 C. I. Lewis’s (1912) response to the paradoxes of the material conditional is his proposal of a strict conditional (aka “strict implication”, section  3).  The strict conditional is no longer truth-functional, however. A closer kin to standard two-valued logic in this regard is the family of three-valued logics originating from the work of Łukasiewicz (1920), which retain truth-functionality. Łukasiewicz’s main motivation was not to deal specifically with conditionals, however, but with the notion of possibility. Łukasiewicz’s conditional is shown in  Table 2.  Łukasiewicz defines validity to be the preservation of the value 1 from premises to conclusion, as in two-valued logic. His operator retains the law of Identity \(A&gt;A\) and various classic schemata (such as Modus Ponens), but it also fails some classical properties (it is non-contractive, for example, failing the inference from \(A&gt;(A&gt;B)\) to \(A&gt;B\)). It agrees with the two-valued conditional on classical input, and it does not block the paradoxes of the material conditional.