Hybrid logic refers to a number of extensions to
propositional modal logic with more expressive power, though still less than
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...
. In
formal logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
, there; is a trade-off between expressiveness and
computational tractability. The history of hybrid logic began with
Arthur Prior
Arthur Norman Prior (4 December 1914 – 6 October 1969), usually cited as A. N. Prior, was a New Zealand–born logician and philosopher. Prior (1957) founded tense logic, now also known as temporal logic, and made important contributi ...
's work in
tense logic.
Unlike ordinary modal logic, hybrid logic makes it possible to refer to states (possible worlds) in
formulas.
This is achieved by a class of formulas called ''nominals'', which are true in exactly one state, and by the use of the @ operator, which is defined as follows:
:''@
i p'' is true
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
''p'' is true in the unique state named by the nominal ''i'' (i.e., the state where ''i'' is true).
Hybrid logics with extra or other operators exist, but @ is more-or-less standard.
Hybrid logics have many features in common with
temporal logics (which sometimes use nominal-like constructs to denote specific points in time), and they are a rich source of ideas for researchers in modern modal logic. They also have applications in the areas of
feature logic,
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
,
proof theory, and the logical analysis of
natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languag ...
. Hybrid logic is also closely connected to
description logic because the use of nominals allows one to perform assertional
ABox reasoning, as well as the more standard terminological
TBox reasoning.
References
Further reading
* P. Blackburn. 2000. Representation, reasoning and relational structures: a hybrid logic manifesto. ''Logic Journal of the IGPL'', 8(3):339-365.
External links
Hybrid Logics' Home PageStanford Encyclopedia of Philosophy entry on Hybrid Logic
Modal logic
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