a singular necessity, which applies to each singular substance, or event. There is one proper noun for each singular event: Leibniz creates a logic of singularity, which Aristotle thought impossible (he considered that there could only be knowledge of generality). 20th century One of the early motivations for the study of many-valued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values: the true, the false and the as-yet-undetermined. This approach was later developed by Arend Heyting and L. E. J. Brouwer; see Łukasiewicz logic. Issues such as this have also been addressed in various temporal logics, where one can assert that "Eventually, either there will be a sea battle tomorrow, or there won't be." (Which is true if "tomorrow" eventually occurs.) The modal fallacy By asserting "A sea-fight must either take place tomorrow or not, but it is not necessary that it should take place tomorrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place tomorrow", Aristotle is simply claiming "necessarily (a or not-a)", which is correct. However, if we then conclude: "If a is the case, then necessarily, a is the case", then this is known as the modal fallacy. Expressed in another way: That is, there are no contingent propositions. Every proposition is either necessarily true or necessarily false. The fallacy arises in the ambiguity of the first premise. If we interpret it close to the English, we get: However, if we recognize that the English expression (i) is potentially misleading, that it assigns a necessity to what is simply nothing more than a necessary condition, then we get instead as our premises: From these latter two premises, one cannot validly infer the conclusion: See also Logical determinism Free will Principle of distributivity Principle of plenitude Truth-value link In Borges' The Garden of Forking Paths, both alternatives happen, thus leading to what Deleuze calls "incompossible worlds" Notes Further reading attempts to reconstruct both Aristotle's and Diodorus' arguments in propositional modal logic Dorothea Frede (1985), "The Sea Battle Reconsidered: A defense of the traditional interpretation", Oxford Studies in Ancient Philosophy 3, 31-87. John MacFarlane (2003), Sea Battles, Futures Contingents, and Relative Truth, The Philosophical Quarterly 53, 321-36 Jules Vuillemin, Le chapitre IX du De Interpretatione d'Aristote - Vers une réhabilitation de l'opinion comme connaissance