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Type of modal logic
In logic, a normal modal logic is a set L of modal formulas such that L contains: All propositional tautologies; All instances of the Kripke schema: ◻
Normal_modal_logic
Type of formal logic
Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fields it is used as a tool for
Modal_logic
One of five systems of modal logic
Langford in their 1932 book Symbolic Logic. It is a normal modal logic, and one of the oldest systems of modal logic of any kind. It is formed with propositional
S5_(modal_logic)
Less-restrictive form of modal logic
A non-normal modal logic is a variant of modal logic that deviates from the basic principles of normal modal logics. Normal modal logics adhere to the
Non-normal_modal_logic
Formal semantics for non-classical logic systems
non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later
Kripke_semantics
Logical operator in modal logic
A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which forms propositions from propositions. In general
Modal_operator
In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem scheme) the duality of the modal operators ◊ A ↔ ¬ ◻ ¬ A {\displaystyle
Classical_modal_logic
In logic, a modal companion of a superintuitionistic (intermediate) logic L is a normal modal logic that interprets L by a certain canonical translation
Modal_companion
Term used to model separate circumstances that cannot exist together
non-normal world. For more discussion of the interpretation of the language of modal logic in models with worlds, see the entries on modal logic and on
Impossible_world
In modal logic, a regular modal logic is a modal logic containing (as axiom or theorem) the duality of the modal operators: ◊ A ↔ ¬ ◻ ¬ A {\displaystyle
Regular_modal_logic
Type of modal logic
Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition
Epistemic_modal_logic
Normal form for modal logic formulas
Modal clausal form, also known as separated normal form by modal levels (SNFml) and Mints normal form, is a normal form for modal logic formulae. Such
Modal_clausal_form
American philosopher and logician (1940–2022)
and original contributions to logic, especially modal logic. His principal contribution is a semantics for modal logic involving possible worlds, now
Saul_Kripke
Family of logics for natural-language and counterfactual conditionals
"basic conditional logic", corresponding to the core selection function semantics and serving as an analogue of the normal modal logic K for conditionals
Conditional_logic
Various systems of symbolic logic
propositional logic (IPC) may be translated into the language of the normal modal logic S4 as follows: ⊥ ∗ = ⊥ A ∗ = ◻ A if A is prime (a positive literal)
Intuitionistic_logic
Boolean algebra with unary operators expressing necessity and possibility modalities
lattice of normal modal logics. Stone's representation theorem can be generalized to the Jónsson–Tarski duality, which ensures that each modal algebra can
Modal_algebra
Modal temporal logic with modalities referring to time
In logic, linear temporal logic or linear-time temporal logic (LTL) is a modal temporal logic with modalities referring to time. In LTL, one can encode
Linear_temporal_logic
Algorithm in modal logic
In modal logic, standard translation is a logic translation that transforms formulas of modal logic into formulas of non-modal first-order logic that
Standard_translation
Topics referred to by the same term
eleventh letter of the English alphabet. K may also refer to: K, a normal modal logic K (programming language), an array processing language developed by
K_(disambiguation)
Provability logic
intensely investigated system in provability logic. Löb's theorem can be proved within normal modal logic using only some basic rules about the provability
Löb's_theorem
American philosopher and logician (1941–2023)
philosopher and logician, known for his work in modal logic, deontic logic, conditional logic, and the logic of agency. He was a long-time member of the Department
Brian_Chellas
Field of philosophical logic
can be used to formalize imperative logic, or directive modality in natural languages. Typically, a deontic logic uses O A {\displaystyle {\mathsf {O}}A}
Deontic_logic
British logician and philosopher (1930 – 1966)
philosopher born in Sheffield, England. He is most well known for his work on modal logic, particularly his joint text with Dana Scott published posthumously (Lemmon
John_Lemmon
System of resource-aware logic
resembling the inference rules governing modalities in sequent calculus formalisations of the normal modal logic S4, and that there is no longer such a
Linear_logic
Type of logical system
first-order logic (FOL), also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system. First-order logic uses quantified
First-order_logic
semantics, also known as Scott–Montague semantics, is a formal semantics for modal logics. It is a generalization, developed independently by Dana Scott and Richard
Neighborhood_semantics
Topics referred to by the same term
four-man sprint kayak K4, a model of the British red telephone box K4, a normal modal logic K4, in graph theory, the complete graph of four vertices K4, in abstract
K4
formula in disjunctive normal form, its dual is a formula in conjunctive normal form.) dynamic modal logic A branch of modal logic that studies necessary
Glossary_of_logic
Pair of logical equivalences
alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators
De_Morgan's_laws
Assignment of meaning to the symbols of a formal language
semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways
Interpretation_(logic)
philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic. The Nasadiya
History_of_logic
Topics referred to by the same term
registration prefix D) Germany (aircraft registration prefix D) D, a normal modal logic D (grade), a below average grade in education D, a brassiere cup size
D_(disambiguation)
Relation that relates every element to some element
which every element has non-empty "predecessor neighborhood". In normal modal logic, the extension of fundamental axiom set K by the serial property results
Serial_relation
Overview of and topical guide to logic
Intuitionistic logic Linear logic Many-valued logic Mathematical logic Metalogic Minimal logic Modal logic Non-Aristotelian logic Non-classical logic Noncommutative
Outline_of_logic
Type of logic regarding reasoning about beliefs
the set of beliefs of c {\displaystyle c} . In doxastic logic, belief is treated as a modal operator. There is complete parallelism between a person
Doxastic_logic
Topics referred to by the same term
variety of modal algebras, also called Interior algebra Tetrahedral symmetry, the symmetric group S4 S4 (modal logic), a normal modal logic S4: Keep away
S4
Theorem in mathematical logic
first-order logics extended with Lindström quantifiers. Lindström's theorem has been extended to various other systems of logic, in particular modal logics by
Lindström's_theorem
(normal and non-normal) modal logic, hybrid logic, first-order modal logic, paraconsistent logic, linear logic, relevance logic, connexive logic, belief
Dialogical_logic
theory. 1963 - Saul Kripke extends his possible-world semantics to normal modal logics. 1965 - Michael D. Morley introduces the beginnings of stable theory
Timeline of mathematical logic
Timeline_of_mathematical_logic
Certain kind of modal formula
In modal logic, Sahlqvist formulas are a certain kind of modal formula with remarkable properties. The Sahlqvist correspondence theorem states that every
Sahlqvist_formula
Kind of proof calculus
reference work on natural deduction, and included applications for modal and second-order logic. In natural deduction, a proposition is deduced from a collection
Natural_deduction
American philosopher (1883–1964)
American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logician, he
C._I._Lewis
American mathematician (born 1938)
{\displaystyle \mathrm {P} \neq \mathrm {NP} } . Proving that GL (the normal modal logic which has the instances of the schema ◻ ( ◻ A → A ) → ◻ A {\displaystyle
Robert_M._Solovay
In logic, general frames (or simply frames) are Kripke frames with an additional structure, which are used to model modal and intermediate logics. The
General_frame
Tool for proving a logical formula
satisfiability of finite sets of formulas of various logics. It is the most popular proof procedure for modal logics. A method of truth trees contains a fixed set
Method_of_analytic_tableaux
Algebraic structure
topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.
Interior_algebra
Relationship between programs and proofs
generalizes to much richer models of computation, and is itself related to modal logic by a natural extension of the Curry–Howard isomorphism). A more radical
Curry–Howard_correspondence
Logical formula with NOT only on variables
In mathematical logic, a formula is in negation normal form (NNF) if the negation operator ( ¬ {\displaystyle \lnot } , not) is only applied to variables
Negation_normal_form
Type of non-monotonic logic
variants of default logic or between default logic and a logic in which a concept similar to extension exists, e.g., models in modal logic; a translation is
Default_logic
Propositional calculus in which there are more than two truth values
Many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in
Many-valued_logic
Finnish and American philosopher and logician (1929–2015)
Introduction to the Logic of the Two Notions ISBN 1-904987-08-7 1969. Models for Modalities: Selected Essays ISBN 978-90-277-0598-3 1973 Logic, Language-Games
Jaakko_Hintikka
logic L with its standard consequence relation ⊢ L {\displaystyle \vdash _{L}} generated by modus ponens and axioms, and we identify a normal modal logic
Admissible_rule
System for reasoning about vagueness
doi:10.1016/j.asoc.2014.10.035. Mironov, A. M. (August 2005). "Fuzzy Modal Logics". Journal of Mathematical Sciences. 128 (6): 3461–3483. doi:10.1007/s10958-005-0281-1
Fuzzy_logic
Reasoning by means of visual representations
(nearly) isomorphic to normal modal logic. Alpha nests in beta and gamma. Beta does not nest in gamma, quantified modal logic being more than even Peirce
Diagrammatic_reasoning
Family of modal logics for agency and choice
STIT logic (from seeing to it that) is a family of modal and branching-time logics for reasoning about agency and choice. A typical STIT operator has
STIT_logic
Approach to the semantics of logic that locates meaning in inferential role
affairs. The framework yields sound and complete semantics for the normal modal logics determined by the axioms K, T, 4, and 5; a subsequent adjustment
Proof-theoretic_semantics
metric temporal logic is defined similarly to linear temporal logic, where a set of non-negative real numbers is added to temporal modal operators U and
Metric_temporal_logic
Inference seeking the simplest and most likely explanation
first-order logic, without requiring any preliminary reduction of formulae into normal forms. These methods have also been extended to modal logic. Abductive
Abductive_reasoning
Type of diagrammatic notation for propositional logic
(nearly) isomorphic to normal modal logic. Alpha nests in beta and gamma. Beta does not nest in gamma, quantified modal logic being more general than
Existential_graph
Subdiscipline of proof theory
[Originally published in Russian in 1968]. "On some calculi of modal logic". The Calculi of Symbolic Logic. Proceedings of the Steklov Institute of Mathematics
Structural_proof_theory
Logical connective OR
classical logic have been noted in cases such as free choice disjunction and simplification of disjunctive antecedents, where certain modal operators
Logical_disjunction
Algebraization of first-order logic
In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic
Predicate_functor_logic
Mathematics notation with operators preceding operands
nonimplication) in propositional logic and Łukasiewicz uses L {\displaystyle L} and M {\displaystyle M} in modal logic. Prefix notation has seen wide application
Polish_notation
Branch of mathematical logic
predicate logic of either the classical or intuitionistic flavour, almost any modal logic, and many substructural logics, such as relevance logic or linear
Proof_theory
Extension of first-order logic with atoms expressing variable dependencies
second-order logic. The dependence atom, or a suitable variant thereof, can be added to the language of modal logic, thus obtaining modal dependence logic. As
Dependence_logic
boolean formulas First-order logic of equality Provability in intuitionistic propositional logic Satisfaction in modal logic S4 First-order theory of the
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Theorem in mathematical logic
Similar constructive proofs may be provided for the basic modal logic K, intuitionistic logic and μ-calculus, with similar complexity measures. Craig interpolation
Craig_interpolation
Branch of medieval logic
medieval logic that was probably aimed at giving accounts of issues similar to modern accounts of reference, plurality, tense, and modality, within an
Supposition_theory
Algebraic concept in measure theory, also referred to as an algebra of sets
arise naturally in modal logic where the points represent the possible worlds in the Kripke semantics of a theory in the modal logic S4, the preorder represents
Field_of_sets
Concept in metaphysics and philosophy
standard (Kripkean) possible-worlds semantics for interpreting quantified modal logic. Counterpart theory still presupposes possible worlds, but differs in
Counterpart_theory
Mathematical use of "for all" and "there exists"
formal analysis. Term logic treated All, Some and No in the 4th century BC, in an account also touching on the alethic modalities. In 1827, George Bentham
Quantifier_(logic)
American scientist (1839–1914)
having wavered earlier as to just how positively real the modalities are. In his 1897 "The Logic of Relatives" he wrote: I formerly defined the possible
Charles_Sanders_Peirce
American philosopher and logician (1940–1996)
Burgess. Kurt Gödel wrote the first paper on provability logic, which applies modal logic—the logic of necessity and possibility—to the theory of mathematical
George_Boolos
Formal logical system
predicate logic (but with a grammar closer to higher-order logics), augmented with imperative, deontic, belief, and modal logic symbols. Formal logic uses
Formal_ethics
Logical connective
propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models. Similar discrepancies
Material_conditional
Overview of and topical guide to discrete mathematics
typical terms of art that may be encountered. Logic – Study of correct reasoning Modal logic – Type of formal logic Set theory – Branch of mathematics that
Outline of discrete mathematics
Outline_of_discrete_mathematics
This is a list of mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and
List of mathematical logic topics
List_of_mathematical_logic_topics
Principle in epistemology
property of some belief systems and of many epistemic modal logics (viz. those which are "normal"). It is the principle that if a subject S {\displaystyle
Epistemic_closure
interpretants. In logic and mathematics the most clarified and most succinct signs for an object are called canonical forms or normal forms. The interpretant
Semiotic theory of Charles Sanders Peirce
Semiotic_theory_of_Charles_Sanders_Peirce
Form of typed lambda calculus
parameter |citeseerx= (help) Borghuis, Tijn (1998). "Modal Pure Type Systems". Journal of Logic, Language and Information. 7 (3): 265–296. doi:10.1023/A:1008254612284
Pure_type_system
Belgian academic
mechanics. He has authored the books Logic and Islam Part I: Faith issues: Answers to current questions; Logic and Islam Part II: Scientific issues;
Magd_Abdel_Wahab
Theoretical framework
world objects and events. It is a graphical representation of modal logic in which modal operators are used to distinguish statement about concepts from
Conceptual_model
semantics, normalisation by evaluation (NBE) is a method of obtaining the normal form of terms in the λ-calculus by appealing to their denotational semantics
Normalisation_by_evaluation
Type of topology in mathematics
of modal logic that an equivalence exists between finite topological spaces and preorders on finite sets (the finite model frames for the modal logic S4)
Alexandrov_topology
Principles in the philosophy of Aristotle
philosophy regards possibility, as studied by modal metaphysics, to be an aspect of modal logic. Modal logic as a named subject owes much to the writings
Potentiality_and_actuality
Epistemological study of Buddhism
of pramāṇa (epistemic tool, valid cognition) and hetu-vidya (reasoning, logic). While the term may refer to various Buddhist systems and views on reasoning
Buddhist_logico-epistemology
Method of logical reasoning
two criteria. Another approach to the analysis of reasoning is that of modal logic, which deals with the distinction between the necessary and the possible
Inductive_reasoning
Branch of linguistics which inquires into the nature of language
language and semanticists make use of propositional, predicate, and modal logics to express their ideas about word meaning. Syntax is the study of language
Theoretical_linguistics
American philosopher
differences between non-relativistic quantum mechanics and quantum field theory, modal semantics for quantum physics and virtue-epistemological theories of warrant
Laura_Ruetsche
American linguist (1931–2005)
some of the deepest problems of philosophy, especially in the field of modal logic. He concentrated on the everyday uses of English as a subject worthy
Julian_C._Boyd
Formal means of expressing grammar
way of expressing grammar, either for natural or formal languages, in a logic programming language such as Prolog. It is closely related to the concept
Definite_clause_grammar
Method of estimating the parameters of a statistical model
many types of models, such as mixture models, the posterior may be multi-modal. In such a case, the usual recommendation is that one should choose the
Maximum a posteriori estimation
Maximum_a_posteriori_estimation
Sentence expressing an 'if-then' relation
Conditional mood Modality Propositional attitude This use of past tense is often called fake past since it does not contribute a normal past tense meaning
Conditional_sentence
Philosophical doctrine
"Testability and Meaning". Meaning and Necessity: A Study in Semantics and Modal Logic (2nd ed.). Chicago: University of Chicago Press. pp. 420–424. Creath
Verificationism
Polish mathematician and philosopher (1922–2014)
Definability: Modal and Intuitionistic Logic. Oxford University Press, Oxford Maksimova, Larisa Lvovna (2004): Definability in Normal Extensions of S4
Andrzej_Grzegorczyk
Playing back a passage of notes
Guerino, Stefan Göller, and Stefan Müller. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Basel: Birkhäuser, 2002. Meyer, Christian
Retrograde_(music)
Short story by L. Sprague de Camp
story in a footnote of his 1968 paper "Counterpart theory and quantified modal logic" as a source for his conception of individuals across possible worlds
The_Wheels_of_If
Typographic character indicating a question (?)
Minkowski's question mark function. In linear logic, the question mark denotes one of the exponential modalities that control weakening and contraction. When
Question_mark
1990s and employs the techniques of proof theory, substructural logic, linear logic, model theory and recursion theory to prove bounds on the expressive
Implicit computational complexity
Implicit_computational_complexity
List of concepts in artificial intelligence
solve problems declaratively based on abductive reasoning. It extends normal logic programming by allowing some predicates to be incompletely defined, declared
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
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