Logical connective

A logical connective forms a compound proposition from component propositions; logical truth and logical falsity are the zero-input cases. Logical negation has one input, while logical conjunction, logical disjunction, and logical implication each have two. In classical propositional logic, these logical connectives are truth-functional: their output value is determined by their input values, as listed in a truth table. This does not characterize every kind of logical connective in every logic. See forall x: Calgary, section 10.1.

Logical implication forms the proposition $A\to B$, read “if $A$, then $B$”. In classical propositional logic, it is false exactly when $A$ is true and $B$ is false. Equivalently, it has the same truth table as $\neg A\vee B$, using logical negation and logical disjunction. A false hypothesis makes the logical implication true regardless of its conclusion; thus it asserts neither the hypothesis nor a causal connection.

For example, for every natural number $n$, divisibility by four implies that $n$ is an even number: $n=4k=2(2k)$. At $n=8$, both statements are true. At $n=6$, the hypothesis is false and the conclusion true; at $n=3$, both are false. The only forbidden case is a multiple of four that is not an even number.

In Intuitionistic propositional logic, a proof of $A\to B$ consists of a construction transforming any proof of $A$ into a proof of $B$. The implication introduction rule proves $B$ under a temporary assumption $A$; the modus ponens rule applies the result when $A$ is available. The classical identification with $\neg A\vee B$ is not generally intuitionistically valid.

The contrapositive of a logical implication $A\to B$ reverses the direction and applies logical negation to both sides: $\neg B\to \neg A$. In classical propositional logic they are equivalent, so either can be proved in place of the other. The contrapositive of “divisibility by four implies being an even number” says that a number that is not an even number cannot be divisible by four.

In Intuitionistic propositional logic, $A\to B$ still implies its contrapositive: assumptions $A$ and $\neg B$ would yield $B$ and hence a contradiction. The reverse direction is not generally valid. From $\neg B\to \neg A$ and $A$, one obtains only $\neg \neg B$; obtaining $B$ additionally requires double-negation elimination.

The converse implication of $A\to B$ is $B\to A$. It exchanges hypothesis and conclusion without logical negation. It need not follow from the original logical implication: $n=2$ is an even number but is not divisible by four. Thus “being an even number implies divisibility by four” is false, although the original implication holds for all natural numbers. Proving both directions establishes an “if and only if” statement; proving a contrapositive establishes the original implication classically, without establishing its converse implication.

A logical implication holds whenever its premise is false. For partial correctness, an execution that never terminates supplies no final state violating the postcondition.

The conjunction $A\wedge B$ holds when both $A$ and $B$ hold.

The disjunction $A\vee B$ holds when at least one of $A$ and $B$ holds. Constructively, a proof records which disjunct holds and provides its proof.

Logical truth is the nullary connective that is always true.

Logical falsity is the nullary connective with no proof.