Consequentia mirabilis

Consequentia mirabilis (Latin for "admirable consequence"), also known as Clavius's Law, is used in traditional and classical logic to establish the truth of a proposition from the inconsistency of its negation. It is thus related to reductio ad absurdum, but it can prove a proposition using just its own negation and the concept of consistency. For a more concrete formulation, it states that if a proposition is a consequence of its negation, then it is true, for consistency. In formal notation:
 * $$(\neg A \to A) \to A $$.

Weaker variants of the principle are provable in minimal logic, but the full principle itself is not provable even in intuitionistic logic.

History
Consequentia mirabilis was a pattern of argument popular in 17th-century Europe that first appeared in a fragment of Aristotle's Protrepticus: "If we ought to philosophise, then we ought to philosophise; and if we ought not to philosophise, then we ought to philosophise (i.e. in order to justify this view); in any case, therefore, we ought to philosophise."

Barnes claims in passing that the term consequentia mirabilis refers only to the inference of the proposition from the inconsistency of its negation, and that the term Lex Clavia (or Clavius' Law) refers to the inference of the proposition's negation from the inconsistency of the proposition.

Minimal logic
The following shows what weak forms of the law still holds in minimal logic, which lacks both excluded middle and the principle of explosion.

Weaker variants
Frege's theorem states
 * $$\big(B \rightarrow (C\to D)\big) \to \Big((B \rightarrow C) \to (B\to D)\Big) $$

For $$D=\bot$$ this is a form of negation introduction, and then for $$B=C$$ and using the law of identity, it reduces to
 * $$\big(B \rightarrow \neg B\big) \rightarrow \neg B$$

Now for $$B=\neg A$$, it follows that $$(\neg A \to \neg \neg A) \to \neg \neg A$$. By implication introduction, this is indeed an equivalence,
 * $$(\neg A \to \neg \neg A) \leftrightarrow \neg \neg A$$

In minimal logic, the first double-negation in the original implication can optionally also be removed, weakening the forward direction statement to $$(\neg A \to A) \to \neg \neg A$$. Here, consequentia mirabilis thus holds whenever $$\neg \neg A \leftrightarrow A$$. As $$E \to \neg \neg A$$ is always also still equivalent to $$\neg \neg (E \to A)$$ in minimal logic, the above also constructively establishes the double negation of consequentia mirabilis. Of course, when adopting the double-negation elimination principle for all propositions, consequentia mirabilis also follows simply because the latter brings minimal logic back to full classical logic.

This shows that already minimal logic validates that a proposition $$A$$ cannot be rejected exactly if this is implied by its negation, and that $$A$$ holds exactly when it is implied by both $$\neg A$$ and $$\neg \neg A$$.

The weak form $$(\neg A\to A) \to \neg (\neg A)$$ can also be seen to be equivalent to the principle of non-contradiction $$\neg\big(A\land\neg A\big)$$. To this end, first note that using modus ponens and implication introduction, the principle is equivalent to $$\neg\big((\neg A\to A)\land(\neg A)\big)$$. The claim now follows from $$(B\to\neg C)\leftrightarrow\neg(B\land C)$$, i.e. the fact that there are equivalent characterizations of two propositions being mutually exclusive.

Equivalence to excluded middle
The negation of any excluded middle disjunction implies the disjunction itself. From the above weak form, it thus follows that the double-negated excluded middle statement is valid, in minimal logic. Likewise, this argument shows how the full consequentia mirabilis implies excluded middle.

The following argument shows that the converse also holds. A principle related to case analysis may be formulated as such: If both $$B$$ and $$C$$ each imply $$A$$, and either of them must hold, then $$A$$ follows. Formally,
 * $$\big((B\to A)\land (C\to A)\land (B\lor C)\big)\to A$$

For $$B=A$$ and $$C=\neg A$$, the principle of identity now entails
 * $$\big((\neg A\to A)\land (A\lor\neg A)\big)\to A$$

Whenever the second conjunct of the antecedent, $$A\lor\neg A$$, is true, the so is the instance of the principle.

Intuitionistic logic
One has that $$(A\to B)\to A$$ implies $$(A\to B)\to(A\land B)$$. By conjunction elimination, this is in fact an equivalence. In particular, one has
 * $$\big(\neg A\to A\big) \leftrightarrow \big(\neg A\to(A \land \bot)\big)$$

The right hand of this also implies $$\neg\neg A$$, which gives another demonstration of how double-negation elimination implies consequentia mirabilis, in minimal logic.

In intuitionistical logic, the principle of explosion itself may be formulated as $$\bot\to (A\land \bot)$$, and therefore $$\bot\leftrightarrow (A\land \bot)$$. So here,
 * $$\big(\neg A\to A\big) \leftrightarrow \neg \neg A$$

Classical logic
It was established how consequentia mirabilis follows from double-negation elimination in minimal logic, and how it is equivalent to excluded middle. Indeed, it may also be established by using the classically valid propositional form of the reverse disjunctive syllogism $$(B\to A)\to (\neg B\lor A)$$ chained together with the double-negation elimination principle in the form $$(\neg \neg A \lor A) \to A $$.

Related to the last intuitionistic derivation given above, consequentia mirabilis also follow as the special case of Pierce's law
 * $$\big((A\to B)\to A\big) \to A$$

for $$B=\bot$$. That article can be consulted for more, related equivalences.