CHAPTER V. DISJUNCTIVE PROPOSITIONS.

the proposition (1) is of the form

but these inferences are not equivalent to the false ones (2) and (3).

We can readily represent disjunctive reasoning by the modus ponendo tollens, when it is valid, by expressing the inconsistency of the alternatives explicitly. Thus if we resort to our instance of

and take

then the premise is apparently of the form

but in reality there is an unexpressed condition that “what is salt is not fresh,” from which follows, by a process of inference to be afterwards described, that “what is fresh is not salt.” We have then, in letter-terms, the two propositions

If we substitute these descriptions in the original proposition, we obtain

uniting B to each side we infer

that is,

I should weary the reader if I attempted to illustrate the multitude of forms which disjunctive reasoning may take; and as in the next chapter we shall be constantly treating the subject, I must here restrict myself to a single instance. A very common process of reasoning consists in the determination of the name of a thing by the successive exclusion of alternatives, a process called by the old name abscissio infiniti. Take the case:

Let us assign the letter-symbols thus—

Assuming that the alternatives copper or gold are intended to be exclusive, as just explained in the case of fresh and salt water, the premises may be stated in the forms