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.^[1] 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

.

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."^[2]

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.^[3]

Derivations

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

Minimal logic

Equivalence to excluded middle

One principle related to case analysis may be formulated as follows: 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 $C=\neg A$ , this entails

{\big (}(\neg A\to A)\land (A\lor \neg A){\big )}\to A

This also means that excluded middle implies consequentia mirabilis.

The converse also holds: Since the double-negation of any excluded middle statement is valid, assuming the negation of excluded middle implies any negated statement also in minimal logic, and by weakening thus also excluded middle itself. Using consequentia mirabilis with excluded middle thus gives the result.

Weaker variants

The following shows what weak forms of the law still holds in minimal logic, which lacks both excluded middle and the principle of explosion. Negation introduction may be formulated as

{\big (}(P\rightarrow Q)\land (P\rightarrow \neg Q){\big )}\rightarrow \neg P

and with the principle of identity follows $(P\rightarrow \neg P)\rightarrow \neg P$ and so in particular

(\neg A\to \neg \neg A)\to \neg \neg A

Here, the first double-negation can optionally also be removed, weakening the statement. Note that since $D\to \neg \neg A$ is here always also still equivalent to $\neg \neg (D\to A)$ , the above also constructively establishes the double negation of consequentia mirabilis.

Consequentia mirabilis thus holds whenever $\neg \neg A\leftrightarrow A$ . When adopting the double-negation elimination principle for all propositions, it follows also simply because the latter brings minimal logic back to full classical logic.

To conclude, here is another argument establishing the weakest variant: Firstly, the principle of non-contradiction $\neg {\big (}A\land \neg A{\big )}$ is equivalent to $\neg {\big (}(\neg A\to A)\land (\neg A){\big )}$ by implication introduction and modus ponens. And now using the equivalent characterization for two mutually exclusive propositions, the latter formula is equivalent to

(\neg A\to A)\to \neg (\neg A)

This result can alternatively also be seen as the special case of

{\big (}(A\to B)\to A{\big )}\to {\big (}(A\to B)\to B{\big )}

for $B=\bot$ . This proposition follows from the propositional form of modus ponens $A\to {\big (}(A\to B)\to B{\big )}$ together with the fact that always $A\to (C\to B)\vdash (C\to A)\to (C\to B)$ .

Intuitionistic logic

The law intuitionistically also implies double-negation elimination: Assuming the $(A\to B)\to A$ , it follows that $(A\to B)\to (A\land B)$ . By conjunction elimination $(A\land B)\to A$ , these are seen to be equivalent. Now as $A\land \bot$ is equivalent to $\bot$ by explosion, the antecedant of the law is intuitionistically equivalent to $\neg \neg A$ .

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.

References

^ Sainsbury, Richard. Paradoxes. Cambridge University Press, 2009, p. 128.
^ Kneale, William (1957). "Aristotle and the Consequentia Mirabilis". The Journal of Hellenic Studies. 77 (1): 62–66. doi:10.2307/628635. JSTOR 628635. S2CID 163283107.
^ Barnes, Jonathan. The Pre-Socratic Philosophers: The Arguments of the Philosophers. Routledge, 1982, p. 217 (p 277 in 1979 edition).

[1] Sainsbury, Richard. Paradoxes. Cambridge University Press, 2009, p. 128.

[Kneale1957-2] Kneale, William (1957). "Aristotle and the Consequentia Mirabilis". The Journal of Hellenic Studies. 77 (1): 62–66. doi:10.2307/628635. JSTOR 628635. S2CID 163283107.

[3] Barnes, Jonathan. The Pre-Socratic Philosophers: The Arguments of the Philosophers. Routledge, 1982, p. 217 (p 277 in 1979 edition).

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