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Mathematical model for deduction or proof systems
A formal system (or deductive system) is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems
Formal_system
Sequence of words formed by specific rules
logic and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems, and mathematical formalism is the philosophy
Formal_language
Mathematical program specifications
formal methods are mathematically rigorous techniques for the specification, development, analysis, and verification of software and hardware systems
Formal_methods
Establishment of a theorem using inference from the axioms
assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is generally effective, but
Formal_proof
This is a list of formal systems, also known as logical calculi. Functional calculus, a way to apply various types of functions to operators Matrix calculus
List_of_formal_systems
Study of correct reasoning
language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates
Logic
Topics referred to by the same term
calculus Formal methods, mathematically based techniques for the specification, development and verification of software and hardware systems Formal specification
Formal
Study of abstract structures described by formal systems
Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems. Whereas the natural sciences
Formal_science
Subfield of mathematics
mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can
Mathematical_logic
Method of deriving conclusions
swapped. They contrast with formal fallacies—invalid argument forms involving logical errors. Logicians construct formal systems to precisely capture and
Rule_of_inference
Rules used for constructing, or transforming the symbols and words of a language
syntax is an arrangement of well-structured entities in the formal languages or formal systems that express something. Syntax is concerned with the rules
Syntax_(logic)
Structure of a formal language
A formal grammar is a set of symbols and the production rules for rewriting some of them into every possible string of a formal language over an alphabet
Formal_grammar
Interrelated entities that form a whole
organizations Formal system Glossary of systems theory Human body § Systems, systems in the human body Market (economics) Meta-system System of systems System of
System
Proving or disproving the correctness of certain intended algorithms
using formal methods of mathematics. Formal verification is a key incentive for formal specification of systems, and is at the core of formal methods
Formal_verification
Limitative results in mathematical logic
related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Mathematical term; concerning axioms used to derive theorems
system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science. It consists of a set of formal statements
Axiomatic_system
Aspect of computer science
computer science, formal specifications are mathematically based techniques whose purpose is to help with the implementation of systems and software. They
Formal_specification
Faulty deductive reasoning due to a logical flaw
In logic and philosophy, a formal fallacy is a pattern of reasoning with a flaw in its logical structure (the logical relationship between the premises
Formal_fallacy
In mathematics, a statement that has been proven
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses
Theorem
Basic framework of mathematics
study of formal theories, in disciplines such as reverse mathematics and computational complexity theory. As noted by Weyl, formal logical systems also run
Foundations_of_mathematics
Token in a mathematical or logical formula
of the language. In a formal system a symbol may be used as a token in formal operations. The set of formal symbols in a formal language is referred to
Symbol_(formal)
incompleteness theorems, which showed that any sufficiently powerful formal system (like those used to express arithmetic) cannot be both complete and
Mathematical_object
DCFS, the International Workshop on Descriptional Complexity of Formal Systems is an annual academic conference in the field of computer science. Beginning
Descriptional Complexity of Formal Systems
Descriptional_Complexity_of_Formal_Systems
Characteristic of some logical systems
metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e
Completeness_(logic)
Logic statement about a formal system proven in a metalanguage
metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within
Metatheorem
Term in logic and deductive reasoning
property of formal deductive systems. An argument is sound if (and only if) it is both valid in form and has no false premises. A formal system is sound
Soundness
Assigning meanings to concepts, symbols, objects
construct a formal system that will serve as a conceptual model of reality. Predictions or other statements drawn from such a formal system mirror or map
Interpretation_(philosophy)
Precisely specified semantic version of a statement
statement is a precisely specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous
Logical_form
Theoretical framework
theory, using tools from mathematical logic. A system that gives meaning to the sentences of a formal language is called a model for the language. If
Conceptual_model
Subdivisions of science defined by their scope
disciplines, are commonly divided into three major groups: Formal sciences: the study of formal systems, such as those under the branches of logic and mathematics
Branches_of_science
Programming language designed 1942 to 1945
term for a formal system—as in Hilbert-Kalkül, the original name for the Hilbert-style deduction system—so Plankalkül refers to a formal system for planning
Plankalkül
Non-contradiction of a theory
{\displaystyle A} under some (specified, possibly implicitly) formal deductive system. The set of axioms A {\displaystyle A} is consistent when there
Consistency
Study of the properties of logical systems
metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical
Metalogic
Approach to systems analyis
distinguish between two strands such theory: formal systems and principally non-formal system. Formal viable system theory is normally referred to as viability
Viable_system_theory
System manipulating symbols as expressions
A physical symbol system (also called a formal system) takes physical patterns (symbols), combining them into structures (expressions) and manipulating
Physical_symbol_system
Relationship where one statement follows from another
any interpretation of the formal system. A formula A {\displaystyle A} is a semantic consequence within some formal system F S {\displaystyle {\mathcal
Logical_consequence
Type system used in computer programming and mathematics
and later rediscovered by Robin Milner. Luis Damas contributed a close formal analysis and proof of the method in his PhD thesis. Among HM's more notable
Hindley–Milner_type_system
Systematic endeavour to gain knowledge
relationships and processes. Formal science is an area of study that generates knowledge using formal systems. A formal system is an abstract structure used
Science
Interdisciplinary study of systems
behaviours and processes or interrelate through formal contextual boundary conditions (attractors). Passive systems are structures and components that are being
Systems_theory
Identifier for Standard Generalized Markup Language
A Formal Public Identifier (FPI) is a short piece of text with a particular structure that may be used to uniquely identify a product, specification or
Formal_Public_Identifier
Topics referred to by the same term
Formal theory can refer to: Another name for a theory which is expressed in formal language An axiomatic system, something representable by symbols and
Formal_theory
Species naming system
taxonomy, binomial nomenclature ("two-term naming system"), also called binary nomenclature, is a formal system of naming species of living things by giving
Binomial_nomenclature
Analysis of computer programs without executing them
original system). Data-flow analysis, a lattice-based technique for gathering information about the possible set of values; Hoare logic, a formal system with
Static_program_analysis
Branch of logic
formal system in which formulas of a formal language are interpreted to represent propositions. This formal language is the basis for proof systems,
Propositional_logic
Study of the semantics, or interpretations, of formal and natural languages
logic, the semantics or formal semantics is the study of the meaning and interpretation of formal languages, formal systems, and (idealizations of) natural
Semantics_(logic)
Expression which is not assigned an interpretation
meaning within a specific formal system. Attempting to assign or use an undefined value within a particular formal system, may produce contradictory
Undefined_(mathematics)
Historic category for ships
to the number of their carriage-mounted guns. The rating system of the Royal Navy formally came to an end in the late 19th century by declaration of
Rating system of the Royal Navy
Rating_system_of_the_Royal_Navy
System which describes the computational effects of computer programs
an effect system is a formal system that describes the computational effects of computer programs, such as side effects. An effect system can be used
Effect_system
Rules to verify computer program correctness
Hoare logic (also known as Floyd–Hoare logic or Hoare rules) is a formal system with a set of logical rules for reasoning rigorously about the correctness
Hoare_logic
Money sent to Bangladesh
the Hundi system are the absence of any transaction charges, its fast delivery and the opportunity to maintain confidentiality. While the formal methods
Remittances_to_Bangladesh
Fundamental theorem in mathematical logic
models, and proof theory, which studies what can be formally proven in particular formal systems. It was first proved by Kurt Gödel in 1929. It was then
Gödel's_completeness_theorem
Statement that is taken to be true
arithmetic because it is satisfied by the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no known
Axiom
Supposition or system of ideas intended to explain something
derived deductively from axioms (basic assumptions) according to a formal system of rules, sometimes as an end in itself and sometimes as a first step
Theory
Base set of symbols with which a language is formed
In formal language theory, an alphabet, often called a vocabulary in the context of terminal and nonterminal symbols, is a non-empty set of indivisible
Alphabet_(formal_languages)
Software research and information organisation
Formal Methods Europe (FME) is an organization whose aim is to encourage the research and application of formal methods for the improvement of software
Formal_Methods_Europe
Mathematical study of the meaning of programming languages
variations in formal semantic systems arise from the choice of supporting mathematical formalism.[citation needed] Some variations of formal semantics include
Semantics (programming languages)
Semantics_(programming_languages)
Theorem that arithmetical truth cannot be defined in arithmetic
any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. In 1931, Kurt Gödel published
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Basic proposition or assumption
Physicists include counterintuitive concepts with reiteration. In a formal logical system—that is, a set of propositions that are consistent with one another—it
First_principle
Puzzle in Douglas Hofstadter's book "Gödel, Escher, Bach"
Bach involving a simple formal system called "MIU". Hofstadter's motivation is to contrast reasoning within a formal system (i.e., deriving theorems)
MU_puzzle
Type of logical system
logic, predicate calculus, or quantificational logic, is a type of formal system. First-order logic uses quantified variables over non-logical objects
First-order_logic
Category of learning situation
structured formal educational environments. Nonformal learning is frequently used to address educational needs that formal education systems may not fully
Nonformal_learning
Study of the scope and nature of logic
philosophy of logic as the discipline investigating the properties of formal logical systems, like consistency and completeness. Various characterizations of
Philosophy_of_logic
Civil service examination system in Imperial China
Starting with the Song dynasty, the imperial examination system became a more formal system and developed into a roughly three-tiered ladder from local
Imperial_examination
Formal study of linguistic meaning
by the meanings of its parts. Propositional and predicate logic are formal systems used to analyze the semantic structure of sentences. They introduce
Formal semantics (natural language)
Formal_semantics_(natural_language)
Mathematical logician and philosopher
incompleteness theorems address limitations of formal axiomatic systems. In particular, they imply that a formal axiomatic system satisfying certain technical conditions
Kurt_Gödel
Programming paradigm based on applying and composing functions
more suited to formal verification. Functional programming has its roots in academia, evolving from the lambda calculus, a formal system of computation
Functional_programming
Concept in mathematical logic
inconsistent. Another common definition, that is equivalent if the formal system satisfies the principle of explosion, requires instead that either φ
Complete_theory
Type of formal logic
modal logic, since what ought to be true can be false. Modal logics are formal systems that include unary operators such as ◊ {\displaystyle \Diamond } and
Modal_logic
Modeling language
Universal Systems Language (USL) is a systems modeling language and formal method for the specification and design of software and other complex systems. It
Universal_Systems_Language
Form of role-playing game for leisure
characterization, and the actions succeed or fail according to a set formal system of rules and guidelines, usually involving randomization (such as through
Tabletop_role-playing_game
Transmission of knowledge and skills
outside the formal schooling system, while informal education is unstructured learning through daily experiences. Formal and non-formal education are
Education
In philosophy, the term formal ontology is used to refer to an ontology defined by axioms in a formal language with the goal to provide an unbiased (domain-
Formal_ontology
Assumption that what is not known to be true is false
The closed-world assumption (CWA), in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is
Closed-world_assumption
1985 book by Robert Rosen
that natural systems, physical things in the world, are modeled by formal systems, which are at their heart mathematical. These formal systems simulate the
Anticipatory_Systems
Quality of an algorithm being correct with respect to a specification
in this way is called program extraction. Hoare logic is a specific formal system for reasoning rigorously about the correctness of computer programs
Correctness (computer science)
Correctness_(computer_science)
1895 allegorical dialogue by Lewis Carroll
regress. However, if a formal system is introduced whereby modus ponens is simply a rule of inference defined within the system, then it can be abided
What the Tortoise Said to Achilles
What_the_Tortoise_Said_to_Achilles
Philippine vocational and skills authority
fields. TVET is classified into two main systems: the formal system and the non-formal system. The formal system is a post-secondary technical education
Technical Education and Skills Development Authority
Technical_Education_and_Skills_Development_Authority
In logic, a statement which is always true
(B\to C))} . Whether a given formula is a tautology depends on the formal system of logic that is in use. For example, the following formula is a tautology
Tautology_(logic)
Ancient Sanskrit grammarian
first context-sensitive formal model of language", showing "many features of a formal, computationally implementable system" comparable to the modern
Pāṇini
Social structure that governs how people work together
to integrate interests, goals, methods, and evaluation systems of both the informal and formal organizations, resulting in greater productivity and satisfaction
Informal_organization
Number words used in the Japanese language
reading) and a native Japanese reading (Kun reading) used somewhat less formally for numbers up to 10. In some cases (listed below) the Japanese reading
Japanese_numerals
1931 paper by Kurt Gödel
"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica
On Formally Undecidable Propositions of Principia Mathematica and Related Systems
On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems
Mathematical-logic system
mathematical logic, the lambda calculus (also written as λ-calculus) is a formal system for expressing computation based on function abstraction and application
Lambda_calculus
American mathematician and logician (1919–2017)
published in 1957. The other was later published in his 1961 book Theory of Formal Systems. While still a student at the University of Chicago, on the basis of
Raymond_Smullyan
Measure of algorithmic complexity
procedure tries every proof until it finds a string and a proof in the formal system S of the formula K ( s ) ≥ L {\displaystyle K(s)\geq L} for some L ≥
Kolmogorov_complexity
Formal system for transcribing expressions into equivalent terms
Jean-Pierre Jouannaud Rewrite Systems, Chapter 6 in Jan van Leeuwen (Ed.), Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics, Elsevier
Abstract_rewriting_system
Structured system of communication
communication systems such as formally defined computer languages used for computer programming. Unlike conventional human languages, a formal language in
Language
Concept that is not defined in terms of previously defined concepts
In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously defined concepts. It
Primitive_notion
Function in mathematical logic
Gödel used a system based on prime factorization. He first assigned a unique natural number to each basic symbol in the formal language of arithmetic
Gödel_numbering
System of formal deduction in logic
Hilbert–Ackermann system, is a type of formal proof system attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied
Hilbert_system
Mathematical theory of data types
logic, and theoretical computer science, type theory is the study of formal systems that classify expressions or mathematical objects by their types. Roughly
Type_theory
German mathematician (1862–1943)
uncertainties, ended in failure. Gödel demonstrated that any consistent formal system that is sufficiently powerful to express basic arithmetic cannot prove
David_Hilbert
Formal system for describing qualifications
A national qualifications framework (NQF) is a formal system describing qualifications. 47 countries participating in the Bologna Process are committed
National qualifications framework
National_qualifications_framework
Set of sentences in a formal language
logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context
Theory_(mathematical_logic)
1938 doctoral thesis by Alan Turing
The thesis is an exploration of formal mathematical systems after Gödel's theorem. Gödel showed that for any formal system S powerful enough to represent
Systems of Logic Based on Ordinals
Systems_of_Logic_Based_on_Ordinals
Overview of and topical guide to logic
the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope
Outline_of_logic
Topics referred to by the same term
not bound to an identifier. Lambda expression in lambda calculus, a formal system in mathematical logic and computer science for expressing computation
Lambda_expression
Formal language used to prove statements
proof system is built to prove statements. A proof system includes the components: Formal language: The set L of formulas admitted by the system, for example
Proof_calculus
Series of interconnected leagues
parts of the country which are not officially part of the system as they do not have formal agreements with other leagues, but are recognised at various
English football league system
English_football_league_system
Claim that human mathematicians are not describable as formal proof systems
Gödel's first incompleteness theorem shows that for any consistent formal system F {\displaystyle F} that allows certain arithmetic operations, there
Penrose–Lucas_argument
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