List of formal systems

This is a list of formal systems, also known as logical calculi.

## Mathematical

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* Functional calculus, a way to apply various types of functions to operators
* Matrix calculus, a specialized notation for multivariable calculus over spaces of matrices
* Umbral calculus, the combinatorics of certain operations on polynomials
* Vector calculus (also called vector analysis), comprising specialized notations for multivariable analysis of vectors in an inner-product space

### Logical

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* Predicate calculus, specifies the rules of inference governing the logic of predicates
* Propositional calculus, specifies the rules of inference governing the logic of propositions

### In theoretical computer science (Formal language)

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* Modal μ-calculus, a common temporal logic used by formal verification methods such as model checking
* Lambda calculus, a formulation of the theory of reflexive functions that has deep connections to computational theory
* Kappa calculus, a reformulation of the first-order fragment of typed lambda calculus
* Rho calculus, introduced as a general means to uniformly integrate rewriting into lambda calculus
* Process calculus, a set of approaches to formulating formal models of concurrent systems
* Ambient calculus, a family of models for concurrent systems based on the concept of agent mobility
* Join calculus, a theoretical model for the design of distributed programming languages
* π-calculus, a formulation of the theory of concurrent, communicating processes, that was invented by Robin Milner
* Relational calculus, a calculus for the relational data model
* Domain relational calculus
* Tuple calculus, inspired the SQL language
* Refinement calculus, a way of refining models of programs into efficient programs

## Other formal systems

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* Formal ethics – Formal logical system

Music is a formal system too. Please have editors illuminate on this.

## See also

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* Formal system – Mathematical model for deduction or proof systems
* Computability:Formal models
* Model of computation – Mathematical model describing how an output of a function is computed given an input
* Calculus (disambiguation)

* v
* t
* e

Mathematical logic

General|

* Axiom
* list
* Cardinality
* First-order logic
* Formal proof
* Formal semantics
* Foundations of mathematics
* Information theory
* Lemma
* Logical consequence
* Model
* Theorem
* Theory
* Type theory

Theorems (list)
and paradoxes|

* Gödel's completeness and incompleteness theorems
* Tarski's undefinability
* Banach–Tarski paradox
* Cantor's theorem, paradox and diagonal argument
* Compactness
* Halting problem
* Lindström's
* Löwenheim–Skolem
* Russell's paradox

Logics|

| Traditional|

* Classical logic
* Logical truth
* Tautology
* Proposition
* Inference
* Logical equivalence
* Consistency
* Equiconsistency
* Argument
* Soundness
* Validity
* Syllogism
* Square of opposition
* Venn diagram

|

Propositional|

* Boolean algebra
* Boolean functions
* Logical connectives
* Propositional calculus
* Propositional formula
* Truth tables
* Many-valued logic
* 3
* finite
* ∞

Predicate|

* First-order
* list
* Second-order
* Monadic
* Higher-order
* Fixed-point
* Free
* Quantifiers
* Predicate
* Monadic predicate calculus

Set theory|

* Set
* hereditary
* Class
* (Ur-)Element
* Ordinal number
* Extensionality
* Forcing
* Relation
* equivalence
* partition
* Set operations:
* intersection
* union
* complement
* Cartesian product
* power set
* identities

Types of sets|

* Countable
* Uncountable
* Empty
* Inhabited
* Singleton
* Finite
* Infinite
* Transitive
* Ultrafilter
* Recursive
* Fuzzy
* Universal
* Universe
* constructible
* Grothendieck
* Von Neumann

Maps and cardinality|

* Function/Map
* domain
* codomain
* image
* In/Sur/Bi-jection
* Schröder–Bernstein theorem
* Isomorphism
* Gödel numbering
* Enumeration
* Large cardinal
* inaccessible
* Aleph number
* Operation
* binary

Set theories|

* Zermelo–Fraenkel
* axiom of choice
* continuum hypothesis
* General
* Kripke–Platek
* Morse–Kelley
* Naive
* New Foundations
* Tarski–Grothendieck
* Von Neumann–Bernays–Gödel
* Ackermann
* Constructive

Formal systems (list),
language and syntax|

* Alphabet
* Arity
* Automata
* Axiom schema
* Expression
* ground
* Extension
* by definition
* conservative
* Relation
* Formation rule
* Grammar
* Formula
* atomic
* closed
* ground
* open
* Free/bound variable
* Language
* Metalanguage
* Logical connective
* ¬
* ∨
* ∧
* →
* ↔
* =
* Predicate
* functional
* variable
* propositional variable
* Proof
* Quantifier
* ∃
* !
* ∀
* rank
* Sentence
* atomic
* spectrum
* Signature
* String
* Substitution
* Symbol
* function
* logical/constant
* non-logical
* variable
* Term
* Theory
* list

Example axiomatic
systems (list)|

* of arithmetic:
* Peano
* second-order
* elementary function
* primitive recursive
* Robinson
* Skolem
* of the real numbers
* Tarski's axiomatization
* of Boolean algebras
* canonical
* minimal axioms
* of geometry:
* Euclidean:
* Elements
* Hilbert's
* Tarski's
* non-Euclidean

* Principia Mathematica

Proof theory|

* Formal proof
* Natural deduction
* Logical consequence
* Rule of inference
* Sequent calculus
* Theorem
* Systems
* axiomatic
* deductive
* Hilbert
* list
* Complete theory
* Independence (from ZFC)
* Proof of impossibility
* Ordinal analysis
* Reverse mathematics
* Self-verifying theories

Model theory|

* Interpretation
* function
* of models
* Model
* equivalence
* finite
* saturated
* spectrum
* submodel
* Non-standard model
* of arithmetic
* Diagram
* elementary
* Categorical theory
* Model complete theory
* Satisfiability
* Semantics of logic
* Strength
* Theories of truth
* semantic
* Tarski's
* Kripke's
* T-schema
* Transfer principle
* Truth predicate
* Truth value
* Type
* Ultraproduct
* Validity

Computability theory|

* Church encoding
* Church–Turing thesis
* Computably enumerable
* Computable function
* Computable set
* Decision problem
* decidable
* undecidable
* P
* NP
* P versus NP problem
* Kolmogorov complexity
* Lambda calculus
* Primitive recursive function
* Recursion
* Recursive set
* Turing machine
* Type theory

Related|

* Abstract logic
* Algebraic logic
* Automated theorem proving
* Category theory
* Concrete/Abstract category
* Category of sets
* History of logic
* History of mathematical logic
* timeline
* Logicism
* Mathematical object
* Philosophy of mathematics
* Supertask

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