Inference of a Partial from Two Partial Identities. However common be the cases of inference already noticed, there is a form occurring almost more frequently, and which deserves much attention, because it occupied a prominent place in the ancient syllogistic system. That system strangely overlooked all the kinds of argument we have as yet considered, and selected, as the type of all reasoning, one which employs two partial identities as premises. Thus from the propositions Sodium is a metal Metals conduct electricity, we may conclude that Sodium conducts electricity. (1) (2) (3) Taking A, B, C to represent the three terms respectively, the premises are of the forms (2) Now for B in (1) we can substitute its expression as given in (2), obtaining Metal metal conducting electricity, we infer (3) (1) (2) Sodium = sodium metal conducting electricity, (3 which, in the elliptical language of common life, becomes "Sodium conducts electricity." The above is a syllogism in the mood called Barbara1 in the truly barbarous language of ancient logicians; and the first figure of the syllogism contained Barbara and three other moods which were esteemed distinct forms of argument. But it is worthy of notice that, without any real change in our form of inference, we readily include these three other moods under Barbara. The negative mood Celarent will be represented by the example Hence Neptune is a planet, (1) (2) No planet has retrograde motion; 1 An explanation of this and other technical terms of the old logic will be found in my Elementary Lessons in Logic, Sixth Edition, 1876; Macmillan. If we put A for Neptune, B for planet, and C for "having retrograde motion," then by the corresponding negative term c, we denote "not having retrograde motion." The premises now fall into the forms and by substitution for B, exactly as before, we obtain What is called in the old logic a particular conclusion may be deduced without any real variation in the symbols. Particular quantity is indicated, as before mentioned (p. 41), by joining to the term an indefinite adjective of quantity, such as some, a part of, certain, &c., meaning that an unknown part of the term enters into the proposition as subject. Considerable doubt and ambiguity arise out of the question whether the part may not in some cases be the whole, and in the syllogism at least it must be understood in this sense.1 Now, if we take a letter to represent this indefinite part, we need make no change in our formulæ to express the syllogisms Darii and Ferio. Consider the example 'Some metals are of less density than water, (1) All bodies of less density than water will float upon the surface of water; hence Some metals will float upon the surface of = body of less density than water, (2) (3) C = = floating on the surface of water then the propositions are evidently as before, A = AC hence A = AB, (1) (2) (3) Thus the syllogism Darii does not really differ from Barbara. If the reader prefer it, we can readily employ a distinct symbol for the indefinite sign of quantity. B and C having the same meanings as before. Then the premises become 1 Elementary Lessons in Logic, pp. 67, 79. The mood Ferio is of exactly the same character as Dari or Barbara, except that it involves the use of a negative term. Take the example, Bodies which are equally elastic in all directions do not doubly refract light; Some crystals are bodies equally elastic in all directions; therefore, some crystals do not doubly refract light. Assigning the letters as follows:— A some crystals, B = bodies equally elastic in all directions, C doubly refracting light, = c = not doubly refracting light. Our argument is of the same form as before, and may be concisely stated in one line, If it is preferred to put PQ for the indefinite some crystals, we have PQ== PQB = PQBc. The only difference is that the negative term c takes the place of C in the mood Darii. Ellipsis of Terms in Partial Identities. = The reader will probably have noticed that the conclusion which we obtain from premises is often more full than that drawn by the old Aristotelian processes. Thus from "Sodium is a metal," and " Metals conduct electricity," we inferred (p. 55) that "Sodium sodium, metal, conducting electricity," whereas the old logic simply concludes that "Sodium conducts electricity." Symbolically, from A = AB, and B = BC, we get A = ABC, whereas the old logic gets at the most A AC. It is therefore well to show that without employing any other principles of inference than those already described, we may infer AC from A = ABC, though we cannot infer the latter A = more full and accurate result from the former. show this most simply as follows: By the first Law of Thought it is evident that AA = AA; We may and if we have given the proposition A = ABC, we may substitute for both the A's in the second side of the above, obtaining ABC. ABC. But from the property of logical symbols expressed in the Law of Simplicity (p. 33) some of the repeated letters may be made to coalesce, and we have A = ABC. C. Substituting again for ABC its equivalent A, we obtain A = AC, the desired result. By a similar process of reasoning it may be shown that we can always drop out any term appearing in one member of a proposition, provided that we substitute for it the whole of the other member. This process was described in my first logical Essay, as Intrinsic Elimination, but it might perhaps be better entitled the Ellipsis of Terms. It enables us to get rid of needless terms by strict substitutive reasoning. Inference of a Simple from Two Partial Identities. Two terms may be connected together by two partial identities in yet another manner, and a case of inference then arises which is of the highest importance. In the two premises A = AB (1) (2) the second member of each is the same; so that we can by obvious substitution obtain A = B. Thus, in plain geometry we readily prove that "Every equilateral triangle is also an equiangular triangle," and we can with equal ease prove that "Every equiangular triangle is an equilateral triangle." Thence by substitution, as explained above, we pass to the simple identity, Equilateral triangle = equiangular triangle. We thus prove that one class of triangles is entirely identical with another class; that is to say, they differ only in our way of naming and regarding them. The great importance of this process of inference arises from the fact that the conclusion is more simple and general than either of the premises, and contains as much information as both of them put together. It is on this account constantly employed in inductive investigation, as will afterwards be more fully explained, and it is the natural mode by which we arrive at a conviction of the truth of simple identities as existing between classes of numerous objects. Inference of a Limited from Two Partial Identities. We have considered some arguments which are of the type treated by Aristotle in the first figure of the syllogism. But there exist two other types of argument which employ a pair of partial identities. If our premises are as shown in these symbols, B = AB B = (1) (2) we may substitute for B either by (1) in (2) or by (2) in (1), and by both modes we obtain the conclusion AB = CB, (3) a proposition of the kind which we have called a limited identity (p. 42). Potassium hence Thus, for example, = potassium metal = water; Potassium potassium capable of floating on Potassium metal = potassium capable of float ing on water. (1) (2) (3) This is really a syllogism of the mood Darapti in the third figure, except that we obtain a conclusion of a more exact character than the old syllogism gives. From the premises "Potassium is a metal" and "Potassium floats on water," Aristotle would have inferred that "Some metals float on water." But if inquiry were made what the "some metals" are, the answer would certainly be "Metal which is potassium." Hence Aristotle's conclusion simply leaves out some of the information afforded in the premises; it |