CHAPTER IV.
DEDUCTIVE REASONING.
The general principle of inference having been explained in the previous chapters, and a suitable system of symbols provided, we have now before us the comparatively easy task of tracing out the most common and important forms of deductive reasoning. The general problem of deduction is as follows:--*From one or more propositions called premises to draw such other propositions as will necessarily be true when the premises are true.* By deduction we investigate and unfold the information contained in the premises; and this we can do by one single rule--*For any term occurring in any proposition substitute the term which is asserted in any premise to be identical with it.* To obtain certain deductions, especially those involving negative conclusions, we shall require to bring into use the second and third Laws of Thought, and the process of reasoning will then be called *Indirect Deduction*. In the present chapter, however, I shall confine my attention to those results which can be obtained by the process of *Direct Deduction*, that is, by applying to the premises themselves the rule of substitution. It will be found that we can combine into one harmonious system, not only the various moods of the ancient syllogism but a great number of equally important forms of reasoning, which had no recognised place in the old logic. We can at the same time dispense entirely with the elaborate apparatus of logical rules and mnemonic lines, which were requisite so long as the vital principle of reasoning was not clearly expressed.
*Immediate Inference.*
Probably the simplest of all forms of inference is that which has been called *Immediate Inference*, because it can be performed upon a single proposition. It consists in joining an adjective, or other qualifying clause of the same nature, to both sides of an identity, and asserting the equivalence of the terms thus produced. For instance, since
Conductors of electricity = Non-electrics,
it follows that
Liquid conductors of electricity = Liquid non-electrics.
If we suppose that
Plants = Bodies decomposing carbonic acid,
it follows that
Microscopic plants = Microscopic bodies decomposing carbonic acid.
In general terms, from the identity
A = B
we can infer the identity
AC = BC.
This is but a case of plain substitution; for by the first Law of Thought it must be admitted that
AC = AC,
and if, in the second side of this identity, we substitute for A its equivalent B, we obtain
AC = BC.
In like manner from the partial identity
A = AB
we may obtain
AC = ABC
by an exactly similar act of substitution; and in every other case the rule will be found capable of verification by the principle of inference. The process when performed as here described will be quite free from the liability to error which I have shown[55] to exist in “Immediate Inference by added Determinants,” as described by Dr. Thomson.[56]
[55] *Elementary Lessons in Logic*, p. 86.
[56] *Outline of the Laws of Thought*, § 87.
*Inference with Two Simple Identities.*
One of the most common forms of inference, and one to which I shall especially direct attention, is practised with two simple identities. From the two statements that “London is the capital of England” and “London is the most populous city in the world,” we instantaneously draw the conclusion that “The capital of England is the most populous city in the world.” Similarly, from the identities
Hydrogen = Substance of least density, Hydrogen = Substance of least atomic weight,
we infer
Substance of least density = Substance of least atomic weight.
The general form of the argument is exhibited in the symbols
B = A (1) B = C (2) hence A = C. (3)
We may describe the result by saying that terms identical with the same term are identical with each other; and it is impossible to overlook the analogy to the first axiom of Euclid that “things equal to the same thing are equal to each other.” It has been very commonly supposed that this is a fundamental principle of thought, incapable of reduction to anything simpler. But I entertain no doubt that this form of reasoning is only one case of the general rule of inference. We have two propositions, A = B and B = C, and we may for a moment consider the second one as affirming a truth concerning B, while the former one informs us that B is identical with A; hence by substitution we may affirm the same truth of A. It happens in this particular case that the truth affirmed is identity to C, and we might, if we preferred it, have considered the substitution as made by means of the second identity in the first. Having two identities we have a choice of the mode in which we will make the substitution, though the result is exactly the same in either case.
Now compare the three following formulæ,
(1) A = B = C, hence A = C (2) A = B ~ C, hence A ~ C (3) A ~ B ~ C, no inference.
In the second formula we have an identity and a difference, and we are able to infer a difference; in the third we have two differences and are unable to make any inference at all. Because A and C both differ from B, we cannot tell whether they will or will not differ from each other. The flowers and leaves of a plant may both differ in colour from the earth in which the plant grows, and yet they may differ from each other; in other cases the leaves and stem may both differ from the soil and yet agree with each other. Where we have difference only we can make no inference; where we have identity we can infer. This fact gives great countenance to my assertion that inference proceeds always through identity, but may be equally well effected in propositions asserting difference or identity.
Deferring a more complete discussion of this point, I will only mention now that arguments from double identity occur very frequently, and are usually taken for granted, owing to their extreme simplicity. In regard to the equivalence of words this form of inference must be constantly employed. If the ancient Greek χαλκός is our *copper*, then it must be the French *cuivre*, the German *kupfer*, the Latin *cuprum*, because these are words, in one sense at least, equivalent to copper. Whenever we can give two definitions or expressions for the same term, the formula applies; thus Senior defined wealth as “All those things, and those things only, which are transferable, are limited in supply, and are directly or indirectly productive of pleasure or preventive of pain.” Wealth is also equivalent to “things which have value in exchange;” hence obviously, “things which have value in exchange = all those things, and those things only, which are transferable, &c.” Two expressions for the same term are often given in the same sentence, and their equivalence implied. Thus Thomson and Tait say,[57] “The naturalist may be content to know matter as that which can be perceived by the senses, or as that which can be acted upon by or can exert force.” I take this to mean--
Matter = what can be perceived by the senses; Matter = what can be acted upon by or can exert force.
[57] *Treatise on Natural Philosophy*, vol. i. p. 161.
For the term “matter” in either of these identities we may substitute its equivalent given in the other definition. Elsewhere they often employ sentences of the form exemplified in the following:[58] “The integral curvature, or whole change of direction of an arc of a plane curve, is the angle through which the tangent has turned as we pass from one extremity to the other.” This sentence is certainly of the form--
The integral curvature = the whole change of direction, &c. = the angle through which the tangent has turned, &c.
[58] *Treatise on Natural Philosophy*, vol. i. p. 6.
Disguised cases of the same kind of inference occur throughout all sciences, and a remarkable instance is found in algebraic geometry. Mathematicians readily show that every equation of the form *y* = *mx* + *c* corresponds to or represents a straight line; it is also easily proved that the same equation is equivalent to one of the general form A*x* + B*y* + C = 0, and *vice versâ*. Hence it follows that every equation of the form in question, that is to say, every equation of the first degree, corresponds to or represents a straight line.[59]
[59] Todhunter’s *Plane Co-ordinate Geometry*, chap. ii. pp. 11–14.
*Inference with a Simple and a Partial Identity.*
A form of reasoning somewhat different from that last considered consists in inference-between a simple and a partial identity. If we have two propositions of the forms
A = B, B = BC,
we may then substitute for B in either proposition its equivalent in the other, getting in both cases A = BC; in this we may if we like make a second substitution for B, getting
A = AC.
Thus, since “The Mont Blanc is the highest mountain in Europe, and the Mont Blanc is deeply covered with snow,” we infer by an obvious substitution that “The highest mountain in Europe is deeply covered with snow.” These propositions when rigorously stated fall into the forms above exhibited.
This mode of inference is constantly employed when for a term we substitute its definition, or *vice versâ*. The very purpose of a definition is to allow a single noun to be employed in place of a long descriptive phrase. Thus, when we say “A circle is a curve of the second degree,” we may substitute a definition of the circle, getting “A curve, all points of which are at equal distances from one point, is a curve of the second degree.” The real forms of the propositions here given are exactly those shown in the symbolic statement, but in this and many other cases it will be sufficient to state them in ordinary elliptical language for sake of brevity. In scientific treatises a term and its definition are often both given in the same sentence, as in “The weight of a body in any given locality, or the force with which the earth attracts it, is proportional to its mass.” The conjunction *or* in this statement gives the force of equivalence to the parenthetic phrase, so that the propositions really are
Weight of a body = force with which the earth attracts it. Weight of a body = weight, &c. proportional to its mass.
A slightly different case of inference consists in substituting in a proposition of the form A = AB, a definition of the term B. Thus from A = AB and B = C we get A = AC. For instance, we may say that “Metals are elements” and “Elements are incapable of decomposition.”
Metal = metal element. Element = what is incapable of decomposition.
Hence
Metal = metal incapable of decomposition.
It is almost needless to point out that the form of these arguments does not suffer any real modification if some of the terms happen to be negative; indeed in the last example “incapable of decomposition” may be treated as a negative term. Taking
A = metal B = element C = capable of decomposition *c* = incapable of decomposition;
the propositions are of the forms
A = AB B = *c*
whence, by substitution,
A = A*c*.
*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 (1) Metals conduct electricity, (2)
we may conclude that
Sodium conducts electricity. (3)
Taking A, B, C to represent the three terms respectively, the premises are of the forms
A = AB (1) B = BC. (2)
Now for B in (1) we can substitute its expression as given in (2), obtaining
A = ABC, (3)
or, in words, from
Sodium = sodium metal, (1) Metal = metal conducting electricity, (2)
we infer
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 Barbara[60] 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
[60] 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.
Neptune is a planet, (1) No planet has retrograde motion; (2) Hence Neptune has not retrograde motion. (3)
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
A = AB (1) B = B*c*, (2)
and by substitution for B, exactly as before, we obtain
A = AB*c*. (3)
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.[61] 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--
[61] *Elementary Lessons in Logic*, pp. 67, 79.
Some metals are of less density than water, (1)
All bodies of less density than water will float upon the surface of water; hence (2)
Some metals will float upon the surface of water. (3)
Let
A = some metals, B = body of less density than water, C = floating on the surface of water
then the propositions are evidently as before,
A = AB, (1) B = BC; (2) hence A = ABC, (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.
Let P = some, Q = metal,
B and C having the same meanings as before. Then the premises become
PQ = PQB, (1) B = BC; (2)
hence, by substitution, as before,
PQ = PQBC. (3)
Except that the formulæ look a little more complicated there is no difference whatever.
The mood Ferio is of exactly the same character as Darii 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,
A = AB = AB*c*.
If it is preferred to put PQ for the indefinite *some crystals*, we have
PQ = PQB = PQB*c*.
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 A = AC from A = ABC, though we cannot infer the latter more full and accurate result from the former. We may show this most simply as follows:--
By the first Law of Thought it is evident that
AA = AA;
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
AA = 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,[62] 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.
[62] *Pure Logic*, p. 19.
*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) B = AB (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 (1) B = CB, (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). Thus, for example,
Potassium = potassium metal (1) Potassium = potassium capable of floating on water; (2)
hence
Potassium metal = potassium capable of floating on water. (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 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, (1) Metals are good conductors of electricity, (2) Good conductors of electricity are useful for telegraphic purposes. (3)
It obviously follows that
Iron is useful for telegraphic purposes. (4)
Now if we take our letters thus,
A = Iron, B = metal, C = good conductor of electricity, D = useful for telegraphic purposes,
the premises will assume the forms
A = AB, (1) B = BC, (2) C = CD. (3)
For B in (1) we can substitute its equivalent in (2) obtaining, as before,
A = ABC.
Substituting for C in this intermediate result its equivalent as given in (3), we obtain the complete conclusion
A = ABCD. (4)
The full interpretation is that *Iron is iron, metal, good conductor of electricity, useful for telegraphic purposes*, which 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; (3)
it will follow that
Common salt does not possess the power of double refraction. (4)
Taking our letter-terms thus,
A = Common salt, B = Sodium chloride, C = Crystallizing in a cubical form, D = Possessing the power of double refraction,
we may state the premises in the forms
A = B, (1) B = BC, (2) C = C*d*. (3)
Substituting by (3) in (2) and then by (2) as thus altered in (1) we obtain
A = BC*d*, (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.[63] 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 substituting for A in the second side of (1) its expression in (2). We thus get
[63] *An Outline of the Necessary Laws of Thought*, Fifth Ed. p. 161.
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) Basalt is not a sedimentary rock, (2)
we ought not to be able to draw any inference concerning the relation between granite and basalt. Taking our letter-terms thus:
A = granite, B = sedimentary rock, C = basalt,
the premises may be expressed in the forms
A ~ B, (1) C ~ B. (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
A = A*b* (1) C = C*b*. (2)
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:
Whatever is not metallic is not capable of powerful magnetic influence, (1) Carbon is not metallic, (2) Therefore, carbon is not capable of powerful magnetic 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. To show this let
A = carbon, B = metallic, C = capable of powerful magnetic influence.
The premises readily take the forms
*b* = *bc*, (1) A = A*b*, (2)
and substitution for *b* in (2) by means of (1) gives the conclusion
A = A*bc*. (3)
Our 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, (1) All metals are elements. (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 premises then become
A = AB, (1) C = CB. (2)
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.
The old fallacy, technically called the *Illicit Process of the Major Term*, is more easy to commit and more difficult to detect than any other breach of the syllogistic rules. In our system it could hardly occur. From the premises
All planets are subject to gravity, (1) Fixed stars are not planets, (2)
we might inadvertently but fallaciously infer that, “Fixed stars are not subject to gravity.” To reduce the premises to symbolic form, let
A = planet B = fixed star C = subject to gravity;
then we have the propositions
A = AC (1) B = B*a*. (2)
The reader will try in vain to produce from these premises by legitimate substitution any relation between B and C; he could not then commit the fallacy of asserting that B is not C.
There remain two other kinds of paralogism, commonly known as the fallacy of Four Terms and the Illicit Process of the Minor Term. They are so evidently impossible while we obey the rule of the substitution of equivalents, that it is not necessary to give any illustrations. When there are four distinct terms in two propositions as in A = B and C = D, there could evidently be no opening for substitution. As to the Illicit Process of the Minor Term it consists in a flagrant substitution for a term of another wider term which is not known to be equivalent to it, and which is therefore not allowed by our rule to be substituted for it.