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The modern symbol for inclusion first appears in Gergonne (1816), who defines it as one idea 'containing' or being 'contained' by another, using the backward letter 'C' to express this. Peirce articulated this clearly in 1870, arguing also that inclusion was a wider concept than equality, and hence a logically simpler one. [ 2 ]
The inclusion relation has a natural interpretation in various Boolean algebras: in the subset algebra, the subset relation; in arithmetic Boolean algebra, divisibility; in the algebra of propositions, material implication; in the two-element algebra, the set { (0,0), (0,1), (1,1) }. Some useful properties of the inclusion relation are:
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
A thought experiment by Aristotle to explore the concept of future contingents and the problem of determinism and free will. Aristotle's theses The formulas ¬ (¬ A → A) and ¬ (A → ¬A) in propositional logic; they are theorems in connexive logic but not in classical logic. [17] [18] [19] See also Boethius' theses. arity
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain. A free logic with the latter property is an inclusive logic.
Inclusion–exclusion illustrated by a Venn diagram for three sets. Generalizing the results of these examples gives the principle of inclusion–exclusion. To find the cardinality of the union of n sets: Include the cardinalities of the sets. Exclude the cardinalities of the pairwise intersections.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
The (restricted) "first-order predicate calculus" is the "system of logic" that adds to the propositional logic (cf Post, above) the notion of "subject-predicate" i.e. the subject x is drawn from a domain (universe) of discourse and the predicate is a logical function f(x): x as subject and f(x) as predicate (Kleene 1967:74). Although Gödel's ...