The axiom, “Equals subtracted from equals leave equal differences,” may be demonstrated from the two axioms in the text. If A = a and B = b, A-B = a-b. For if not, let A-B = a-b+c. Then since B = b, adding equals to equals, A = a+c. But A = a. Therefore a = a+c, which is impossible.
This proposition having been demonstrated, we may, by means of it, demonstrate the following: “If equals be added to unequals, the sums are unequal.” If A = a and B not = b, A+B is not = a+b. For suppose it to be so. Then, since A = a and A+B = a+b, subtracting equals from equals, B = b; which is contrary to the hypothesis.
So again, it may be proved that two things, one of which is equal and the other unequal to a third thing, are unequal to one another. If A = a and A not = B, neither is a = B. For suppose it to be equal. Then since A = a and a = B, and since things equal to the same thing are equal to one another A = B; which is contrary to the hypothesis.
A writer to whom I have several times referred, gives as the definition of an impossibility, that which there exists in the world no cause adequate to produce. This definition does not take in such impossibilities as these—that two and two should make five; that two straight lines should inclose a space; or that any thing should begin to exist without a cause. I can think of no definition of impossibility comprehensive enough to include all its varieties, except the one which I have given: viz., An impossibility is that, the truth of which would conflict with a complete induction, that is, with the most conclusive evidence which we possess of universal truth.
As to the reputed impossibilities which rest on no other grounds than our ignorance of any cause capable of producing the supposed effects; very few of them are certainly impossible, or permanently incredible. The facts of traveling seventy miles an hour, painless surgical operations, and conversing by instantaneous signals between London and New York, held a high place, not many years ago, among such impossibilities.
“E, ex, extra, extraneus, étranger, stranger.”
Another etymological example sometimes cited is the derivation of the English uncle from the Latin avus. It is scarcely possible for two words to bear fewer outward marks of relationship, yet there is but one step between them, avus, avunculus, uncle. So pilgrim, from ager: per agrum, peragrinus, peregrinus, pellegrino, pilgrim. Professor Bain gives some apt examples of these transitions of meaning. “The word ‘damp’ primarily signified moist, humid, wet. But the property is often accompanied with the feeling of cold or chilliness, and hence the idea of cold is strongly suggested by the word. This is not all. Proceeding upon the superadded meaning, we speak of damping a man’s ardor, a metaphor where the cooling is the only circumstance concerned; we go on still further to designate the iron slide that shuts off the draft of a stove, ‘the damper,’ the primary meaning being now entirely dropped. ‘Dry,’ in like manner, through signifying the absence of moisture, water, or liquidity, is applied to sulphuric acid containing water, although not thereby ceasing to be a moist, wet, or liquid substance.” So in the phrases, dry sherry, or Champagne.
“ ‘Street,’ originally a paved way, with or without houses, has been extended to roads lined with houses, whether paved or unpaved. ‘Impertinent’ signified at first irrelevant, alien to the purpose in hand: through which it has come to mean, meddling, intrusive, unmannerly, insolent.” (Logic, ii., 173, 174.)
“An advocate,” says Mr. De Morgan (Formal Logic, p. 270), “is sometimes guilty of the argument à dicto secundum quid ad dictum simpliciter: it is his business to do for his client all that his client might honestly do for himself. Is not the word in italics frequently omitted? Might any man honestly try to do for himself all that counsel frequently try to do for him? We are often reminded of the two men who stole the leg of mutton; one could swear he had not got it, the other that he had not taken it. The counsel is doing his duty by his client, the client has left the matter to his counsel. Between the unexecuted intention of the client, and the unintended execution of the counsel, there may be a wrong done, and, if we are to believe the usual maxims, no wrong-doer.”
The same writer justly remarks (p. 251) that there is a converse fallacy, à dicto simpliciter ad dictum secundum quid, called by the scholastic logicians fallacia accidentis; and another which may be called à dicto secundum quid ad dictum secundum alterum quid (p. 265). For apt instances of both, I must refer the reader to Mr. De Morgan’s able chapter on Fallacies.