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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

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

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

Formal ethics – formal logical system for describing and evaluating the "form" as opposed to the "content" of ethical principles

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

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 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