Chapter III: Critique of Formalist Logic

[Sidenote: _Intrinsic impossibility of formal Logic._]

From the fact that in the verbal form all distinctions (pure concepts, and empirical and abstract concepts, distinct concepts and opposite concepts) are indistinguishable, and on the other hand all identities, such as that of concept, definition and demonstration, appear differentiated or capable of differentiation, we can deduce the impossibility of constructing logical Science by means of an analysis of the verbal form. The condemnation of all _formal_ Logic is thus pronounced.

[Sidenote: _Its nature._]

This Logic has been variously called _Aristotelian, peripatetic, scholastic,_ after its authors and historical representatives; _syllogistic,_ from the doctrine that forms its principal content; _formal,_ from its pretensions to philosophic purity; _empirical,_ by those who tried to drive it back to its place; and although this last name is correct, it would be better to call it _formal,_ and still better, _verbal,_ to indicate of what the empiricism to which it is desired to allude, chiefly consists. Indeed, if empiricism be marked by its limiting itself to single representations, regrouping them in types and arranging them in classes, there is no doubt that that method of treatment is empirical, which takes the logical function, not in the eternal peculiarity of its character as thought of the universal, but only in its various particular translations or manifestations, in which it acquires contingent characteristics. Since these contingent characteristics come to it, in the first place, from the verbal form, it can well be called verbalism. Owing to its verbalism, too, it has happened, that over and above the grammars of individual languages, there has been conceived as existing a _general, rational_ and _logical_ Grammar; and this hybrid science, which is no longer grammar and arose from logical assumptions, has developed in such a way as to be indistinguishable from empirical or verbal Logic.

[Sidenote: _Its partial justification._]

Certainly, as mere empiricism, this so-called Logic could not be condemned. And Hegel was not wrong in remarking that if people are interested in establishing that there are sixty species of parrots and one hundred and thirty-seven of veronica, it is not clear why it should be of less interest to establish the various forms of the judgment and of the syllogism. That discipline has its utility as mere empiricism, and it may be useful to any one to employ in certain cases the terminology in which an affirmation is characterized as positive or as merely negative, as particular or as universal, as a judgment that awaits reasoning and demonstration, as an immediate inference, enthymeme or sorites, as a conclusive or an inconclusive, or as a correct or an incorrect syllogism, and so on. It is also comprehensible how, as mere empiricism, it assumed a _normative_ character, and was translated into _rules_; rules, which are valid within their own sphere, neither more nor less than are all empirical rules.

[Sidenote: _Its error._]

But it does not limit itself to acting simply as an empirical description, nor even as a simple technique; it usurps a much more lofty office. Just as Rhetoric and Grammar, innocent and useful so long as they limit themselves to the functions of convenient grouping and convenient terminology, become false and harmful when they assume the attitude of sciences of absolute values, and must then be resolved into, and replaced by Æsthetic; so empirical or verbal Logic becomes transformed into error when it claims to give the laws of thought, or the thought of thought, which cannot be other than the concept of the concept. It is not, then, _formal,_ as it boasts itself to be, because the only logical form is the universal, and this alone is the object of logical investigation; but it is falsely formal, since it relies upon contingencies, and must, therefore, be called _formalist._ We reject it here exclusively in its formalist aspect; that is to say, in so far as it is a complex of empirical distinctions that wish to pass as rational and usurp the place of true rationality.

[Sidenote: _Its traditional constitution._]

Several of such empirical distinctions, such as the distinction between thought and principle of thought, truth and reason of truth, judgments and syllogisms, and such-like, have been recorded and criticized; we shall proceed to mention others, when suitable opportunities occur. Here it will be well to refer to the general physiognomy and structure of that Logic, as it was embodied for centuries in the schools and still persists in treatises.

[Sidenote: _The three logical forms._]

Its point of departure is the external distinction between words and connections of words, which belongs properly to Grammar. But words are then treated by it as concepts, and connections of words, as judgments. Thus it obtains the identification of the concept with the abstract and mutilated grammatical word and arrives at the monstrous determination of the concepts as things which are not in themselves either true or false. Thus, again, by constantly calling upon the connections of the concepts for succour, it succeeds in distinguishing the judgment from the mere proposition. A double criterion is constantly adopted in establishing these and other fundamental forms: the verbal and the logical; and formalist Logic oscillates equivocally between the two different determinations; whence the alternating appearance of truth and of falsehood, with which its distinctions present themselves. The syllogism, which should be the third fundamental form, is conceived as the connection of three distinct judgments; but if it yet retains its importance and preponderance over two-membered forms or over serial forms of more than three propositions and judgments, this is really because to the distinction and enumeration of the three propositions there is added the criterion of the concept as a nexus, or as a triunity of universal, particular and singular.

[Sidenote: _The theories of the concept and of the judgment._]

The three fundamental forms have been reduced by some logicians to two, by others; amplified to four or to five, by adding to them the perceptive form or the definitive and systematic form. These restrictions and amplifications have always encountered resistance, because it was justly felt that in this way one form of empiricism was being mingled with another: the verbal form with empirical distinctions drawn from other presuppositions. But in determining in particular the three fundamental forms, formalist Logic has not been able to restrict itself to the mere distinction of words and propositions, artificially placed in relation with the pure concept; but has been obliged to draw from other sources. The concepts are variously classified, sometimes from the verbal point of view, as _identical, equivalent, equivocal, anonymous_ and _synonymous_; sometimes from the logical point of view, as _distinct, disparate, contrary_ or _contradictory_; sometimes from the psychological point of view, as _incomplete_ and _complete, obscure_ and _clear,_ the concepts further always being understood as names, so that, for example, distinct concepts are indifferently philosophically distinct concepts, and empirically distinct concepts; and the contraries are both the philosophical contraries and those empirically so-called. The same has occurred in the classification of judgments where sometimes the determinations of the concept are taken as foundation and the judgments distinguished as _universal particular_ and _individual;_ sometimes the intrinsic dialectic nature of the concept, and they are distinguished as _affirmative, negative_ and _indeterminate_ or _infinite_; sometimes the stages passed through in the search for truth, and they are distinguished into _categorical, hypothetical_ and _disjunctive,_ or _apodeictic, assertory_ and _problematic._ And these forms have further always been understood verbally. "Universality" is the "totality" empirically designated by the word, and not true universality; and "individuality," on the contrary, is not only the individuality of the representation, but also the single particularity of the distinct concept; "affirmative" is differentiated from "negative" by accidental grammatical form, and not because that unique act which is thought, at once affirmation and negation (as the will is both love and hatred) can be truly divided.

[Sidenote: _The theory of the syllogism._]

The classification of syllogisms, founded exactly upon the empirical conception of the judgment as the copulation of a _subject_ and a _predicate_ affords a suitable parallel to this method of treatment of the judgment; subject and predicate being understood in an empirical and grammatical manner, whence they are also discovered in those verbal affirmations, in which they are not distinct, because they are identical, as in the case of the judgment of definition. For empirical Logic, in the judgment: "The will is the practical form of the spirit," "will" is subject and "practical form" predicate in the same way as in "Peter is a man," "Peter" is subject, and "man" predicate. From the distinction between subject and predicate, arise the four _figures_ of the syllogism; the criterion being the position of the middle term in the two premisses of the three propositions of which the syllogism is formed. If the middle term be subject in the first premiss and predicate in the second, we have the first figure; if it be predicate in both, the second; if it be subject in both, the third; if it be predicate in the first and subject in the second, the fourth figure ("_sub-prae,_ turn _prae-prae,_ turn _sub-sub,_ turn _prae-sub")._ But in order to deduce the moods of each figure recourse is then had to another criterion, indeed to two other criteria; that is, to the empirical distinctions of judgments into universal and particular, and into affirmative and negative, with the four consequent determinations into universal-affirmative judgments (A), universal-negative (E), particular-affirmative (I), and particular-negative (O). Thus, in the first figure, two universal affirmative premisses constitute the first mood, and the conclusion is universal affirmative _(Barbara)_; two premisses, both universal, but one affirmative and the other negative, constitute the second, and the conclusion is universal negative _(celarent)_; two premisses, one universal affirmative and the other particular affirmative, constitute the third mood, and the conclusion is particular affirmative _(darii);_ two premisses, one universal negative and one particular affirmative, constitute the fourth mood, and the conclusion is particular negative _(ferio)._ And so on.

[Sidenote: _Spontaneous reductions to the absurd of formal Logic._]

This is not the occasion to go on expounding in its other particulars this construction, of which we have given an example, for it is very well known: nor to attach importance to criticizing it, since' its foundations themselves have already been shown to be false and its hybrid genesis explained. Verbal Logic, which vaunts itself as rational, carries its own caricature in itself, namely the creation of _Sophisms_; because, since it seeks the force of thought in words, it cannot prevent sophistical ability from making use, in its turn, of words, in order capriciously to create thoughts and forms of thought. Thus verbal Logic, in order to combat sophisms, is constrained hastily and eagerly to abandon simple verbal connections, and to take refuge in concepts and connections of concepts thought in words; that is to say, neither more nor less than to negate the formalist point of view. And with analogous self-irony it renounces that point of view and dissolves itself, when it tries to refute the fourth figure of the syllogism, or to reduce the second, third and fourth to the first, as the only real figure, and then the first to a connection of three concepts; not to mention the permanent self-irony and patent demonstration of falsity involved in the logical deduction of the figures of the syllogism which it makes from a series of moods, recognized as _not conclusive._

[Sidenote: _Mathematical Logic or Logistic._]

Formalist Logic has been the object of many violent attacks from the Renaissance onwards; but it cannot be said that it has been struck in its essential part, because up to the present, the principle itself, or the incoherence from which it springs, has not been attacked. Several attempts at reform have followed and still follow; they have all of them the same defect, which is the wish to reform formal Logic without issuing from its circle, and without refuting its tacit presumption-- the pretension of obtaining thought in words, concepts in propositions. The most considerable attempt of the kind that has been made, which has many zealous followers in our day, is _mathematical Logic,_ also called _calculatory, algebraical, algorhythmic, symbolic, a new analytic,_ or a _Logical calculus or Logistic._

[Sidenote: _Its non-mathematical character._]

It is admitted by those who profess it and is for the rest evident from the definitions of Logistic that have been given, that it has nothing in common with mathematics, for although the majority of its cultivators are mathematicians and use is made of the phraseology usual in Mathematics, and it is directed toward Mathematics, in certain of its practical intentions, there is nothing intrinsically mathematical in it. Logistic is a science which deals, not with quantity alone, but with _quantity and quality together_; it is a science of _things in general_; it is _universal mathematics,_ containing also, subordinated to itself, the mathematical sciences properly so-called, but not coinciding with these. It means to be, not mathematics, but _a general science of thought._

[Sidenote: _Example of its mode of treatment._]

But the "thought" of Logistic is nothing but the "verbal proposition," which, in fact, supplies its starting-point. What the proposition is; whether it be possible truly to distinguish the proposition we call "verbal" from all the others, poetical, musical, pictorial; whether the verbal proposition does not bear indistinctly in itself, a series of very diverse spiritual formations, from poetry to mathematics, from history and philosophy to the natural sciences; what language is and what the concept is--these and all other questions concerning the forms of the spirit and the nature of thought, remain altogether extraneous to Logistic and do not disturb it in its work. The propositions (the concept of the proposition remaining an unexplained presupposition) can be indicated by _p, q,_ etc.; the relation of implication of one proposition in another can be indicated by the sign _⊃,_ hence an isolated proposition is "that which implies itself" _(p.⊃.q.)._ By following a method such as this, many distinctions of the traditional formalist Logic are eliminated, and in compensation for this, new ones are added and old and new are dressed in a new phraseology. The logical _sum a + b_ is the smallest concept, which contains the other two _a_ and _b_ and is what was previously called the "sphere of the concept"; the logical _product a x b_ indicates the greater concept contained in _a_ and in _b,_ and answers to that which was previously called "comprehension." There are also new or renovated laws, like the law of _identity,_ by force of which, in Logic (differently from Algebra), _a + a + a ... = a;_ by which it is desired to signify this profound truth, that the repetition of one and the same concept as many times as one wishes, always gives the same concept;--the law of _commutation,_ by which _ab = ba_;--or that of absorption, by which _a(a + b) = a;_ or--(the convention being that the negation of a concept is indicated by placing against it a vertical line) the other beautiful laws and formulæ: _a + a | = a| (a | )a = a; aa | = o._ This is a charming amusement for those who have a taste for it.

[Sidenote: _Identity of nature of Logistic with formalist Logic._]

Thus it is seen that if the words and the formulæ be somewhat different, the nature of mathematical Logic in no respect differs from that of formalist Logic. Where the new Logic contradicts the old, it is not possible to say which of the two is right; as of two people walking side by side over insecure ground, it is impossible to say which of the two walks securely. The very doctrine of the _quantification of the predicate_ (which has been the leaven of the reform) in no wise alters the traditional manner of conceiving the judgment, with the corresponding arbitrary manner of distinguishing subject and predicate. It simply establishes a convention with the object of being able to symbolize, with the sign of equality, the subject and the predicate:--the subject being included in the predicate, is part of it: "men are mortal" equals: "men are some mortals"; and so, "men" being indicated with _a_ and "some mortals" with _b,_ the judgment can be symbolized: _a = b._ For us, it is indifferent whether the modes of the syllogism be the 64 and the 19 recognized as valid by traditional Logic, or the 12 affirmative and the 24 negative of Hamilton's Logic, which distinguishes four classes of affirmative and four of negative propositions. It is indifferent whether the methods of conversion be three or two or one. It is indifferent whether logical laws or principles be enumerated as two, three, five or ten. Since we do not accept the point of departure, it is impossible for us, far from admitting the development, even to discuss it; save to demonstrate that from capricious choice comes capricious choice, as we have made sufficiently clear in our treatment of formalist Logic. Mathematical Logic is a new manifestation of this formalist Logic, involving a great change in traditional formulæ, but none in the intimate substance of that pretended science of thought.

[Sidenote: _Practical aspect of Logistic._]

As the _science of thought,_ Logistic is a laughable thing; worthy, for that matter, of the brains that conceive and advocate it, which are the same that are promulgating a new Philosophy of language, indeed a new Æsthetic, with their insipid theories of the _universal Language._ As a formula of _practical utility_ it is not incumbent upon us to examine it here; all the more since we have already had occasion to give our opinion upon this subject. In the time of Leibnitz, fifty years later in the last days of Wolffianism; a century ago in Hamilton's time; forty years ago in the time of Jevons and of others; and finally now, when Peano, Boole, and Couturat are flourishing, these new arrangements are offered on the market. But every one has always found them too costly and complicated, so that they have not hitherto been generally used. Will they be so in the future? The practical work of persuasion, proper to the commercial traveller seeking purchasers of a new product, and the foresight of the merchant or manufacturer as to the fortune that may await that product, are not pertinent to Philosophy; which, being disinterested, could here, at the most, reply with words of benevolent patience: "If they be roses, they will bloom."