Minimal logic

Like intuitionistic logic, minimal logic can be formulated in the language using an implication ${\displaystyle \to }$, a conjunction ${\displaystyle \land }$, a disjunction ${\displaystyle \lor }$, and falsum or absurdity ${\displaystyle \bot }$ as the basic connectives. Negation ${\displaystyle \neg A}$ is treated as an abbreviation for ${\displaystyle A\to \bot }$. Minimal logic is axiomatized as the positive fragment of intuitionistic logic.

Adding the double negation law ${\displaystyle \neg \neg A\to A}$ to minimal logic brings the calculus back to classical logic:

• Excluded middle, ${\displaystyle A\lor \neg A}$, is then the equivalent to the rejection of ${\displaystyle \neg (A\lor \neg A)}$ and is achieved using disjunction introduction on both sides.
• Explosion, ${\displaystyle (A\land \neg A)\to B}$, then follows from proof-by-contradiction ${\displaystyle (C\to (A\land \neg A))\to \neg C}$ using ${\displaystyle C=\neg B}$.

The propositional form of modus ponens,

${\displaystyle (A\land (A\to B))\to B,}$

is clearly valid also in minimal logic.

Constructively, ${\displaystyle \bot }$ represents a proposition for which there can be no reason to believe it. To prove propositions of the form ${\displaystyle \neg A}$, one shows that assuming ${\displaystyle A}$ leads to a contradiction, ${\displaystyle A\to \bot }$. With the principle of explosion this stated as

${\displaystyle (A\land (A\to \bot ))\to B,}$

the principle of explosion expresses that to derive any proposition ${\displaystyle B}$ one may also do this by deriving absurdity ${\displaystyle \bot }$. This principle is rejected in minimal logic. This means the formula does not axiomatically hold for arbitrary ${\displaystyle A}$ and ${\displaystyle B}$.

As minimal logic represents only the positive fragment of intuitionistic logic, it is a subsystem of intuitionistic logic and is strictly weaker.

Practically, this enables the disjunctive syllogism the intuitionistic context:

${\displaystyle ((A\lor B)\land (A\to \bot ))\to B.}$

Given a constructive proof of ${\displaystyle A\lor B}$ and constructive rejection of ${\displaystyle A}$, the principle of explosion unconditionally allows for the positive case choice of ${\displaystyle B}$. This is because if ${\displaystyle A\lor B}$ was proven by proving ${\displaystyle B}$ then ${\displaystyle B}$ is already proven, while if ${\displaystyle A\lor B}$ was proven by proving ${\displaystyle A}$, then ${\displaystyle B}$ also follows if the system allows for explosion.

Note that with ${\displaystyle \bot }$ taken for ${\displaystyle B}$ in the modus ponens expression, the law of non-contradiction

${\displaystyle (A\land (A\to \bot ))\to \bot ,}$

i.e. ${\displaystyle \neg (A\land \neg A)}$, can still be proven in minimal logic. Moreover, any formula using only ${\displaystyle \land ,\lor ,\to }$ is provable in minimal logic if and only if it is provable in intuitionistic logic.

There are semantics of minimal logic that mirror the frame-semantics of intuitionistic logic, see the discussion of semantics in paraconsistent logic. Here the valuation functions assigning truth and falsity to propositions can be subject to less constraints.

• Intuitionistic logic
• Paraconsistent logic
• Implicational propositional calculus
• List of logic systems

• A.S. Troelstra and H. Schwichtenberg, 2000, Basic Proof Theory, Cambridge University Press, ISBN 0521779111