Derivation of the Principle of Sufficient Reason from the Law of Non-Contradiction

Many common assertions of fact are possibilities that are not necessarily true. We may be practically justified in making assumptions about facts (as possibilities), but we are not justified in asserting that a particular possibility is true (a fact) without a sufficient reason; this would imply not only that the possibility is true, but that we know that the possibility is true. I will show that the latter, second-order claim violates the law of non-contradiction, and is therefore necessarily false (as an assertion of knowledge about facts).

The principle of sufficient reason can be expressed as follows: for every fact F, there must be a sufficient reason that F is true. It follows that knowledge that F is true obtains only in virtue of knowledge of a sufficient reason that F is true. A sufficient reason for knowledge that F is true is such that it precludes the possibility that F is false.

Let P signify the knowledge that F is true. The law of non-contradiction: P cannot be true and false at the same time and in the same respect, or ¬(P ∧ ¬P). To assert that ‘P is true’ without a sufficient reason implies that ‘P is true without a sufficient reason’, therefore any claim can be true without sufficient reason, therefore the negation of P can also be true without sufficient reason, therefore contradiction.

Similarly, to assert that ‘P is true’ without a proof implies that ‘P is true without a proof’, therefore anything can be true without a proof, therefore the negation of P can also be true without a proof, therefore contradiction.

Another way, unsoundness of truth-claims implies that unknowns are known, or that something untrue is true. In short, any assertion of fact that violates the principle of sufficient reason is self-negating.

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The Principle of Sufficient Reason REVISED (16.3.2025)

The sense of ‘sufficiency’ in the principle of sufficient reason (PSR) is not arbitrary: a sufficient reason warrants that something is a fact. Sufficiency qualifies reasons as corresponding to facts; it is not the case that facts are qualified as corresponding to reasons. For example, if it is raining then the ground is wet: rain is a sufficient reason for the ground being wet, but ‘the ground is wet’ does not imply that ‘it is raining’. There may be facts for which there are no reasons (why is there something rather than nothing?), but it cannot be the case that there is a sufficient reason and no corresponding fact. The interpretation that was historically assumed (that Fact implies Sufficient Reason) is therefore back to front. It should be understood that a reason is sufficient only if it implies a fact: R(x)→F(x). ‘It is raining’ is a sufficient reason for the ground being wet only if raining implies that the ground is wet.

It follows that a sufficient reason for ‘knowing that P’ implies ‘knowing that P’, but ‘knowing that P’ also implies a sufficient reason for ‘knowing that P’, where P includes facts for which there are no reasons. The sufficiency of reason for ‘knowing that P’ overrides the insufficiency of reason for P just with respect to knowing. We can thus assert the knowledge that ‘there is something rather than nothing’ or ‘consciousness is self-evident’ because consciousness is self-evident and it implies that there is something rather than nothing.

Due to the bi-implication between knowledge and the sufficiency of reasons for knowledge, for every fact of knowing that P, there must be a sufficient reason for knowing that P. To assert the knowledge that P without a sufficient reason for the knowledge, implies that P is known without a sufficient reason for knowing, therefore any claim can be known without a sufficient reason for knowing, therefore the negation of P can also be known without a sufficient reason, therefore contradiction. The original formulation of PSR applies to knowledge (but not to all facts that can be known) because knowledge implies certainty which implies the sufficiency of reasons whereas facts do not.