Intuitionistic logic
Disjunction vs. implication

Already minimal logic proves excluded middle equivalent to consequentia mirabilis, an instance of Peirce's law. Now akin to modus ponens, clearly (ϕ ∨ ψ) → ((ϕ → ψ) → ψ) {\displaystyle (\phi \lor \psi)\to ((\phi \to \psi)\to \psi)} already in minimal logic, which is a theorem that does not even involve negations. In classical logic, this implication is in fact an equivalence. With taking ϕ {\displaystyle \phi } to be of the form ψ → φ {\displaystyle \psi \to \varphi }, excluded middle together with explosion is seen to entail Peirce's law.

In intuitionistic logic, one obtains variants of the stated theorem involving ⊥ {\displaystyle \bot }, as follows. Firstly, note that two different formulas for ¬ (ϕ ∧ ψ) {\displaystyle \neg (\phi \land \psi)} mentioned above can be used to imply (¬ ϕ ∨ ¬ ψ) → (ϕ → ¬ ψ) {\displaystyle (\neg \phi \vee \neg \psi)\to (\phi \to \neg \psi)}. The latter are forms of the disjunctive syllogism for negated propositions, ¬ ψ {\displaystyle \neg \psi }. A strengthened form still holds in intuitionistic logic:

As in previous sections, the positions of ¬ ϕ {\displaystyle \neg \phi } and ϕ {\displaystyle \phi } may be switched, giving a stronger principle than the one mentioned in the introduction. So, for example, intuitionistically "Either P {\displaystyle P} or Q {\displaystyle Q} " is a stronger propositional formula than "If not P {\displaystyle P}, then Q {\displaystyle Q} ", whereas these are classically interchangeable. The implication cannot generally be reversed, as that immediately implies excluded middle.

Non-contradiction and explosion together also prove the stronger variant (¬ ϕ ∨ ψ) → (¬ ¬ ϕ → ψ) {\displaystyle (\neg \phi \lor \psi)\to (\neg \neg \phi \to \psi)}. And this shows how excluded middle for ψ {\displaystyle \psi } implies double-negation elimination for it. For a fixed ψ {\displaystyle \psi }, this implication cannot generally be reversed. (However, as ¬ ¬ (ψ ∨ ¬ ψ) {\displaystyle \neg \neg (\psi \lor \neg \psi)} is always constructively valid, it follows that assuming double-negation elimination for all such disjunctions implies classical logic also.)

Of course the formulas established here may be combined to obtain yet more variations. For example, the disjunctive syllogism as presented generalizes to

If some term exists at all, the antecedent here even implies ∃ x (ϕ (x) → φ) {\displaystyle \exists x{\big (}\phi (x)\to \varphi {\big)}}, which in turn itself also implies the conclusion here (this is again the very first formula mentioned in this section).

The bulk of the discussion in these sections applies just as well to just minimal logic. But as for the disjunctive syllogism with general ψ {\displaystyle \psi }, minimal logic can at most prove (¬ ϕ ∨ ψ) → (¬ ¬ ϕ → ψ ′) {\displaystyle (\neg \phi \lor \psi)\to (\neg \neg \phi \to \psi ')} where ψ ′ {\displaystyle \psi '} denotes ¬ ¬ ψ ∧ (ψ ∨ ¬ ψ) {\displaystyle \neg \neg \psi \land (\psi \lor \neg \psi)}. The conclusion here can only be simplified to ψ {\displaystyle \psi } using explosion.
change_case:capital_word_frequency[8], combination:two_responses