In
mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, 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.
What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for ...
. For example, the
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...
S4 is characterized by the class of
topological boolean algebras—that is, boolean algebras with an
interior operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are d ...
. Other modal logics are characterized by various other algebras with operators. The class of
boolean algebras characterizes
classical propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
, and the class of
Heyting algebras 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-algebra In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation \oplus, a unary operation \neg, and the constant 0, satisfying certain axioms. MV-algebras are the algebraic semantics of Łukasiew ...
s are the algebraic semantics of
Łukasiewicz logic.
See also
*
Algebraic semantics (computer science)
*
Lindenbaum–Tarski algebra
Further reading
* (2nd published by
ASL in 2009
open accessat
Project Euclid
*
*
* Good introduction for readers with prior exposure to
non-classical logics 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 themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study ...
; 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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