In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of Łukasiewicz logic.

See also

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Further reading

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  • Josep Maria Font; Ramón Jansana (1996). A general algebraic semantics for sentential logics. Springer-Verlag. ISBN 9783540616993. (2nd published by ASL in 2009) open access at Project Euclid
  • W.J. Blok; Don Pigozzi (1989). Algebraizable logics. American Mathematical Society. ISBN 0821824597.
  • Janusz Czelakowski (2001). Protoalgebraic logics. Springer. ISBN 9780792369400.
  • J. Michael Dunn; Gary M. Hardegree (2001). Algebraic methods in philosophical logic. Oxford University Press. ISBN 9780198531920. Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra; the book covers these prerequisites at length. The book, however, has been criticized for poor and sometimes incorrect presentation of abstract algebraic logic results. [1]


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Intuitionistic logic

Several systems of semantics for intuitionistic logic have been studied. One of these semantics mirrors classical Boolean-valued semantics but uses Heyting

Boolean algebra

value of the Boolean term corresponding to the formula. In classical semantics, only the two-element Boolean algebra is used, while in Boolean-valued

Truth value

classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantics of logical connectives are

Classical logic

admits other semantics. In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra;

Boolean-valued model

Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of

Outline of logic

form (Boolean algebra) Boolean conjunctive query Boolean-valued model Boolean domain Boolean expression Boolean ring Boolean function Boolean-valued function

Short-circuit evaluation

evaluation, or McCarthy evaluation (after John McCarthy) is the semantics of some Boolean operators in some programming languages in which the second argument

Principle of bivalence

intended semantics of classical logic is bivalent, but this is not true of every semantics for classical logic. In Boolean-valued semantics (for classical