even leaves us open to interpret the some metals in a wider sense than we are warranted in doing. From these distinct defects of the old syllogism the process of substitution is free, and the new process only incurs the possible objection of being tediously minute and accurate. Miscellaneous Forms of Deductive Inference. The more common forms of deductive reasoning having been exhibited and demonstrated on the principle of substitution, there still remain many, in fact an indefinite number, which may be explained with nearly equal ease. Such as involve the use of disjunctive propositions will be described in a later chapter, and several of the syllogistic moods which include negative terms will be more conveniently treated after we have introduced the symbolic use of the second and third laws of thought. We sometimes meet with a chain of propositions which allow of repeated substitution, and form an argument called in the old logic a Sorites. Take, for instance, the premises Iron is a metal, Metals are good conductors of electricity, (1) (2) A = Iron, B = metal, = C good conductor of the premises will assume the forms A = AB, (1) (2) (3) For B in (1) we can obtaining, as before, Α = substitute its equivalent in (2) Substituting for C in this intermediate result its equivalent as given in (3), we obtain the complete conclusion The full interpretation is that Iron is iron, metal, good conductor of electricity, useful for telegraphic purposes, which ABC. is abridged in common language by the ellipsis of the circumstances which are not of immediate importance. Instead of all the propositions being exactly of the same kind as in the last example, we may have a series of premises of various character; for instance, Common salt is sodium chloride, (1) Sodium chloride crystallizes in a cubical form, (2) What crystallizes in a cubical form does not possess the power of double refraction; it will follow that (3) Common salt does not possess the power of double Taking our letter-terms thus, A Common salt, = D = (4) Possessing the power of double refraction, we may state the premises in the forms (3) Substituting by (3) in (2) and then by (2) as thus altered in (1) we obtain A = BCd, (4) which is a more precise version of the common conclusion. We often meet with a series of propositions describing the qualities or circumstances of the one same thing, and we may combine them all into one proposition by the process of substitution. This case is, in fact, that which Dr. Thomson has called "Immediate Inference by the sum of several predicates," and his example will serve my purpose well. He describes copper as A metal-of a red colour-and disagreeable smell-and taste-all the preparations of which are poisonous-which is highly malleable ductile-and tenacious-with a specific gravity of about 8.83." If we assign the letter A to copper, and the succeeding letters of the alphabet in succession to the series of predicates, we have nine distinct statements, of the form A = AB (1) A = AC (2) A AD (3) ..... A AK (9). We can readily combine these propositions into one by = = 1 An Outline of the Necessary Laws of Thought, Fifth Ed. p. 161. substituting for A in the second side of (1) its expression in (2). We thus get A = ABC, and by repeating the process over and over again we obviously get the single proposition A = ABCD .. JK. But Dr. Thomson is mistaken in supposing that we can obtain in this manner a definition of copper. Strictly speaking, the above proposition is only a description of copper, and all the ordinary descriptions of substances in scientific works may be summed up in this form. Thus we may assert of the organic substances called Paraffins that they are all saturated hydrocarbons, incapable of uniting with other substances, produced by heating the alcoholic iodides with zinc, and so on. It may be shown that no amount of ordinary description can be equivalent to a definition of any substance. Fallacies. I have hitherto been engaged in showing that all the forms of reasoning of the old syllogistic logic, and an indefinite number of other forms in addition, may be readily and clearly explained on the single principle of substitution. It is now desirable to show that the same principle will prevent us falling into fallacies. So long as we exactly observe the one rule of substitution of equivalents it will be impossible to commit a paralogism, that is to break any one of the elaborate rules of the ancient system. The one new rule is thus proved to be as powerful as the six, eight, or more rules by which the correctness of syllogistic reasoning was guarded. It was a fundamental rule, for instance, that two negative premises could give no conclusion. If we take the propositions Granite is not a sedimentary rock, (1) (2) we ought not to be able to draw any inference concerning the relation between granite and basalt. letter-terms thus: = A = granite, B sedimentary rock, the premises may be expressed in the forms Taking our C = basalt, C ~ B. (1) (2) We have in this form two statements of difference; but the principle of inference can only work with a statement of agreement or identity (p. 63). Thus our rule gives us no power whatever of drawing any inference; this is exactly in accordance with the fifth rule of the syllogism. It is to be remembered, indeed, that we claim the power of always turning a negative proposition into an affirmative one (p. 45); and it might seem that the old rule against negative premises would thus be circumvented. Let us try. The premises (1) and (2) when affirmatively stated take the forms The reader will find it impossible by the rule of substitution to discover a relation between A and C. Three terms occur in the above premises, namely A, b, and C; but they are so combined that no term occurring in one has its exact equivalent stated in the other. No substitution can therefore be made, and the principle of the fifth rule of the syllogism holds true. Fallacy is impossible. It would be a mistake, however, to suppose that the mere occurrence of negative terms in both premises of a syllogism renders them incapable of yielding a conclusion. The old rule informed us that from two negative premises no conclusion could be drawn, but it is a fact that the rule in this bare form does not hold universally true; and I am not aware that any precise explanation has been given of the conditions under which it is or is not imperative. Consider the following example: A Whatever is not metallic is not capable of power- netic influence. (3) Here we have two distinctly negative premises (1) and (2), and yet they yield a perfectly valid negative conclusion (3). The syllogistic rule is actually falsified in its bare and general statement. In this and many other cases we can convert the propositions into affirmative ones which will yield a conclusion by substitution without any difficulty. Card with all of mayor or man To show this let A = carbon, B = metallic, C = capable of powerful magnetic influence. The premises readily take the forms (2) and substitution for b in (2) by means of (1) gives the conclusion A = Abc. (3) Cur principle of inference then includes the rule of negative premises whenever it is true, and discriminates correctly between the cases where it does and does not hold true. The paralogism, anciently called the Fallacy of Undistributed Middle, is also easily exhibited and infallibly avoided by our system. Let the premises be Hydrogen is an element, All metals are elements. (1) (2) According to the syllogistic rules the middle term "element " is here undistributed, and no conclusion can be obtained; we cannot tell then whether hydrogen is or is not a metal. Represent the terms as follows A = hydrogen, B = element, C = metal. The reader will here, as in a former page (p. 62), find it impossible to make any substitution. The only term which occurs in both premises is B, but it is differently combined in the two premises. For B we must not substitute A, which is equivalent to AB, not to B. Nor must we confuse together CB and AB, which, though they contain one common letter, are different aggregate terms. The rule of substitution gives us no right to decompose combinations; and if we adhere rigidly to the rule, that if two terms are stated to be equivalent we may substitute one for the other, we cannot commit the fallacy. It is apparent that the form of premises stated above is the same as that which we obtained by translating two negative premises into the affirmative form. |