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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
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
Formal systems of logic that significantly differ from standard logical systems
intuitionistic logic have an equivalent theorem in the classical modal logic S4. The result has been generalized to superintuitionistic logics and extensions
Non-classical_logic
Phenomenon in natural language
Q)} This symbolic logic formula above is not valid in classical modal logic: Adding this principle as an axiom to standard modal logics would allow one
Free_choice_inference
2001 textbook by Graham Priest
wide range of topics including modal logic, intuitionistic logic, many-valued logic, relevant logic, and fuzzy logic. The book has been published in
An Introduction to Non-Classical Logic
An_Introduction_to_Non-Classical_Logic
Application of logical methods to philosophical problems
systems like modal logic. Some theorists conceive of philosophical logic in a broader sense as the study of the scope and nature of logic in general. In
Philosophical_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
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
Formal semantics for non-classical logic systems
for 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
One of five systems of modal logic
In logic and philosophy, S5 is one of five systems of modal logic proposed by Clarence Irving Lewis and Cooper Harold Langford in their 1932 book Symbolic
S5_(modal_logic)
Extension of modal logic
In logic, philosophy, and theoretical computer science, dynamic logic is an extension of modal logic capable of encoding properties of computer programs
Dynamic_logic_(modal_logic)
Phenomenon whereby language is used to discuss possible situations
on modal logic. In these approaches, modal expressions such as must and can are analyzed as quantifiers over a set of possible worlds. In classical modal
Modality_(semantics)
Modal logic
Provability logic is a branch of proof theory and a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The
Provability_logic
Study of correct reasoning
ethics, and epistemology. Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new
Logic
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 logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name
List_of_logic_symbols
logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science. A modal logic
Multimodal_logic
Bearer of truth values
Logicians examine the relation between different modal propositions. For example, classical modal logic states that a proposition is necessarily true if
Proposition
Various systems of symbolic logic
logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by
Intuitionistic_logic
Family of logics for natural-language and counterfactual conditionals
logics are intended to capture the meaning and patterns of inference associated with natural language conditionals more faithfully than the classical
Conditional_logic
System of resource-aware logic
Linear logic is a substructural logic proposed by French logician Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the
Linear_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
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
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
Study of the scope and nature of logic
of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But
Philosophy_of_logic
Boolean algebra with unary operators expressing necessity and possibility modalities
x, y in A. Modal algebras provide models of propositional modal logics in the same way as Boolean algebras are models of classical logic. In particular
Modal_algebra
Concept in modal logic
possibly F. It is equivalent in a classical modal logic (but not necessarily in other formulations of modal logic) to ∃ x ◻ F x → ◻ ∃ x F x {\displaystyle
Buridan_formula
Family of formal knowledge representation
exist. For example, a description logic might be combined with a modal temporal logic such as linear temporal logic. Philosophy portal Formal concept
Description_logic
Method of deriving conclusions
instantiation. Modal logics are formal systems that extend propositional logic and first-order logic with additional operators. Alethic modal logic introduces
Rule_of_inference
Type of fallacy in modal logic
The modal fallacy or modal scope fallacy is a type of formal fallacy that occurs in modal logic. It is the fallacy of placing a proposition in the wrong
Modal_fallacy
Theory of logic to account for observations from quantum theory
Quantum logic embeds into linear logic and the modal logic B. Indeed, modern logics for the analysis of quantum computation often begin with quantum logic, and
Quantum_logic
Extension of propositional modal logic
propositional modal logic (with many modalities) by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic. The
Modal_μ-calculus
Formal semantics based on algebras
Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and
Algebraic semantics (mathematical logic)
Algebraic_semantics_(mathematical_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
non-standard logic Logics that diverge from or extend classical logic, including non-classical logics, many-valued logics, and modal logics, among others
Glossary_of_logic
Reasoning about equations with free variables
logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses
Algebraic_logic
Modality in linguistics
Alethic modality (from Greek ἀλήθεια = truth) is a linguistic modality that indicates modalities of truth, in particular the modalities of logical necessity
Alethic_modality
Formal statement in logic
of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For
Strict_conditional
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)
American philosopher and logician (1940–2022)
semantics) for modal logics. Kripke semantics is a formal semantics for non-classical logic systems. It was first made for modal logics, and later adapted
Saul_Kripke
Logical connective OR
from classical logic have been noted in cases such as free choice disjunction and simplification of disjunctive antecedents, where certain modal operators
Logical_disjunction
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
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
Founder(s): K F. Gödel (1933) showed that intuitionistic logic can be embedded into modal logic S4. (other systems) Interpretation (Gödel): ◻ P {\displaystyle
Constructive_logic
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
Subfield of mathematics
recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The method of forcing is employed in set theory, model theory, and recursion
Mathematical_logic
Propositional logic extending intuitionistic logic
properties. The logics are partially ordered under containment. A logic is stronger than another iff it contains the other. Classical logic is the strongest
Intermediate_logic
Branch of logic
Propositional logic is a branch of classical logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes
Propositional_logic
Class of non-classical logics
Deviant logic is a type of logic incompatible with classical logic. Philosopher Susan Haack uses the term deviant logic to describe certain non-classical systems
Deviant_logic
Approach to predicate logic
such fine logical structures like modal, temporal, dynamic, epistemic ones). In order to achieve its special goal, logic was forced to develop its own formal
Intensional_logic
Ancient philosophy
"Stoic modal logic is not a logic of modal propositions (e.g., propositions of the type 'It is possible that it is day' ...) ... instead, their modal theory
Stoicism
System of logic in mathematics and philosophy
defined in the early 20th century by Jan Łukasiewicz as a three-valued modal logic; it was later generalized to n-valued (for all finite integers n) as
Łukasiewicz_logic
Argument whose conclusion must be true if its premises are
propositional logic, they are tautologies. A statement can be called valid, i.e. logical truth, in some systems of logic like in modal logic if the statement
Validity_(logic)
Translation of a text into a logical system
logic translations that convert formulas from one logical system into another, for example, from modal logic to first-order logic. This form of logic
Logic_translation
Propositional calculus in which there are more than two truth values
(i.e., true and false) for any proposition. Classical two-valued logic may be extended to n-valued logic for n greater than 2. Those most popular in the
Many-valued_logic
Concept of philosophy and logic used to express modal claims
possible worlds. For instance, in the relational semantics for classical propositional modal logic, the formula ◊ P {\displaystyle \Diamond P} (read as "possibly
Possible_world
Formal system of logic
logic" is assumed in some context to refer to classical higher-order logic. However, modal higher-order logic has been studied as well. According to several
Higher-order_logic
Possible truths which are not necessary
In logic, contingency is the feature of a statement making it neither necessary nor impossible. Contingency is a fundamental concept of modal logic. Modal
Contingency_(philosophy)
Study of the semantics, or interpretations, of formal and natural languages
(developed by Saul Kripke and others for modal logic and related systems), algebraic semantics (connecting logic to abstract algebra), and game semantics
Semantics_(logic)
Formal argument for the existence of God
attempted to clarify with his ontological argument. The argument uses modal logic, which deals with statements about what is necessarily true or possibly
Gödel's_ontological_proof
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
System including an indeterminate value
contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false. Emil Leon
Three-valued_logic
Philosophical view
Noneism, also known as modal Meinongianism (named after Alexius Meinong), is a theory in logic and metaphysics. It holds that some things do not exist
Noneism
Logical proposition that cannot be disproved
as such, is often thought of as the broadest type of possibility. In modal logic, a logical proposition is possible if it is true in some possible world
Logical_possibility
Field of philosophical and mathematical logic studying agency and action
case. Agentive logics generalise modal logic by adding modalities indexed to agents and to actions. Typical examples include: STIT logics (from sees to
Agentive_logic
System for reasoning about vagueness
and lack certainty. Fuzzy logic has been applied to many fields, from control theory to artificial intelligence. Classical logic only permits conclusions
Fuzzy_logic
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
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
Paradox about the perception of probability
appeal to cumulative non-monotonic logics, Horacio Arlo-Costa's (2007) use of minimal model (classical) modal logics, and Joe Halpern's (2003) use of first-order
Lottery_paradox
Topics referred to by the same term
classical propositional calculus that uses only the material conditional connective Strict conditional or strict implication, a connective of modal logic
Implication
Approach to logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to
Term_logic
Family of modal logics that extend provability logic
Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical
Interpretability_logic
deductive systems for propositional logics. Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent and
List of axiomatic systems in logic
List_of_axiomatic_systems_in_logic
Methods in artificial intelligence research
(AI), symbolic artificial intelligence (also known as classical artificial intelligence or logic-based artificial intelligence) is a collection of methods
Symbolic artificial intelligence
Symbolic_artificial_intelligence
Term used to model separate circumstances that cannot exist together
advent of possible world semantics for modal logic, as well as world based semantics for non-classical logics, but have yet to find the ubiquitous acceptance
Impossible_world
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
Kind of non-classical logic
new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication
Relevance_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
Framework in logic and natural language semantics
Irene Heim Modal logic Scope (formal semantics) Veltman, Frank (1996). "Defaults in Update Semantics" (PDF). Journal of Philosophical Logic. 25 (3). doi:10
Dynamic_semantics
and non-normal) modal logic, hybrid logic, first-order modal logic, paraconsistent logic, linear logic, relevance logic, connexive logic, belief revision
Dialogical_logic
Japaridze's polymodal logic (GLP) is a system of provability logic with infinitely many provability modalities. This system has played an important role
Japaridze's_polymodal_logic
T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval [0, 1] for the
T-norm_fuzzy_logics
Symbol connecting formulas in logic
conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as truth functions, though they
Logical_connective
Type of non-monotonic logic
majority of cases but not always. A classical example is: “birds typically fly”. This rule can be expressed in standard logic either by “all birds fly”, which
Default_logic
Approach to formal semantics
interpretations for various logical systems, including classical logic, intuitionistic logic, linear logic, and modal logic. The approach bears conceptual resemblances
Game_semantics
Aspect of mathematical logic
of classical first-order logic, and revived relation algebra, whose models include all well-known axiomatic set theories. Classical algebraic logic, which
Abstract_algebraic_logic
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
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
American logician (1925–1973)
as "reductions" of deontic logic to alethic modal logic. This is misleading at best, however, since alethic modal logics generally do not contain anything
Alan_Ross_Anderson
Philosophical concept
Modal realism is the view propounded by the philosopher David Lewis that all possible worlds are real in the same way as is the actual world: they are
Modal_realism
System of formal deduction in logic
propositional logics – or two – with generalisation, to handle predicate logics, as well – and several infinite axiom schemas. Hilbert systems for alethic modal logics
Hilbert_system
Characteristic of some logical systems
theorem, and neither is its negation). In superintuitionistic and modal logics, a logic is structurally complete if every admissible rule is a derivable
Completeness_(logic)
Function in logic
thus every compound statement is a truth function. On the other hand, modal logic is non-truth-functional. A logical connective is truth-functional if
Truth_function
German-born British philosopher (born 1960)
proposed logic is weaker than classical logic and stronger than intuitionistic logic. It is a modal companion to the superintuitionistic logic QH+KF. Determinism
Susanne_Bobzien
Dutch professor, philosopher and logician
Institute for Logic, Language and Computation in September 2014. The Logic of Time, Reidel, Dordrecht, 1983 Modal Logic and Classical Logic, Bibliopolis
Johan_van_Benthem_(logician)
View that there are statements that are both true and false
dialetheism on the basis that, in traditional systems of logic (e.g., classical logic and intuitionistic logic), every statement becomes a theorem if a contradiction
Dialetheism
Formal study of linguistic meaning
intensionality, modality, negation, plural expressions, and the influence of contextual factors. Formal semantics is relevant to various fields. In logic and computer
Formal semantics (natural language)
Formal_semantics_(natural_language)
Technique in mathematical logic
interpretation Modal companion Glivenko 1929. Troelstra & Van Dalen 1988, Ch. 2, Sec. 3. Sørensen, Morten Heine; Urzyczyn, Paweł (2006). "6. Classical logic and
Double-negation_translation
Argument for the existence of God
ontological argument was formulated by Kurt Gödel in private notes, using modal logic. Although he never published or publicly presented it, a version was
Ontological_argument
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
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