Logic: Deductive and Inductive Part 5

there is here no alternative predication, but only an Exponible equivalent to _No A is B, and No C is D_. And if we write--

_Either A is not B, or C is not D,_

this is affirmative as to the alternation, and is for all methods of treatment equivalent to A.

Logicians are divided in opinion as to the interpretation of the conjunction 'either, or'; some holding that it means 'not both,' others that it means 'it may be both.' Grammatical usage, upon which the question is sometimes argued, does not seem to be established in favour of either view. If we say _A man so precise in his walk and conversation is either a saint or a consummate hypocrite_; or, again, _One who is happy in a solitary life is either more or less than man_; we cannot in such cases mean that the subject may be both. On the other hand, if it be said that _the author of 'A Tale of a Tub' is either a misanthrope or a dyspeptic_, the alternatives are not incompatible. Or, again, given that _X. is a lunatic, or a lover, or a poet_, the three predicates have much congruity.

It has been urged that in Logic, language should be made as exact and definite as possible, and that this requires the exclusive interpretation 'not both.' But it seems a better argument, that Logic (1) should be able to express all meanings, and (2), as the science of evidence, must not a.s.sume more than is given; to be on the safe side, it must in doubtful cases a.s.sume the least, just as it generally a.s.sumes a preindesignate term to be of particular quant.i.ty; and, therefore 'either, or' means 'one, or the other, or both.'

However, when both the alternative propositions have the same subject, as _Either A is B, or A is C_, if the two predicates are contrary or contradictory terms (as 'saint' and 'hypocrite,' or 'saint' and 'not-saint'), they cannot in their nature be predicable in the same way of the same subject; and, therefore, in such a case 'either, or' means one or the other, but not both in the same relation. Hence it seems necessary to admit that the conjunction 'either, or' may sometimes require one interpretation, sometimes the other; and the rule is that it implies the further possibility 'or both,' except when both alternatives have the same subject whilst the predicates are contrary or contradictory terms.

If, then, the disjunctive _A is either B or C_ (_B_ and _C_ being contraries) implies that both alternatives cannot be true, it can only be adequately rendered in hypotheticals by the two forms--(1) _If A is B, it is not C_, and (2)_If A is not B, it is C_. But if the disjunctive _A is either B or C_ (_B_ and _C_ not being contraries) implies that both may be true, it will be adequately translated into a hypothetical by the single form, _If A is not B, it is C_. We cannot translate it into--_If A is B, it is not C_, for, by our supposition, if '_A is B_'

is true, it does not follow that '_A is C_' must be false.

Logicians are also divided in opinion as to the function of the hypothetical form. Some think it expresses doubt; for the consequent depends on the antecedent, and the antecedent, introduced by 'if,' may or may not be realised, as in _If the sky is clear, the night is cold_: whether the sky is, or is not, clear being supposed to be uncertain. And we have seen that some hypothetical propositions seem designed to draw attention to such uncertainty, as--_If there is a resisting medium in s.p.a.ce_, etc. But other Logicians lay stress upon the connection of the clauses as the important matter: the statement is, they say, that the consequent may be inferred from the antecedent. Some even declare that it is given as a necessary inference; and on this ground Sigwart rejects particular hypotheticals, such as _Sometimes when A is B, C is D_; for if it happens only sometimes the connexion cannot be necessary. Indeed, it cannot even be probably inferred without further grounds. But this is also true whenever the antecedent and consequent are concerned with different matter. For example, _If the soul is simple, it is indestructible_. How do you know that? Because _Every simple substance is indestructible_. Without this further ground there can be no inference. The fact is that conditional forms often cover a.s.sertions that are not true complex propositions but a sort of euthymemes (chap.

xi. -- 2), arguments abbreviated and rhetorically disguised. Thus: _If patience is a virtue there are painful virtues_--an example from Dr.

Keynes. Expanding this we have--

Patience is painful; Patience is a virtue: ? Some virtue is painful.

And then we see the equivocation of the inference; for though patience be painful _to learn_, it is not painful _as a virtue_ to the patient man.

The hypothetical, '_If Plato was not mistaken poets are dangerous citizens_,' may be considered as an argument against the laureateship, and may be expanded (informally) thus: 'All Plato's opinions deserve respect; one of them was that poets are bad citizens; therefore it behoves us to be chary of encouraging poetry.' Or take this disjunctive, '_Either Bacon wrote the works ascribed to Shakespeare, or there were two men of the highest genius in the same age and country_.'

This means that it is not likely there should be two such men, that we are sure of Bacon, and therefore ought to give him all the glory. Now, if it is the part of Logic 'to make explicit in language all that is implicit in thought,' or to put arguments into the form in which they can best be examined, such propositions as the above ought to be a.n.a.lysed in the way suggested, and confirmed or refuted according to their real intention.

We may conclude that no single function can be a.s.signed to all hypothetical propositions: each must be treated according to its own meaning in its own context.

-- 5. As to Modality, propositions are divided into Pure and Modal. A Modal proposition is one in which the predicate is affirmed or denied, not simply but _c.u.m modo_, with a qualification. And some Logicians have considered any adverb occurring in the predicate, or any sign of past or future tense, enough to const.i.tute a modal: as 'Petroleum is _dangerously_ inflammable'; 'English _will be_ the universal language.'

But far the most important kind of modality, and the only one we need consider, is that which is signified by some qualification of the predicate as to the degree of certainty with which it is affirmed or denied. Thus, 'The bite of the cobra is _probably_ mortal,' is called a Contingent or Problematic Modal: 'Water is _certainly_ composed of oxygen and hydrogen' is an a.s.sertory or Certain Modal: 'Two straight lines _cannot_ enclose a s.p.a.ce' is a Necessary or Apodeictic Modal (the opposite being inconceivable). Propositions not thus qualified are called Pure.

Modal propositions have had a long and eventful history, but they have not been found tractable by the resources of ordinary Logic, and are now generally neglected by the authors of text-books. No doubt such propositions are the commonest in ordinary discourse, and in some rough way we combine them and draw inferences from them. It is understood that a combination of a.s.sertory or of apodeictic premises may warrant an a.s.sertory or an apodeictic conclusion; but that if we combine either of these with a problematic premise our conclusion becomes problematic; whilst the combination of two problematic premises gives a conclusion less certain than either. But if we ask 'How much less certain?' there is no answer. That the modality of a conclusion follows the less certain of the premises combined, is inadequate for scientific guidance; so that, as Deductive Logic can get no farther than this, it has abandoned the discussion of Modals. To endeavour to determine the degree of certainty attaching to a problematic judgment is not, however, beyond the reach of Induction, by a.n.a.lysing circ.u.mstantial evidence, or by collecting statistics with regard to it. Thus, instead of 'The cobra's bite is _probably_ fatal,' we might find that it is fatal 80 times in 100. Then, if we know that of those who go to India 3 in 1000 are bitten, we can calculate what the chances are that any one going to India will die of a cobra's bite (chap. xx.).

-- 6. Verbal and Real Propositions.--Another important division of propositions turns upon the relation of the predicate to the subject in respect of their connotations. We saw, when discussing Relative Terms, that the connotation of one term often implies that of another; sometimes reciprocally, like 'master' and 'slave'; or by inclusion, like species and genus; or by exclusion, like contraries and contradictories.

When terms so related appear as subject and predicate of the same proposition, the result is often tautology--e.g., _The master has authority over his slave; A horse is an animal; Red is not blue; British is not foreign_. Whoever knows the meaning of 'master,' 'horse,' 'red,'

'British,' learns nothing from these propositions. Hence they are called Verbal propositions, as only expounding the sense of words, or as if they were propositions only by satisfying the forms of language, not by fulfilling the function of propositions in conveying a knowledge of facts. They are also called 'a.n.a.lytic' and 'Explicative,' when they separate and disengage the elements of the connotation of the subject.

Doubtless, such propositions may be useful to one who does not know the language; and Definitions, which are verbal propositions whose predicates a.n.a.lyse the whole connotations of their subjects, are indispensable instruments of science (see chap. xxii.).

Of course, hypothetical propositions may also be verbal, as _If the soul be material it is extended_; for 'extension' is connoted by 'matter'; and, therefore, the corresponding disjunctive is verbal--_Either the soul is not material, or it is extended_. But a true divisional disjunctive can never be verbal (chap. xxi. -- 4, rule 1).

On the other hand, when there is no such direct relation between subject and predicate that their connotations imply one another, but the predicate connotes something that cannot be learnt from the connotation of the subject, there is no longer tautology, but an enlargement of meaning--e.g., _Masters are degraded by their slaves; The horse is the n.o.blest animal; Red is the favourite colour of the British army; If the soul is simple, it is indestructible_. Such propositions are called Real, Synthetic, or Ampliative, because they are propositions for which a mere understanding of their subjects would be no subst.i.tute, since the predicate adds a meaning of its own concerning matter of fact.

To any one who understands the language, a verbal proposition can never be an inference or conclusion from evidence; nor can a verbal proposition ever furnish grounds for an inference, except as to the meaning of words. The subject of real and verbal propositions will inevitably recur in the chapters on Definition; but tautologies are such common blemishes in composition, and such frequent pitfalls in argument, that attention cannot be drawn to them too early or too often.

CHAPTER VI

CONDITIONS OF IMMEDIATE INFERENCE

-- 1. The word Inference is used in two different senses, which are often confused but should be carefully distinguished. In the first sense, it means a process of thought or reasoning by which the mind pa.s.ses from facts or statements presented, to some opinion or expectation. The data may be very vague and slight, prompting no more than a guess or surmise; as when we look up at the sky and form some expectation about the weather, or from the trick of a man's face entertain some prejudice as to his character. Or the data may be important and strongly significant, like the footprint that frightened Crusoe into thinking of cannibals, or as when news of war makes the city expect that Consols will fall. These are examples of the act of inferring, or of inference as a process; and with inference in this sense Logic has nothing to do; it belongs to Psychology to explain how it is that our minds pa.s.s from one perception or thought to another thought, and how we come to conjecture, conclude and believe (_cf._ chap. i. -- 6).

In the second sense, 'inference' means not this process of guessing or opining, but the result of it; the surmise, opinion, or belief when formed; in a word, the conclusion: and it is in this sense that Inference is treated of in Logic. The subject-matter of Logic is an inference, judgment or conclusion concerning facts, embodied in a proposition, which is to be examined in relation to the evidence that may be adduced for it, in order to determine whether, or how far, the evidence amounts to proof. Logic is the science of Reasoning in the sense in which 'reasoning' means giving reasons, for it shows what sort of reasons are good. Whilst Psychology explains how the mind goes forward from data to conclusions, Logic takes a conclusion and goes back to the data, inquiring whether those data, together with any other evidence (facts or principles) that can be collected, are of a nature to warrant the conclusion. If we think that the night will be stormy, that John Doe is of an amiable disposition, that water expands in freezing, or that one means to national prosperity is popular education, and wish to know whether we have evidence sufficient to justify us in holding these opinions, Logic can tell us what form the evidence should a.s.sume in order to be conclusive. What _form_ the evidence should a.s.sume: Logic cannot tell us what kinds of fact are proper evidence in any of these cases; that is a question for the man of special experience in life, or in science, or in business. But whatever facts const.i.tute the evidence, they must, in order to prove the point, admit of being stated in conformity with certain principles or conditions; and of these principles or conditions Logic is the science. It deals, then, not with the subjective process of inferring, but with the objective grounds that justify or discredit the inference.

-- 2. Inferences, in the Logical sense, are divided into two great cla.s.ses, the Immediate and the Mediate, according to the character of the evidence offered in proof of them. Strictly, to speak of inferences, in the sense of conclusions, as immediate or mediate, is an abuse of language, derived from times before the distinction between inference as process and inference as result was generally felt. No doubt we ought rather to speak of Immediate and Mediate Evidence; but it is of little use to attempt to alter the traditional expressions of the science.

An Immediate Inference, then, is one that depends for its proof upon only one other proposition, which has the same, or more extensive, terms (or matter). Thus that _one means to national prosperity is popular education_ is an immediate inference, if the evidence for it is no more than the admission that _popular education is a means to national prosperity:_ Similarly, it is an immediate inference that _Some authors are vain_, if it be granted that _All authors are vain_.

An Immediate Inference may seem to be little else than a verbal transformation; some Logicians dispute its claims to be called an inference at all, on the ground that it is identical with the pretended evidence. If we attend to the meaning, say they, an immediate inference does not really express any new judgment; the fact expressed by it is either the same as its evidence, or is even less significant. If from _No men are G.o.ds_ we prove that _No G.o.ds are men_, this is nugatory; if we prove from it that _Some men are not G.o.ds_, this is to emasculate the sense, to waste valuable information, to lose the commanding sweep of our universal proposition.

Still, in Logic, it is often found that an immediate inference expresses our knowledge in a more convenient form than that of the evidentiary proposition, as will appear in the chapter on Syllogisms and elsewhere.

And by transforming an universal into a particular proposition, as _No men are G.o.ds_, therefore, _Some men are not G.o.ds_,--we get a statement which, though weaker, is far more easily proved; since a single instance suffices. Moreover, by drawing all possible immediate inferences from a given proposition, we see it in all its aspects, and learn all that is implied in it.

A Mediate Inference, on the other hand, depends for its evidence upon a plurality of other propositions (two or more) which are connected together on logical principles. If we argue--

No men are G.o.ds; Alexander the Great is a man; ? Alexander the Great is not a G.o.d:

this is a Mediate Inference. The evidence consists of two propositions connected by the term 'man,' which is common to both (a Middle Term), mediating between 'G.o.ds' and 'Alexander.' Mediate Inferences comprise Syllogisms with their developments, and Inductions; and to discuss them further at present would be to antic.i.p.ate future chapters. We must now deal with the principles or conditions on which Immediate Inferences are valid: commonly called the "Laws of Thought."

-- 3. The Laws of Thought are conditions of the logical statement and criticism of all sorts of evidence; but as to Immediate Inference, they may be regarded as the only conditions it need satisfy. They are often expressed thus: (1) The principle of Ident.i.ty--'_Whatever is, is_'; (2) The principle of Contradiction--'_It is impossible for the same thing to be and not be_'; (3) The principle of Excluded Middle--'_Anything must either be or not be_.' These principles are manifestly not 'laws' of thought in the sense in which 'law' is used in Psychology; they do not profess to describe the actual mental processes that take place in judgment or reasoning, as the 'laws of a.s.sociation of ideas' account for memory and recollection. They are not natural laws of thought; but, in relation to thought, can only be regarded as laws when stated as precepts, the observance of which (consciously or not) is necessary to clear and consistent thinking: e.g., Never a.s.sume that the same thing can both be and not be.

However, treating Logic as the science of thought only as embodied in propositions, in respect of which evidence is to be adduced, or which are to be used as evidence of other propositions, the above laws or principles must be restated as the conditions of consistent argument in such terms as to be directly applicable to propositions. It was shown in the chapter on the connotation of terms, that terms are a.s.sumed by Logicians to be capable of definite meaning, and of being used univocally in the same context; if, or in so far as, this is not the case, we cannot understand one another's reasons nor even pursue in solitary meditation any coherent train of argument. We saw, too, that the meanings of terms were related to one another: some being full correlatives; others partially inclusive one of another, as species of genus; others mutually incompatible, as contraries; or alternatively predicable, as contradictories. We now a.s.sume that propositions are capable of definite meaning according to the meaning of their component terms and of the relation between them; that the meaning, the fact a.s.serted or denied, is what we are really concerned to prove or disprove; that a mere change in the words that const.i.tute our terms, or of construction, does not affect the truth of a proposition as long as the meaning is not altered, or (rather) as long as no fresh meaning is introduced; and that if the meaning of any proposition is true, any other proposition that denies it is false. This postulate is plainly necessary to consistency of statement and discourse; and consistency is necessary, if our thought or speech is to correspond with the unity and coherence of Nature and experience; and the Laws of Thought or Conditions of Immediate Inference are an a.n.a.lysis of this postulate.

-- 4. The principle of Ident.i.ty is usually written symbolically thus: _A is A; not-A is not-A_. It a.s.sumes that there is something that may be represented by a term; and it requires that, in any discussion, _every relevant term, once used in a definite sense, shall keep that meaning throughout_. Socrates in his father's workshop, at the battle of Delium, and in prison, is a.s.sumed to be the same man denotable by the same name; and similarly, 'elephant,' or 'justice,' or 'fairy,' in the same context, is to be understood of the same thing under the same _suppositio_.

But, further, it is a.s.sumed that of a given term another term may be predicated again and again in the same sense under the same conditions; that is, we may speak of the ident.i.ty of meaning in a proposition as well as in a term. To symbolise this we ought to alter the usual formula for Ident.i.ty and write it thus: _If B is A, B is A; if B is not-A, B is not-A_. If Socrates is wise, he is wise; if fairies frequent the moonlight, they do; if Justice is not of this world, it is not.

_Whatever affirmation or denial we make concerning any subject, we are bound to adhere to it for the purposes of the current argument or investigation._ Of course, if our a.s.sertion turns out to be false, we must not adhere to it; but then we must repudiate all that we formerly deduced from it.

Again, _whatever is true or false in one form of words is true or false in any other_: this is undeniable, for the important thing is ident.i.ty of meaning; but in Formal Logic it is not very convenient. If Socrates is wise, is it an ident.i.ty to say 'Therefore the master of Plato is wise'; or, further that he 'takes enlightened views of life'? If _Every man is fallible_, is it an identical proposition that _Every man is liable to error_? It seems pedantic to demand a separate proposition that _Fallible is liable to error_. But, on the other hand, the insidious subst.i.tution of one term for another speciously identical, is a chief occasion of fallacy. How if we go on to argue: therefore, _Every man is apt to blunder, p.r.o.ne to confusion of thought, inured to self-contradiction_? Practically, the subst.i.tution of ident.i.ties must be left to candour and good-sense; and may they increase among us. Formal Logic is, no doubt, safest with symbols; should, perhaps, content itself with A and B; or, at least, hardly venture beyond Y and Z.

-- 5. The principle of Contradiction is usually written symbolically, thus: _A is not not-A_. But, since this formula seems to be adapted to a single term, whereas we want one that is applicable to propositions, it may be better to write it thus: _B is not both A and not-A_. That is to say: _if any term may be affirmed of a subject, the contradictory term may, in the same relation, be denied of it_. A leaf that is green on one side of it may be not-green on the other; but it is not both green and not-green on the same surface, at the same time, and in the same light.

If a stick is straight, it is false that it is at the same time not-straight: having granted that two angles are equal, we must deny that they are unequal.

But is it necessarily false that the stick is 'crooked'; must we deny that either angle is 'greater or less' than the other? How far is it permissible to subst.i.tute any other term for the formal contradictory?

Clearly, the principle of Contradiction takes for granted the principle of Ident.i.ty, and is subject to the same difficulties in its practical application. As a matter of fact and common sense, if we affirm any term of a Subject, we are bound to deny of that Subject, in the same relation, not only the contradictory but all synonyms for this, and also all contraries and opposites; which, of course, are included in the contradictory. But who shall determine what these are? Without an authoritative Logical Dictionary to refer to, where all contradictories, synonyms, and contraries may be found on record, Formal Logic will hardly sanction the free play of common sense.

The principle of Excluded Middle may be written: _B is either A or not-A_; that is, _if any term be denied of a subject, the contradictory term may, in the same relation, be affirmed_. Of course, we may deny that a leaf is green on one side without being bound to affirm that it is not-green on the other. But in the same relation a leaf is either green or not-green; at the same time, a stick is either bent or not-bent. If we deny that A is greater than B, we must affirm that it is not-greater than B.

Whilst, then, the principle of Contradiction (that 'of contradictory predicates, one being affirmed, the other is denied ') might seem to leave open a third or middle course, the denying of both contradictories, the principle of Excluded Middle derives its name from the excluding of this middle course, by declaring that the one or the other must be affirmed. Hence the principle of Excluded Middle does not hold good of mere contrary terms. If we deny that a leaf is green, we are not bound to affirm it to be yellow; for it may be red; and then we may deny both contraries, yellow and green. In fact, two contraries do not between them cover the whole predicable area, but contradictories do: the form of their expression is such that (within the _suppositio_) each includes all that the other excludes; so that the subject (if brought within the _suppositio_) must fall under the one or the other.

It may seem absurd to say that Mont Blanc is either wise or not-wise; but how comes any mind so ill-organised as to introduce Mont Blanc into this strange company? Being there, however, the principle is inexorable: Mont Blanc is not-wise.

In fact, the principles of Contradiction and Excluded Middle are inseparable; they are implicit in all distinct experience, and may be regarded as indicating the two aspects of Negation. The principle of Contradiction says: _B is not both A and not-A_, as if _not-A_ might be nothing at all; this is abstract negation. But the principle of Excluded Middle says: _Granting that B is not A, it is still something_--namely, _not-A_; thus bringing us back to the concrete experience of a continuum in which the absence of one thing implies the presence of something else. Symbolically: to deny that B is A is to affirm that B is not A, and this only differs by a hyphen from B is not-A.

These principles, which were necessarily to some extent antic.i.p.ated in chap. iv. -- 7, the next chapter will further ill.u.s.trate.

-- 6. But first we must draw attention to a maxim (also already mentioned), which is strictly applicable to Immediate Inferences, though (as we shall see) in other kinds of proof it may be only a formal condition: this is the general caution _not to go beyond the evidence_.

An immediate inference ought to contain nothing that is not contained (or formally implied) in the proposition by which it is proved. With respect to quant.i.ty in denotation, this caution is embodied in the rule 'not to distribute any term that is not given distributed.' Thus, if there is a predication concerning 'Some S,' or 'Some men,' as in the forms I. and O., we cannot infer anything concerning 'All S.' or 'All men'; and, as we have seen, if a term is given us preindesignate, we are generally to take it as of particular quant.i.ty. Similarly, in the case of affirmative propositions, we saw that this rule requires us to a.s.sume that their predicates are undistributed.