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# The theory of quaternialty

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It is well-known that every involution in a logical or mathematical system gives rise to a theory of duality; for example, negation in the sentential calculus and predicate calculus, complementation in the calculus of classes, complementation and conversion in the calculus of relations, etc. The purpose of this note is to call attention to the fact that every involution in a logical or mathematical system gives rise to a theory of quaternality and that the square of quaternality, of which the classical squares of opposition are special cases, provides a diagrammatic representation for much of the theory of quaternality. For the sake of definiteness we expose the theory of quaternality in the context of the lower predicate calculus.

Let ϕ and ψ be formulas of the lower predicate calculus. The constants {T and F} {∧ and ∨} {→ and ⇷} {← and ⇸}{↔ and ⇹} {↑ and ↓} {∧ (all) and ∨ (some)} are dual. The constant ~ is self-dual. The negational of ϕ, denoted ϕN, is the formula obtained from ϕ by interchanging negated and unnegated variables (that is, sentence variables and predicate variables) and by interchanging dual constants. The contradual of ϕ, denoted ϕC, is the formula obtained from ϕ by interchanging negated and unnegated variables. The dual of ϕ, denoted ϕD, is the formula obtained from ϕ by interchanging dual constants. The four formulas ϕ, ϕN, ϕC, ϕD, are quaternals of each other.

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1 The author is indebted to the referee for a reference to Schmidt, Arnold, Systematische Basisreduktion der Modalitäten bei Idempotenz der positiven Grundmodalitäten, Mathematische Annalen, vol. 122 (1950), pp. 7189 (reviewed, XVI 230; cf. p. 231, lines 1–3), in which the word “quaternal” is used in a sense consistent with the present usage, and for the suggestion that “square of quaternality,” etc., replace the author's original “square of duality,” etc.

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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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