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.

Proponents of paraconsistent logic attempt to circumvent the law of non-contradiction, claiming that classical logic is not fit for effective reasoning under the realistic conditions of vagueness, ambiguity and uncertainty. This view appears to be motivated by misinterpretation of classical logic and is typically exemplified as equivocation between incompatibility (statements that are mutually exclusive but need not be held true at the same time and in the same respect) and contradiction (true and false at the same time and in the same respect). For example, that Newtonian physics and General Relativity yield different predictions for the same system does not entail a contradiction; difference, ambiguity, uncertainty, vagueness, incompatibility or opposition do not of themselves violate the law of non-contradiction. It is possible to consistently ‘work’ with incompatible propositions in ‘classical’ logic provided they are not both regarded as true at the same time and in the same respect; the law of non-contradiction is preserved by means of Disjunction (x OR ¬x), while remaining uncertain which hypothesis is true, instead of (x AND ¬x) being presumed true because of uncertainty (non sequitur). We can construct paradoxes in ‘classical’ logic as problems that are yet to be consistently solved, hidden relations to be revealed, rather than accept them as contradictory ‘solutions’, let alone as true statements about ‘contradictory reality’, which, as demonstrated above, amounts to nihilism about identity.

Some paraconsistent logics may be only superficially non-classical but are fully consistent with the fundamental laws in their meaningful application (Preservationism). Others (Dialetheism) maintain that the Liar paradox is an example of a ‘true contradiction’. Like all paradoxes, “This sentence is false” is a meaningless expression. The law of identity is violated by equivocating between the identity of the sentence “This sentence is false” and the word “sentence” preceded by the word “This”. It can be demonstrated that these two instances of ‘sentence’ are not the same identity, therefore no contradiction occurs. In fact, the term “This sentence” does not refer to anything at all: substitution of the whole sentence for every recurrent instance of “This sentence” results in infinite regress (cannot be completed) and in an empty subject: “(((((…) is false) is false) is false)…)”.

Crucially, there can be no meaningful debate with someone who rejects any of the fundamental laws. Their statements cannot be interpreted to mean what they are normally taken to mean since they could also be intended to mean the opposite, or something altogether different. Someone who rejects any of the fundamental laws cannot be understood; their words have no meaning, they renounce their voice.

“To be, and not to be, that is and is not the question”…