But if with the Hegelian criticism formalist Logic was surpassed by a truly philosophical Logic, and thereupon lost all its importance, it cannot be said that it was definitely dissolved. In Hegel himself there remain traces of it in certain divisions of the forms of judgment and of syllogism, which he either accepts and corrects or creates anew. Definitive criticism demanded that in any case the error peculiar to this empiricism should be recognized. This error consists in confusing language and thought, taking thought as language, and therefore also language as thought. Hegel could not effect this criticism, for he was logistic as regards the theory of language, conceiving it to be a complex of logical and universal elements.[14] Hence the coincidence between the forms of language and those of thought did not seem to him irrational, provided that both were taken in their true connection. The revival of the Philosophy of language, begun by Vico and carried on by Hamann and by Herder, and then again by Humboldt, remained unknown to him or had no influence upon him. Nor, to tell the truth, has it influenced even later Logic, for had it acquired this knowledge, it would have been freed for ever from formalism or verbalism and have possessed a method and a power of application to the nature of the problems that belong to it. Just a trace of serious discussion (but made rather in the interest of the Philosophy of language than in that of Logic) appears in the polemic between Steinthal and Becker concerning the relations between Logic and Grammar.[15]
Formalist Logic in Herbart, in Schopenhauer, in Hamilton.
For this reason, formalist Logic has continued to exist (with difficulty if you will, but yet to exist) in the nineteenth century. From Kant it had received with the name formal a new baptism and a new legitimization. Among post-Kantians Herbart clung closely to it, though he somewhat simplified it, and hostile as he was to all transcendental Logic, he continued to conceive it as the sole instrument of thought. Schopenhauer held logical forms to be a good parallel to rhetorical forms, and limited himself to proposing some slight remodelling of the former: for example, to consider judgments as always universal (both those called by that name and particular and singular judgments as well), and to explain hypothetical and disjunctive judgments as pronounced upon the comparison of two or more categorical judgments. From the syllogism, which he defined as "a judgment drawn from two other judgments, without the intervention of new conditions," he dropped the fourth figure, but he proclaimed the first three to be "ectypes" of three real and essentially different operations of thought.[16] Kant's teaching was followed in England by Hamilton. Hamilton insisted upon the purely hypothetical character of logical reasonings; he excluded from Logic discussions of possibility and impossibility and of the modalities, and declared that the intrusion into that science of the concepts of perfect or imperfect induction, which refer to material differences and are therefore extralogical,[17] was a fundamental error. In this way he reacted against inductive Logic, which, in his country especially, had prevailed against formalist Logic or had strangely accompanied it. He persuaded himself that he could perfect the latter, by simplifying the doctrine of the judgment, by means of what is called the quantification of the predicate.[18]
More recent theories.
Later logicians continued to employ these partial and superficial modifications. Trendelenburg, as has been mentioned, believed that he could make progress by referring the thing to its beginning, that is, by turning from Aristotelianism to Aristotle, and owing to the curious influence of a thought of Hegel, he assigned to logic and reality a common foundation which, for him, was not the Idea, but Movement. Lotze reduced the forms of judgments to three only, according to the variations of the copula: categorical, hypothetical and disjunctive judgments; and he made impersonal judgments precede categorical. By this last class he vainly sought to satisfy the desire for a theoretic form which is presupposed in properly logical thought, and it is yet to seek. Lotze always had at bottom an intellectualistic concept of language: poetry and art seemed to him to be directed, not to contemplation and expression, but to emotion and to feelings of pleasure and pain. He could not therefore recognize the primitive theoretic form in art, in intuition, in pure expressiveness. Drobisch, the Herbartian, revealed formalism in all its crudity, beginning with the affirmation that "there are certainly necessary judgments and syllogisms, but no necessary concepts." Sigwart reformed the classification of judgments (of denomination, of property and activity, impersonal, of relation, abstract, narrative and explicative), and retouched that of syllogisms. Wundt, accepting the old tripartition of logical forms, also attempts new sub-divisions, distinguishing judgments for example, according to their subject, into indeterminate, singular and plural; according to their predicate, into narrative, descriptive and explicative; according to their relation, into judgments of identity, superordination, subordination, co-ordination and dependence; and into negative predications and negative oppositions. Brentano's reform does not in general abandon the formalist circle; hence, having assigned the quantity of judgments to their matter, he limits himself to dividing them into affirmative and negative; among immediate inferences he accepts only the inference ad contradictoriam; among the laws of the syllogism he denies the law ex mere negativis, maintaining indeed that ex mere affirmativis nil sequitur; he defends, as the law of all syllogisms, that of quaternio terminorum, which used to pass for the sign of the sophism; and he further abolishes the vain distinctions of figures and moods.
Mathematical Logic.
Opposed as radical innovators to these logicians, who work more or less with traditional formulas, are the mathematical logicians, who follow, not philosophy, but certain fictions of the Leibnitzian philosophy. George Bentham, De Morgan, Boole, Jevons, Grassman and now several in England, in France, in Germany and in Italy (Peano), have been and are representative of this tendency. They are innovators only in a manner of speaking, for they are ultra-reactionaries, far more formalist than the formalist Aristotle. They are dissatisfied with the divisions made by him, not because they are toe numerous and arbitrary, but because they are toe few and still bear some traces of rationality They strive to the uttermost to provide a theory of thought, from which all thought is absent This kind of Logic has been well defined by Windelband as "Logic of the green cloth."[19]
Inexact idea of language among mathematical logicians and intuitionists.
These logicians have naturally inherited the other fiction of Leibnitz, namely that of the possibility of a constant and universal language,[20] thus revealing another reason for their aberration, and the usual support of the whole formalist error—ignorance of the alogical nature of language. The nature of language remains obscure from another point of view, even to the modern intuitionists (Bergson). They continue to regard as language, not language in its simplicity, but the intellectualist procedure (classificatory and abstractive) which falsifies the continuous in the discontinuous, breaks up duration, and builds a fictitious world upon the real world. They are therefore ultimately led to attribute the value of a pure expression of reality to music, as though music were not language, and true language (not the intellectualist discourse which they accept in place of it) were not essentially music, that is to say, poetry. For the intellectualists also, a Logic (were they to resolve upon constructing one) would be nothing but formalist.