|Aristotle Contemplating Homer, by Rembrandt|
Aristotle advertises the Principle of Non-Contradiction, in the Metaphysics (Bk. IV, ch.3-6), as the first, most certain, and unhypothetical principle upon which all demonstration depends. He writes: “[T]he most certain principle of all is that regarding which it is impossible to be mistaken; for such a principle must be both the best known… and non-hypothetical.” In contrast to hypotheses, unhypothetical principles are not conjectured as merely possible, but are rather meant to be known immediately and necessarily through some direct noetic intuition: Aristotle’s allusion to unhypothetical principles at Metaphysics 1005b14 plausibly alludes to Plato’s unhypothetical principles of knowledge prescinding from the supreme principle of the Good at Republic 510b7-511b7. Where hypotheses are only possibly true and must be discursively demonstrated from some higher and prior conditions, unhypothetical principles are - by definition - fundamental, indemonstrable, and discursively incontrovertible. The incontrovertibility of these unhypothetical principles might suggest that Aristotle, no less than Plato, has quite dogmatically demanded that we accept an indefensible assertion. Aristotle’s indirect argument for the Principle of Non-Contradiction, however, crucially relies upon classical Platonic principles of dialectic, noesis, and emanation.
Aristotle vehemently repudiates his ‘uneducated’ critics, who ask for the principle of non-contradiction to be demonstrated, for failing to realize that not every principle can be proven, for the simple reason that any attempt to demonstrate everything would produce an infinite regress of demonstrations. (For a critical discussion of Aristotle’s abhorrence of infinity, see my Theological History of Infinity) Although he declines to elaborate what this education is meant to consist in, judging by the introduction of unhypothetical principles in the Republic, we may guess that Aristotle intends it to resemble something of what Plato had described in the Republic, consisting in a dialectical ascent, from less to more certain hypotheses through, up the ladder of discursive reasoning until it can recognize the first unhypothetical principle of non-contradiction. Although Aristotle admits that unhypothetical principles cannot be demonstrated by any direct deduction from premises to conclusion, he nonetheless contends that these first principles of philosophy can only be corroborated by an indirect argument, in which Platonic dialectic is formalized for the purpose of refuting any contrary denial of the principle of non-contradiction. However, under closer scrutiny it appears that without the hypothetical and discursive construction of the cosmos through Platonic dialectics, Aristotle’s indirect argument quickly collapses into ampliative circularity.
Aristotle first argues that the principle of non-contradiction is held to be the most certain of all principles, that “answers to the definition” of a hypothetical principle, because “it is impossible for anyone” to affirm and deny the same thing. He suggests that the Principle of Non-Contradiction is meant to be indirectly proven from the impossibility of affirmative and negative opinions in Heraclitean flux: for if Heraclitean flux were extended to all opinions, and all opinions were simultaneously affirmed and denied, then any and all opinions would be both true and false; true by default; and no determination could ever be made between truth and falsity. To preserve the possibility of determining truth, Aristotle rejects this kind of ‘pan-inconsistency’ of all opinions. But this rejection of paninconsistency, in which all opinions are inconsistently true and false, does not immediately also imply the rejection of paraconsistency, in which the truth and falsity of some opinions does not produce an explosive inconsistency of all opinions, triviality, or trivialism. Indeed, Paulla Gottlieb has indicated that Aristotle even appears to endorse at least some paraconsistent syllogistic conclusion at Prior Analytics II 15 64a15 when he writes: “Consequently it is possible that opposites may lead to a conclusion, though not always or in every mood, but only if the terms subordinate to the middle are such that they are either identical or related as whole to part.”
Aristotle’s most famous indirect argument is from the impossibility of meaningfully denying the Principle of Non-Contradiction: for if a sceptic of the Principle of Non-Contradiction makes any significant statement, then the critic must have, in the very act of making this statement, already presupposed the principle of non-contradiction. There are two stages to this argument: analytic and hypothetical. First, if every meaningful statement possesses some determinate shape of signification, then, in the very act of making a meaningful statement, even the sceptic must analytically presuppose the possibility of some determination. Second, such a determination is only possible on the hypothesis that some determination is true and its contrary determination is false. But since this truth and falsity of contrary determinations is only possible on the further hypothesis that for any determination that is true, its contrary cannot be true but must be false, it seems that for every act of determinate signification we must presuppose the lawful prohibition of the coincidence of contrary determinations.
The first analytic stage of the argument is undoubtedly true, but the second hypothetical stage of the argument is problematic for three reasons: first, because it surreptitiously deploys discursive and hypothetical arguments to indirectly demonstrate what is advertised as a non-discursive and explicitly unhypothetical principle; second, because it neglects any explanation of the source, scope, and specificity of its determinations; and third because this entire hypothetical inference remains ampliative. An argument is ampliative if its premises and rules are insufficient for the necessary demonstration of its conclusions. Hypothetical inferences are always ampliative, and insufficiently demonstrative, for the simple reason that there could always be alternative hypotheses that remain to be considered. Plato deliberately deploys hypothetical arguments to produce contrary theses and motivate his dialectic, but Aristotle’s didactic demonstrations are meant to be necessary, and which have neglected to consider alternative formulations for the scope of non-contradiction: specifically whether non-contradiction implies the rejection all inconsistencies (i.e. paninconsistency) or merely some instances of inconcistency that do not entail triviality (i.e. paraconsistency).
The greatest difficulty for Aristotle is that his entire indirect argument seems to have circuitously presupposed the very possibility of a determinate disjunction that it has been developed demonstrate. For every indirect argument involves a disjunctive syllogism (i.e. A v B; ¬ B, TF: A); the function of disjunction (i.e. A v B) already involves at least two determinate disjuncta (i.e. A & B); and these disjuncta are each, in turn, meant to be determined as distinct disjuncts by the Principle of Excluded Middle (i.e. Ex(Fx v ¬Fx)). But if Aristotle maintains that the Principle of Excluded middle presupposes determination, and determination presupposes the Principle of Non-Contradiction, then he has circuitously presupposed this very principle for the purposes of demonstrating it. Where Plato’s supreme unhypothetical principle of the Good is meant to be virtuously circular, because it is the emanative beginning and assimilative end of all discursive reasoning, the circularity of Aristotle’s indirect demonstration for the fundamental unhypothetical Principle of Non-Contradiction surely compromises its purely formal status as a first principle of reason.
For the purpose of distancing himself from any dialectical ascent towards the Good, Aristotle transposes the question of contradiction from polysemous words to putatively unambiguous and punctiliar facts. Any finite number of meanings can be reduced to individual meanings, and individual meanings can be self-identically represented as an abstract variable according to a formal notation convention. Where Plato had formalized Socratic dialectic into a written dialogue, Aristotle formalized Plato’s dialogue into didactic instruction. Since, however, even the words used in didactic instruction can be ambiguous, and liable to contradiction, Aristotle adopts a formalized language with symbols that appear simply identical to themselves. This formal notation is meant to prevent contradictions between terms, but inadvertently engenders an even broader conflict between formality and physicality: for once all terms have been formulated into a consistent system, then this formal system remains distinct and indifferent to the physical world.
Aristotle points to the true purpose of his indirect argument when he contends that the ultimate implication of rejecting the Principle of Non-Contradiction must be to reject all determination, individuation, and definition, and thus to entirely “do away with substance and essence.” His purpose, in defending the principle of non-contradiction, is thus, not merely to preserve the possibility of any determinatively significant statements, but rather, and more metaphysically, to preserve the real definition of essences and the very determinate shape of all substances. For if significant statements did not presuppose determination, and determination did not presuppose non-contradiction, then there could be neither determinations, nor essences, nor substances, and all essences would be reduced to a swamp of accidental attributes, wherein nothing could ever endure and never be known.
Aristotle’s Principle of Non-Contradiction is thus meant to preserve the determinacy of signs, substances and essences. The indirect argument declines to specify whether the principle applies to all instances of contradiction, i.e. paninconsistency, or only some instances of contradiction, i.e. paraconsistency. Aristotle can only prohibit any possibility of contradiction by making the further assumption that a prohibition on the totality of all inconsistencies also implies a prohibition on any particular inconsistency. However, this assumption requires the principles that obtain in every part to be identified with those of the whole and vice versa, so that the whole cosmos may be lawfully regulated, or nomologically constrained, by one self-identical and supreme principle.
The self-identity of forms is, moreover, only possible by the imposition of some superior paradigm of identity, unicity, and individuality. The Principle of Non-Contradiction thus requires that we hypothesize a supreme principle of identity, unicity, and individuality to nomologically constrain the parts by the whole. However, Aristotle cannot admit this hypothetical presupposition without compromising its status as an unhypothetical first principle of reason. Once Aristotle has formalized dialectics into didactic instruction, and further formalized this didactic instruction into various formulae, he can no longer ascend the dialectical ladder from hypotheses, to refutation, to the supreme principles, and his indirect argument for the Principle of Non-Contradiction ineluctably collapses into ampliative circularity. Were the Principle of Non-Contradiction to have, more Platonically emanated from the first principle of the Good through the second principle of Intellect (Nous), then Aristotle might more plausibly preserve all determinations in the virtuous circle of Platonic emanation and assimilation.
See also my lecture Plato, Logic, and Ontology:
See also my critical commentary on Aristotle' criticisms of Plato's mathematical Ideas:
See also my critical commentary on Aristotle' criticisms of Plato's mathematical Ideas: