult than in the case of the dictum de omni et nullo to evolve this principle immediately out of the three laws of thought.
APPENDIX C.
A GENERALIZATION OF LOGICAL PROCESSES IN THEIR APPLICATION TO COMPLEX PROPOSITIONS.[465]
CHAPTER I.
THE COMBINATION OF TERMS.
[465] The following pages deal with problems that have ordinarily been relegated to symbolic logic. They do not, however, treat of symbolic logic directly, if that term is understood in its ordinary sense, namely, as designating that branch of the science in which symbols of operation are used. Of course in a broad sense all formal logic is symbolic.
424. Complex Terms.—A simple term may be defined as a term which does not consist of a combination of other terms. We denote a simple term by a single letter; for example, A, P, X. The combination of simple terms yields a complex term; and the combination may be either conjunctive or alternative.
A complex term resulting from the conjunctive combination of other terms may be called a conjunctive term, and it will be found convenient to denote such a term by the simple juxtaposition of the other terms involved.[466] This kind of combination is sometimes called determination; and we may speak of the elements combined in a conjunctive term as the determinants of that term. Thus, A and B are the determinants of the conjunctive term AB.