An account of implication in which conditional formulae of the form $\phi \mathcal{⊰}\psi$ are intended to capture the notion of the consequent $\psi$ necessarily following from the antecedent $\phi$ with respect to some notion of necessity. The modern formal account of strict implication was discovered by the philosopher Clarence Irving Lewis (1883–1964), who defined a strict conditional $\phi \mathcal{⊰}\psi$ by appealing to a primitive notion of impossibility, so that $\phi \mathcal{⊰}\psi$ is true precisely when it is impossible that $\phi$ is true while $\psi$ is false. In modern notation, this is captured by the formula $\square \neg \left(\phi \wedge \neg \psi \right)$. Strict implication was offered as a resolution to so-called paradoxes of the material conditional according to which $A$ entails $\psi \to \phi$, and $\neg A$ entails $\phi \to \psi$. A strict conditional $\phi \mathcal{⊰}\psi$ purports to resolve the putative deficiencies of the material conditional by asserting an element of necessity. On the material conditional reading, the sentence

• • That the Earth has one natural satellite implies that the War of 1812 lasted two and a half years.

is true because the consequent is true. However, it is reasonable to believe that the War of 1812 could have dragged on through 1817 and that this possibility could have obtained without necessitating any changes to the celestial bodies. Hence, the antecedent appears consistent with the negation of the consequent, whence ‘The Earth has one natural satellite’ does not strictly imply ‘The War of 1812 lasted two years’.