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.