It is a peculiarity, which distinguishes the proofs of transcendental synthetical propositions from those of all other a priori synthetical cognitions, that reason, in the case of the former, does not apply its conceptions directly to an object, but is first obliged to prove, a priori, the objective validity of these conceptions and the possibility of their syntheses. This is not merely a prudential rule, it is essential to the very possibility of the proof of a transcendental proposition. If I am required to pass, a priori, beyond the conception of an object, I find that it is utterly impossible without the guidance of something which is not contained in the conception. In mathematics, it is a priori intuition that guides my synthesis; and, in this case, all our conclusions may be drawn immediately from pure intuition. In transcendental cognition, so long as we are dealing only with conceptions of the understanding, we are guided by possible experience. That is to say, a proof in the sphere of transcendental cognition does not show that the given conception (that of an event, for example) leads directly to another conception (that of a cause)- for this would be a saltus which nothing can justify; but it shows that experience itself, and consequently the object of experience, is impossible without the connection indicated by these conceptions. It follows that such a proof must demonstrate the possibility of arriving, synthetically and a priori, at a certain knowledge of things, which was not contained in our conceptions of these things. Unless we pay particular attention to this requirement, our proofs, instead of pursuing the straight path indicated by reason, follow the tortuous road of mere subjective association. The illusory conviction, which rests upon subjective causes of association, and which is considered as resulting from the perception of a real and objective natural affinity, is always open to doubt and suspicion. For this reason, all the attempts which have been made to prove the principle of sufficient reason, have, according to the universal admission of philosophers, been quite unsuccessful; and, before the appearance of transcendental criticism, it was considered better, as this principle could not be abandoned, to appeal boldly to the common sense of mankind (a proceeding which always proves that the problem, which reason ought to solve, is one in which philosophers find great difficulties), rather than attempt to discover new dogmatical proofs.
But, if the proposition to be proved is a proposition of pure reason, and if I aim at passing beyond my empirical conceptions by the aid of mere ideas, it is necessary that the proof should first show that such a step in synthesis is possible (which it is not), before it proceeds to prove the truth of the proposition itself. The so-called proof of the simple nature of the soul from the unity of apperception, is a very plausible one. But it contains no answer to the objection, that, as the notion of absolute simplicity is not a conception which is directly applicable to a perception, but is an idea which must be inferred- if at all- from observation, it is by no means evident how the mere fact of consciousness, which is contained in all thought, although in so far a simple representation, can conduct me to the consciousness and cognition of a thing which is purely a thinking substance. When I represent to my mind the power of my body as in motion, my body in this thought is so far absolute unity, and my representation of it is a simple one; and hence I can indicate this representation by the motion of a point, because I have made abstraction of the size or volume of the body. But I cannot hence infer that, given merely the moving power of a body, the body may be cogitated as simple substance, merely because the representation in my mind takes no account of its content in space, and is consequently simple. The simple, in abstraction, is very different from the objectively simple; and hence the Ego, which is simple in the first sense, may, in the second sense, as indicating the soul itself, be a very complex conception, with a very various content. Thus it is evident that in all such arguments there lurks a paralogism. We guess (for without some such surmise our suspicion would not be excited in reference to a proof of this character) at the presence of the paralogism, by keeping ever before us a criterion of the possibility of those synthetical propositions which aim at proving more than experience can teach us. This criterion is obtained from the observation that such proofs do not lead us directly from the subject of the proposition to be proved to the required predicate, but find it necessary to presuppose the possibility of extending our cognition a priori by means of ideas. We must, accordingly, always use the greatest caution; we require, before attempting any proof, to consider how it is possible to extend the sphere of cognition by the operations of pure reason, and from what source we are to derive knowledge, which is not obtained from the analysis of conceptions, nor relates, by anticipation, to possible experience. We shall thus spare ourselves much severe and fruitless labour, by not expecting from reason what is beyond its power, or rather by subjecting it to discipline, and teaching it to moderate its vehement desires for the extension of the sphere of cognition.
The first rule for our guidance is, therefore, not to attempt a transcendental proof, before we have considered from what source we are to derive the principles upon which the proof is to be based, and what right we have to expect that our conclusions from these principles will be veracious. If they are principles of the understanding, it is vain to expect that we should attain by their means to ideas of pure reason; for these principles are valid only in regard to objects of possible experience. If they are principles of pure reason, our labour is alike in vain. For the principles of reason, if employed as objective, are without exception dialectical and possess no validity or truth, except as regulative principles of the systematic employment of reason in experience. But when such delusive proof are presented to us, it is our duty to meet them with the non liquet of a matured judgement; and, although we are unable to expose the particular sophism upon which the proof is based, we have a right to demand a deduction of the principles employed in it; and, if these principles have their origin in pure reason alone, such a deduction is absolutely impossible. And thus it is unnecessary that we should trouble ourselves with the exposure and confutation of every sophistical illusion; we may, at once, bring all dialectic, which is inexhaustible in the production of fallacies, before the bar of critical reason, which tests the principles upon which all dialectical procedure is based. The second peculiarity of transcendental proof is that a transcendental proposition cannot rest upon more than a single proof. If I am drawing conclusions, not from conceptions, but from intuition corresponding to a conception, be it pure intuition, as in mathematics, or empirical, as in natural science, the intuition which forms the basis of my inferences presents me with materials for many synthetical propositions, which I can connect in various modes, while, as it is allowable to proceed from different points in the intention, I can arrive by different paths at the same proposition.
But every transcendental proposition sets out from a conception, and posits the synthetical condition of the possibility of an object according to this conception. There must, therefore, be but one ground of proof, because it is the conception alone which determines the object; and thus the proof cannot contain anything more than the determination of the object according to the conception. In our Transcendental Analytic, for example, we inferred the principle: Every event has a cause, from the only condition of the objective possibility of our conception of an event. This is that an event cannot be determined in time, and consequently cannot form a part of experience, unless it stands under this dynamical law. This is the only possible ground of proof; for our conception of an event possesses objective validity, that is, is a true conception, only because the law of causality determines an object to which it can refer. Other arguments in support of this principle have been attempted- such as that from the contingent nature of a phenomenon; but when this argument is considered, we can discover no criterion of contingency, except the fact of an event- of something happening, that is to say, the existence which is preceded by the non-existence of an object, and thus we fall back on the very thing to be proved. If the proposition: "Every thinking being is simple," is to be proved, we keep to the conception of the ego, which is simple, and to which all thought has a relation. The same is the case with the transcendental proof of the existence of a Deity, which is based solely upon the harmony and reciprocal fitness of the conceptions of an ens realissimum and a necessary being, and cannot be attempted in any other manner.
This caution serves to simplify very much the criticism of all propositions of reason. When reason employs conceptions alone, only one proof of its thesis is possible, if any. When, therefore, the dogmatist advances with ten arguments in favour of a proposition, we may be sure that not one of them is conclusive. For if he possessed one which proved the proposition he brings forward to demonstration- as must always be the case with the propositions of pure reason- what need is there for any more? His intention can only be similar to that of the advocate who had different arguments for different judges; this availing himself of the weakness of those who examine his arguments, who, without going into any profound investigation, adopt the view of the case which seems most probable at first sight and decide according to it.
The third rule for the guidance of pure reason in the conduct of a proof is that all transcendental proofs must never be apagogic or indirect, but always ostensive or direct. The direct or ostensive proof not only establishes the truth of the proposition to be proved, but exposes the grounds of its truth; the apagogic, on the other hand, may assure us of the truth of the proposition, but it cannot enable us to comprehend the grounds of its possibility. The latter is, accordingly, rather an auxiliary to an argument, than a strictly philosophical and rational mode of procedure. In one respect, however, they have an advantage over direct proofs, from the fact that the mode of arguing by contradiction, which they employ, renders our understanding of the question more clear, and approximates the proof to the certainty of an intuitional demonstration.
The true reason why indirect proofs are employed in different sciences is this. When the grounds upon which we seek to base a cognition are too various or too profound, we try whether or not we may not discover the truth of our cognition from its consequences. The modus ponens of reasoning from the truth of its inferences to the truth of a proposition would be admissible if all the inferences that can be drawn from it are known to be true; for in this case there can be only one possible ground for these inferences, and that is the true one. But this is a quite impracticable procedure, as it surpasses all our powers to discover all the possible inferences that can be drawn from a proposition. But this mode of reasoning is employed, under favour, when we wish to prove the truth of an hypothesis; in which case we admit the truth of the conclusion- which is supported by analogy- that, if all the inferences we have drawn and examined agree with the proposition assumed, all other possible inferences will also agree with it. But, in this way, an hypothesis can never be established as a demonstrated truth. The modus tollens of reasoning from known inferences to the unknown proposition, is not only a rigorous, but a very easy mode of proof. For, if it can be shown that but one inference from a proposition is false, then the proposition must itself be false. Instead, then, of examining, in an ostensive argument, the whole series of the grounds on which the truth of a proposition rests, we need only take the opposite of this proposition, and if one inference from it be false, then must the opposite be itself false; and, consequently, the proposition which we wished to prove must be true.
The apagogic method of proof is admissible only in those sciences where it is impossible to mistake a subjective representation for an objective cognition. Where this is possible, it is plain that the opposite of a given proposition may contradict merely the subjective conditions of thought, and not the objective cognition; or it may happen that both propositions contradict each other only under a subjective condition, which is incorrectly considered to be objective, and, as the condition is itself false, both propositions may be false, and it will, consequently, be impossible to conclude the truth of the one from the falseness of the other.
In mathematics such subreptions are impossible; and it is in this science, accordingly, that the indirect mode of proof has its true place. In the science of nature, where all assertion is based upon empirical intuition, such subreptions may be guarded against by the repeated comparison of observations; but this mode of proof is of little value in this sphere of knowledge. But the transcendental efforts of pure reason are all made in the sphere of the subjective, which is the real medium of all dialectical illusion; and thus reason endeavours, in its premisses, to impose upon us subjective representations for objective cognitions. In the transcendental sphere of pure reason, then, and in the case of synthetical propositions, it is inadmissible to support a statement by disproving the counter-statement. For only two cases are possible; either, the counter-statement is nothing but the enouncement of the inconsistency of the opposite opinion with the subjective conditions of reason, which does not affect the real case (for example, we cannot comprehend the unconditioned necessity of the existence of a being, and hence every speculative proof of the existence of such a being must be opposed on subjective grounds, while the possibility of this being in itself cannot with justice be denied); or, both propositions, being dialectical in their nature, are based upon an impossible conception. In this latter case the rule applies: non entis nulla sunt predicata; that is to say, what we affirm and what we deny, respecting such an object, are equally untrue, and the apagogic mode of arriving at the truth is in this case impossible. If, for example, we presuppose that the world of sense is given in itself in its totality, it is false, either that it is infinite, or that it is finite and limited in space. Both are false, because the hypothesis is false. For the notion of phenomena (as mere representations) which are given in themselves (as objects) is self-contradictory; and the infinitude of this imaginary whole would, indeed, be unconditioned, but would be inconsistent (as everything in the phenomenal world is conditioned) with the unconditioned determination and finitude of quantities which is presupposed in our conception.
The apagogic mode of proof is the true source of those illusions which have always had so strong an attraction for the admirers of dogmatical philosophy. It may be compared to a champion who maintains the honour and claims of the party he has adopted by offering battle to all who doubt the validity of these claims and the purity of that honour; while nothing can be proved in this way, except the respective strength of the combatants, and the advantage, in this respect, is always on the side of the attacking party. Spectators, observing that each party is alternately conqueror and conquered, are led to regard the subject of dispute as beyond the power of man to decide upon. But such an opinion cannot be justified; and it is sufficient to apply to these reasoners the remark:
Non defensoribus istis
Each must try to establish his assertions by a transcendental deduction of the grounds of proof employed in his argument, and thus enable us to see in what way the claims of reason may be supported. If an opponent bases his assertions upon subjective grounds, he may be refuted with ease; not, however to the advantage of the dogmatist, who likewise depends upon subjective sources of cognition and is in like manner driven into a corner by his opponent. But, if parties employ the direct method of procedure, they will soon discover the difficulty, nay, the impossibility of proving their assertions, and will be forced to appeal to prescription and precedence; or they will, by the help of criticism, discover with ease the dogmatical illusions by which they had been mocked, and compel reason to renounce its exaggerated pretensions to speculative insight and to confine itself within the limits of its proper sphere- that of practical principles.