In
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, algebraic semantics is a
formal semantics based on algebras studied as part of
algebraic logic
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with Free variables and bound variables, free variables.
What is now usually called classical algebraic logic focuses on the identification and algebraic de ...
. For example, the
modal logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields
it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
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 algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' call ...
s propositional
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
.
MV-algebras are the algebraic semantics of
Łukasiewicz logic
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued modal logic;Łukasiewicz J., 1920, O logice trójwartościow ...
.
See also
*
Algebraic semantics (computer science)
In computer science, algebraic semantics is a formal approach to programming language theory that uses algebraic methods for defining, specifying, and reasoning about the behavior of programs. It is a form of axiomatic semantics that provides a ...
*
Lindenbaum–Tarski algebra
Further reading
* (2nd published by
ASL in 2009
open accessat
Project Euclid
Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent a ...
*
*
* Good introduction for readers with prior exposure to
non-classical logic
Non-classical logics (and sometimes alternative logics or non-Aristotelian logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this ...
s but without much background in
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
and/or
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures.
For instance, rather than considering groups or rings as the object of stud ...
; the book covers these prerequisites at length. The book, however, has been criticized for poor and sometimes incorrect presentation of abstract algebraic logic results
Mathematical logic
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