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# Principle of Explosion

The principle of explosion is a logical rule according to which ‘from contradiction anything follows’ (ex contradictione sequitur quodlibet), including its negation. I discuss the implications of the principle vis-a-vis non-classical logics.

The principle is commonly proven in the following way:

1. Assume that ‘P is true’.

2. Assume that ‘P is not true’ is also true.

(1 and 2 imply the rejection of the law of non-contradiction.

3. Introduce C = ‘P is true OR unicorns exist’. ‘P is true’ (from 1) implies that C is true irrespective of whether unicorns exist.

4. C and ‘P is not true’ (from 2) imply that unicorns exist.

Formally: P ∧ ¬P: P → P ∨ Q, (P ∨ Q) ∧ ¬P → Q

The proof attempts to comply with the law of non-contradiction at every step except the truth-value of P, but since Q can be any proposition whatsoever, even a contradiction, any contradiction is also implicitly proven. The proof also contradicts the logical theorems (=, ∧, ∨, →) necessary to construct the proof, therefore negates itself. The meaning of ‘from contradiction anything follows’ must be taken absolutely: ‘from contradiction everything follows, including the negation of everything’, therefore no distinction of identity is possible, therefore no meaning. Another way, from contradiction, nonsense follows.

To formalise this result explicitly one could modify C: ‘P is true OR (everything is true AND everything is not true)’.

It is impossible to circumvent any of the fundamental laws (identity, non-contradiction, excluded middle) and retain the capacity for meaning. This can be demonstrated in several ways, but the most intuitive demonstration may prioritise the law of identity: everything is identical to itself and only to itself. Without adhering to the law of identity, nothing could be said to be itself or a definite something, therefore no identification of objects, meanings, relations or terms would be possible. Furthermore, the law of identity entails that nothing can be both identical and not identical to itself (the law of non-contradiction) and nothing can be partly identical and partly non-identical to itself (the law of excluded middle). To reject non-contradiction or excluded middle amounts to implicitly rejecting identity/=, therefore to reject any of the laws amounts to rejecting them all.