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Style of formal logical argumentation
logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard
Sequent_calculus
Type of logical system
sequent calculus was developed to study the properties of natural deduction systems. Instead of working with one formula at a time, it uses sequents,
First-order_logic
Logical proof involving antecedents and consequents
is almost always associated with the conceptual framework of sequent calculus. Sequents are best understood in the context of the following three kinds
Sequent
In structural proof theory, the nested sequent calculus is a reformulation of the sequent calculus to allow deep inference. Alwen Tiu; Egor Ianovski;
Nested_sequent_calculus
System of resource-aware logic
intuitions. Proof-theoretically, it derives from an analysis of classical sequent calculus in which uses of (the structural rules) contraction and weakening are
Linear_logic
Various systems of symbolic logic
Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system that is sound and complete
Intuitionistic_logic
Relationship between programs and proofs
known as lambda calculus. Actually, Howard's first formulation of the isomorphism was referred to (a variant of) Gentzen's sequent calculus. The observation
Curry–Howard_correspondence
Subdiscipline of proof theory
theory comes from a technical notion introduced in the sequent calculus: the sequent calculus represents the assertion made at any stage of an inference
Structural_proof_theory
Kind of proof calculus
deduction. For this reason he introduced his alternative system, the sequent calculus, for which he proved the Hauptsatz both for classical and intuitionistic
Natural_deduction
Algebraic manipulation of "true" and "false"
is sequent calculus, which has two sorts, propositions as in ordinary propositional calculus, and pairs of lists of propositions called sequents, such
Boolean_algebra
Branch of logic
Weisstein, Eric W. "Sequent Calculus". Wolfram MathWorld. Retrieved 9 August 2025. "Interactive Tutorial of the Sequent Calculus". logitext.mit.edu. Retrieved
Propositional_logic
Formal language used to prove statements
radically different logics. For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic
Proof_calculus
Cirquent calculus (circuit sequent calculus) is a proof calculus that combines aspects of sequent calculus and boolean circuits. Its proof-objects are
Cirquent_calculus
Branch of mathematical logic
analytic proof was introduced by Gentzen for the sequent calculus, where he proved that the sequent calculus of classical and intuitionistic logics are cut-free
Proof_theory
Theorem in formal logic
Hauptsatz) is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in part I of his landmark
Cut-elimination_theorem
German mathematician (1909–1945)
of mathematics, proof theory, especially on natural deduction and sequent calculus. He died of starvation in a Czech prison camp in Prague in 1945. Gentzen
Gerhard_Gentzen
that CoS does not distinguish sequents and formulas, but uses a single object to do the job of both in a sequent calculus. Specifically, a structure can
Calculus_of_structures
System of formal deduction in logic
any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if one is interested only
Hilbert_system
when axioms are reached forms the sub-family of uniform proofs. A sequent calculus is said to have the focusing property when focused proofs are complete
Focused_proof
In sequent calculus, the completeness of atomic initial sequents states that initial sequents A ⊢ A (where A is an arbitrary formula) can be derived from
Completeness of atomic initial sequents
Completeness_of_atomic_initial_sequents
Topics referred to by the same term
Look up sequent in Wiktionary, the free dictionary. A sequent is a formalized statement of provability used within sequent calculus. Sequent may also refer
Sequent_(disambiguation)
Branch of mathematics
propositional calculus, Ricci calculus, calculus of variations, lambda calculus, sequent calculus, and process calculus. Furthermore, the term calculus has variously
Calculus
Topics referred to by the same term
Proof calculus, a framework for expressing systems of logical inference Sequent calculus, a proof calculus for first-order logic Cirquent calculus, a proof
Calculus_(disambiguation)
general idea in structural proof theory that breaks with the classical sequent calculus by generalising the notion of structure to permit inference to occur
Deep_inference
Less-restrictive form of modal logic
which contains the congruence rule in its Hilbert calculus or the E rule in its sequent calculus upon the corresponding proof systems for classical propositional
Non-normal_modal_logic
Rule of mathematical logic
is an inference rule of a sequent calculus that does not refer to any logical connective but instead operates on the sequents directly. Structural rules
Structural_rule
Establishment of a theorem using inference from the axioms
or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof.
Formal_proof
Reasoning about equations with free variables
obtained by matrix multiplication using Boolean arithmetic. An example of calculus of relations arises in erotetics, the theory of questions. In the universe
Algebraic_logic
Subfield of mathematics
Hilbert-style deduction systems, systems of natural deduction, and the sequent calculus developed by Gentzen. The study of constructive mathematics, in the
Mathematical_logic
Mathematical-logic system
In mathematical logic, the lambda calculus (also written as λ-calculus) is a formal system for expressing computation based on function abstraction and
Lambda_calculus
Fragment of first-order logic
monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic (also called predicate calculus) in which all relation
Monadic_predicate_calculus
point combinator SKI combinator calculus B, C, K, W system SECD machine Graph reduction machine Sequent, sequent calculus Natural deduction Intuitionistic
List of functional programming topics
List_of_functional_programming_topics
Inference rule
In mathematical logic, the cut rule is an inference rule of sequent calculus. It is a generalisation of the classical modus ponens inference rule. The
Cut_rule
British mathematician and logician
St Andrews. He is known for his discovery in 1992 of a terminating sequent calculus for intuitionistic propositional logic. His Erdős number was 3. Roy
Roy_Dyckhoff
Argument that leads to a logical absurdity
then P {\displaystyle P} may be concluded." In sequent calculus the principle is expressed by the sequent Γ , ¬ ¬ P ⊢ P , Δ {\displaystyle \Gamma ,\lnot
Reductio_ad_absurdum
Computation model defining an abstract machine
or simply a universal machine). Another mathematical formalism, lambda calculus, with a similar "universal" nature was introduced by Alonzo Church. Church's
Turing_machine
Extension of lambda calculus
simply typed lambda calculus is to intuitionistic propositional logic. Typed lambda-mu calculus can be presented in sequent calculus: Γ , x : τ ⊢ x : τ
Lambda-mu_calculus
Fundamental result of mathematical logic
y_{n})F(y_{1},\ldots ,y_{n})} is valid, then by completeness of cut-free sequent calculus, which follows from Gentzen's cut-elimination theorem, there is a cut-free
Herbrand's_theorem
Method of deriving conclusions
underlying logical reasoning. Sequent calculi, another approach, introduce sequents as formal representations of arguments. A sequent has the form A 1 , … ,
Rule_of_inference
Set of sentences in a formal language
These include Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method and resolution. A formula A is a syntactic consequence
Theory_(mathematical_logic)
Theory of logic to account for observations from quantum theory
formulations include propositions derivable via a natural deduction, sequent calculus or tableaux system. Despite the relatively developed proof theory,
Quantum_logic
Limitative results in mathematical logic
JSTOR 2695030. Zach, Richard (2003). "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program" (PDF). Synthese. 137 (1).
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Basic framework of mathematics
tacitly assumed to be definitive until the introduction of infinitesimal calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. This
Foundations_of_mathematics
Extension of linear logic
the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties (a family of cyclic
Noncommutative_logic
Class of formal logics
Stoic logic. The two were sometimes seen as irreconcilable. Leibniz's calculus ratiocinator can be seen as foreshadowing classical logic. Bernard Bolzano
Classical_logic
Mathematical logic concept
non- P {\displaystyle P} s." The transposition rule may be expressed as a sequent: ( P → Q ) ⊢ ( ¬ Q → ¬ P ) , {\displaystyle (P\to Q)\vdash (\neg Q\to \neg
Contraposition
Syntactically correct logical formula
however, to be considered solely as a formula. The formulas of propositional calculus, also called propositional formulas, are expressions such as ( A ∧ ( B
Well-formed_formula
Impossible task in computing
a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis. The origin
Entscheidungsproblem
Type of infinite structure
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
O-minimal_theory
Statement that is taken to be true
are also used in the predicate calculus, but additional logical axioms are needed to include a quantifier in the calculus. Axiom of equality. Let L {\displaystyle
Axiom
Number of arguments required by a function
NOT operators are examples of unary operators. All functions in lambda calculus and in some functional programming languages (especially those descended
Arity
Formal system of logic
standard semantics does not admit an effective, sound, and complete proof calculus. The model-theoretic properties of HOL with standard semantics are also
Higher-order_logic
Relationship where one statement follows from another
Logic gate Logical graph Peirce's law Probabilistic logic Propositional calculus Sole sufficient operator Strawson entailment Strict conditional Tautology
Logical_consequence
various kinds of networks as opposed to the flat tree structures of sequent calculus. To distinguish the real proof nets from all the possible networks
Geometry_of_interaction
Line-by-line system for natural deduction proofs
logic in undergraduate education. Natural deduction Frederic Fitch Sequent calculus Proof theory Hilbert system Suppes–Lemmon notation Fitch 1952. Suppes
Fitch_notation
Topics referred to by the same term
Assignment (law) In logic, the antecedent and succedent of a sequent in sequent calculus are called cedents. In insurance, a reinsured. This disambiguation
Cedent
Problem in computer science
Church published his proof of the undecidability of a problem in the lambda calculus. Turing's proof was published later, in January 1937. Since then, many
Halting_problem
Branch of non-classical logic
significant substructural logics are relevance logic and linear logic. In a sequent calculus, one writes each line of a proof as Γ ⊢ Σ {\displaystyle \Gamma \vdash
Substructural_logic
Mathematical model for deduction or proof systems
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Formal_system
Theorem in mathematical logic
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Lindström's_theorem
Input to a mathematical function
argument to a function Propositional function – Expression in propositional calculus Type signature – Defines the inputs and outputs for a function, subroutine
Argument_of_a_function
Paradox in set theory
type theory The Kleene–Rosser paradox, showing that the original lambda calculus is inconsistent, by means of a self-negating statement The smallest uninteresting
Russell's_paradox
from regular proof calculi such as the natural deduction calculus and the sequent calculus, where these phenomena are present. Proof nets were introduced
Proof_net
Logic principle
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Extensionality
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Mathematical_object
Function that preserves distinctness
other methods of proving that a function is injective. For example, in calculus if f {\displaystyle f} is a differentiable function defined on some interval
Injective_function
Collection of mathematical objects
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Set_(mathematics)
Branch of mathematics that studies sets
mathematicians had struggled with the concept of infinity. With the development of calculus in the late 17th century, philosophers began to generally distinguish between
Set_theory
Mathematical proof expressed visually
Philosophy of mathematics Proof theory – Branch of mathematical logic Visual calculus – Visual mathematical proofs Dunham 1994, p. 120 Weisstein, Eric W. "Proof
Proof_without_words
Additional mathematical object
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Mathematical_structure
Axioms for the natural numbers
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Peano_axioms
Diagram that shows all possible logical relations between a collection of sets
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Venn_diagram
Function, homomorphism, or morphism
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Map_(mathematics)
Logical connective AND
Peano–Russell notation – Notation used in mathematical logic Propositional calculus – Branch of logicPages displaying short descriptions of redirect targets
Logical_conjunction
Family of formalisms in natural language syntax
T::=\Gamma \,\!} for certain sequents T ← Γ {\displaystyle T\leftarrow \Gamma } that are derivable in the Lambek calculus. Of course, there are infinitely
Categorial_grammar
Field in logic and theoretical computer science
P(\phi ,x)} accepts. Examples of propositional proof systems include sequent calculus, resolution, cutting planes and Frege systems. Strong mathematical
Proof_complexity
mathematics and logic to define functions, sets, and series. sequent In sequent calculus, a formal representation of a logical deduction, consisting of
Glossary_of_logic
Yes-or-no question that cannot ever be solved by a computer
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Undecidable_problem
Statement in a metalanguage
any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only
Judgment_(mathematical_logic)
Mathematical set containing no elements
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Empty_set
Topics referred to by the same term
(former NASDAQ ticker lk) System LK, in mathematics, the classical sequent calculus LK (spacecraft), a Soviet lunar lander LK (index mark code), county
LK
Standard system of axiomatic set theory
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Zermelo–Fraenkel_set_theory
Fundamental theorem in mathematical logic
[citation needed] We first fix a deductive system of first-order predicate calculus, choosing any of the well-known equivalent systems. Gödel's original proof
Gödel's_completeness_theorem
Form of logic that allows quantification over predicates
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Second-order_logic
Proof by Alan Turing
general process for determining whether a given formula U of the functional calculus K is provable. (ibid.) Both Lemmas #1 and #2 are required to form the necessary
Turing's_proof
Formal verification tool
the KeY system lies a first-order theorem prover based on a sequent calculus. A sequent is of the form Γ ⊢ Δ {\displaystyle \Gamma \vdash \Delta } where
KeY
Set of elements in any of some sets
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Union_(set_theory)
Measure of algorithmic complexity
Vitányi 1997". Tromp, John. "John's Lambda Calculus and Combinatory Logic Playground". Tromp's lambda calculus computer model offers a concrete definition
Kolmogorov_complexity
Set whose elements all belong to another set
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Subset
Inference seeking the simplest and most likely explanation
proof-theoretical abduction method for first-order classical logic based on the sequent calculus and a dual one, based on semantic tableaux (analytic tableaux) have
Abductive_reasoning
Token in a mathematical or logical formula
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Symbol_(formal)
Any one of the distinct objects that make up a set in set theory
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Element_of_a_set
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Diagonal_intersection
Non-contradiction of a theory
propositional calculus was proved by Paul Bernays in 1918[citation needed] and Emil Post in 1921, while the completeness of (first order) predicate calculus was
Consistency
3-volume treatise on mathematics, 1910–1913
inverse is the null (empty) set. When applied to relations in section ✱23 CALCULUS OF RELATIONS, the symbols "⊂", "∩", "∪", and "–" acquire a dot: for example:
Principia_Mathematica
Mathematical concept
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Transfinite_induction
Hilbert-style deductive systems for propositional logics. Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent
List of axiomatic systems in logic
List_of_axiomatic_systems_in_logic
Set of the elements not in a given subset
relations and the algebra of sets are the elementary operations of the calculus of relations. In the LaTeX typesetting language, the command \setminus
Complement_(set_theory)
Ordered listing of items in collection
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Enumeration
Study of computable functions and Turing degrees
Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence
Computability_theory
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