¹ The discussion of the semantical dimension of linguistic analysis presented here is necessarily sketchy and inadequate. For a more profound, yet informal, presentation see G. Bergmann, “Pure Semantics, Sentences and Propositions”, Mind, 1944, 238–258.

² That this notion of deduction based on scope is very restrictive, and not simply a matter of indicative logic rewritten, may be seen by considering derivations, valid in sentential calculus, but necessarily invalid in an imperative logic. For example, in the sentential calculus p ∩ (p v q) is a formula. In an imperative logic this would lead to the absurdity that to tell someone to ‘mail the letter’ would be to tell him to ‘mail the letter or burn the letter'.

³ This ascription of only three degrees to V_c and two values to V_p is quite arbitrary, and there is no reason why matrix definitions could not be given for any number of degrees or values, even infinite if desired. Moreover, one sees that several value systems could be related instead of only two. Since I believe the applicability of a two-valued language to be a basic factual feature of our world which no philospher can ignore, I should want to keep, even in such a polydimensional logic, at least one value dimension with only two values. This dimension would have a unique position within the calculus.

⁴ Jan Lukasiewicz, “Die Logik und das Grundlagenproblem”, Les Entretiens de Zurich sur les Fondements et la Méthode des Sciences Mathématiques, pp. 82–100, Leemann frères and Cie, Zurich (1941). This is an excellent presentation of the status of three-valued logics at the time of writing. Much of the three-valued portion of the present article stems directly from this article.

⁵ H. M. Sheffer, “A Set of Five Independent Postulates for Boolean Algebras, with Applications to Logical Constants”, Trans. Amer. Math. Soc., 14 (1913) pp. 481–88.

⁶ E. L. Post, “Introduction to a General Theory of Elementary Propositions”, Amer. Jour. of Math., 43 (1921) pp. 163–85.

⁷ The distinction between the two occurrences of ‘not’ here is superficially similar to the distinction in a sentence of the following kind; ‘Mr. X did not say the sun was not hot.’ If we symbolize the sentence, ‘Mr. X did say the sun was hot’ by ‘X', we may distinguish between two types of negation, call them ‘n', and ‘N'. ‘nX’ will be, ‘Mr. X did say the sun was not hot', whereas ‘NX’ will be, ‘Mr. X did not say the sun was hot'. The original sentence with both negatives would become, ‘NnX'.

⁸ The transformation rule here formulated is commonly called the rule of detachment. In this informal presentation, no mention has been made of other transformation rules such as the rules of substitution, and the rules of re-writing.

⁹ This particular axiom set is due to Jan Lukasiewicz, “Untersuchungen über den Aussagenkalkül”, Compte Renkus des Séances de la Sociéte des Sciences et des Lettres de Varsovie, 23 (1930) class III.

¹⁰ The class (Imp) is identical in structure with the axioms of three-valued sentential calculus presented by Lukasiewicz in the article mentioned in note 4.