In Principia Mathematica (1910) at *2.1, Whitehead and Russell formulated the canonical, although not universally accepted, proof of the law of excluded middle from the definition at *1.01 (p → q = ¬p ∨ q) and the theorem of self-implication at *2.08 (p → p). If ‘p implies q’ necessitates either not-p or q, then by substituting p → p for p → q yields p → p = ¬p ∨ p and thus necessitates the law of excluded middle. The problem with this proof is that *1.01 is not a valid representation of the exclusive disjunction entailed by the law of excluded middle insofar as its validity is subject to the law of non-contradiction. For simple implication (if p then q) we need not accept that not-p precludes q, in the sense that ‘if it is raining’ implies ‘the ground is wet’, but ‘if it is not raining’ does not preclude the ground being wet, which is a non-exclusive possibility ¬p ∧ q that implies contradiction ¬p ∧ p under the substitution with p → p. On this interpretation *2.1 is not a valid proof, as it does not hold for all p. Conversely, if bi-implication/equivalence (if and only if p then q) were intended at *1.01 then it would not account for all the possibilities of implication, thus proving nothing.

The following is possibly a novel proof of the law of excluded middle. I begin by demonstrating the logical equivalence of the law of non-contradiction and the law of identity, then the equivalence of the law of identity and the law of excluded middle. A proof from negation to contradiction is then constructed to close the logical chain.

The relationship between the law of non-contradiction and the law of identity can be formulated as follows:

∀x¬(x∧¬x) → ∀x¬(T(x)=T(¬x)) → ∀x(T(x)=T(x)) → ∀x(x=x)

For all x, x and not-x cannot both be true (i.e. cannot have the same truth-value), which implies that the truth-value of x is equal/identical only to the truth-value of x, which can be expressed as x is equal/identical to x, since x signifies anything whatsoever. Conversely, the identity relation x = x implies that the truth-value of x is equal/identical only to the truth-value of x, which implies that the truth-value of x and the truth-value of not-x cannot be identical, which implies that, for all x, x and not-x cannot both be true.

The relationship between the law of identity and the law of excluded middle can be formulated as follows:

∀x(x=x) → ¬∃x(¬x=x) → ∀x¬∃y(¬y=x.¬y=¬x) → ∀x(¬x∨x)

The law of identity implies that, for all x, there can be no third term that is neither x nor not-x, since the third term implies the identity of x and not-x, therefore non-identity of x. The law of identity implies the law of excluded middle. Conversely, the law of excluded middle implies that there is no third term that is neither x nor not-x, which implies that x cannot be identical to not-x, which implies that x is identical only to itself.

The law of excluded middle can be proven by deriving contradiction from the negation of excluded middle:

¬∀x(¬x∨x) → ∃x∃y(x∨y∨¬x) → ∃x∃y(y→(¬x∧¬¬x)) → ∃x(¬x∧x) → ⊥

The negation of the law of excluded middle implies a term y that is neither x nor not-x, therefore y implies not-x and not-not-x, therefore contradiction. The law of excluded middle is implied by the law of non-contradiction, which also closes the loop of interdependence of the three laws.

The intuitionist school of logic, who reject the law of the excluded middle, may object that the above proof involves ‘double negation elimination’ (¬¬x→x), a principle that the intuitionists also reject. This is not a serious obstacle, since the proof can be constructed in a way that avoids double negation elimination. We can simply regard ¬x as a variable in its own right, such that ¬x=z, in which case y→(z∧¬z), therefore still contradiction.

¬∀x(¬x∨x) → ∃x∃y(x∨y∨¬x) → ∃x∃y(y→(¬x∧¬¬x)).∃z(¬x=z) → ∃z(z∧¬z) → ⊥

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