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

Mathematical

• 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

• 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)

• 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

• Formal ethics – formal logical system for describing and evaluating the "form" as opposed to the "content" of ethical principlesPages displaying wikidata descriptions as a fallback

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

See also

• 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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