Seeking references about the Principle Of Implosion (verum ex quodlibet), $B\vDash A\vee\neg A $

Question

A well-known valid principle in classical logic is the Principle of Explosion, also called "ex falso quodlibet". This principle states that everything follows from a contradiction:

$$A \wedge \neg A \vDash B$$

Everything about this principle intrigues me, but yesterday when I was reading some articles on this subject, I came across something I had never heard of before: the principle of implosion (verum ex quodlibet). If I properly understood it states that a tautology follows from anything:

$$B \vDash A \vee \neg A $$

Because I am very curious about learning more about this so called principle of implosion, I am looking for more information, articles or books on this subject.

Answer 1

SeeMedieval Theories of Consequence and John Buridan(ca.1300-ca.1360)'s Tractatus de consequentiis for the definition of consequence in terms of truth-preservation, and the formulation of some general principles following from the definition, such as that :

from every impossible proposition any other follows and every necessary proposition follows from any other (First conclusion).

See also another edition: Jean Buridan’s Logic: The Treatise on Supposition The Treatise on Consequences (Reidel, 1985): First theorem.

Answer 2

(I can't give you any references, but the following may help clarify your thinking on this topic.)

The Principle of EXPLOSION

$~~~~ A\land \neg A \implies B$

The Truth Table:

Proof based on a form of natural deduction:

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The Principle of IMPLOSION

$~~~~B\implies (A\implies B)$

The Truth Table:

Proof based on a form of natural deduction:

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(Related topic)

The Principle of VACUOUS TRUTH

$~~~~ A\implies [\neg A \implies B]$

The Truth Table:

Proof based on a form of natural deduction: