 Infinite-valued logics are then developed to solve the sorites paradox in a variety of ways. As with all many-valued logics, the connectives and validity can be defined variously, giving rise to a number of distinct logics. A standard proposal proceeds by way of the continuum-valued, truth-functional semantics of Łukasiewicz (see Haack 1974: Appendix). As with the three-valued case, the type of response offered to the paradox also crucially depends on the definition of validity. Where validity is defined as preservation of designated value and only the maximum value is designated, the conditional sorites admits of a type (2) response, as in Hyde 2008. However, reinstating the validity of classical laws on this general approach would require designating more than the maximum value, and a type (3) response results. In contrast, Machina (1976) suggests defining validity as preservation of the lowest degree of truth possessed by any of the argument’s premises. On this approach, the conditional sorites is invalid and thus a type (3) response again ensues. Edgington (1996) expounds a distinctly different non-truth-functional degree theory that preserves the principle of bivalence and classical logic. On this approach the conditional form of the sorites is valid and a type (2) response is advocated. Smith (2008) combines a non-bivalent, truth-functional degree theory with classical logic through a distinctive definition of validity. Smith’s unique approach provides another type (2) response to the paradox.