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Characteristic of some logical systems
example, Gödel's completeness theorem establishes semantic completeness for first-order logic. A formal system S is strongly complete or complete in the strong
Completeness_(logic)
Concept in mathematical logic
completeness means that every possible logic gate can be realized as a network of gates of the types prescribed by the set. In particular, all logic gates
Functional_completeness
Fundamental theorem in mathematical logic
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability
Gödel's_completeness_theorem
Topics referred to by the same term
up completeness, complete, completed, or incompleteness in Wiktionary, the free dictionary. Complete may refer to: Completeness (logic) Completeness of
Completeness
Subfield of mathematics
proved the completeness theorem, which establishes a correspondence between syntax and semantics in first-order logic. Gödel used the completeness theorem
Mathematical_logic
Overview of and topical guide to logic
Classical logic Computability logic Deontic logic Dependence logic Description logic Deviant logic Doxastic logic Epistemic logic First-order logic Formal
Outline_of_logic
Device performing a Boolean function
(FPGA) Flip-flop (electronics) Functional completeness Integrated injection logic Karnaugh map Combinational logic List of 4000 series integrated circuits
Logic_gate
Logical gate whose output is false if all its inputs are true
functional completeness. NAND gates with two or more inputs are available as integrated circuits in transistor–transistor logic, CMOS, and other logic families
NAND_gate
Type of logical system
first-order and higher-order logic was not well understood until metalogical ideas and results arrived, such as Gödel's completeness theorem in 1929. By the
First-order_logic
Topics referred to by the same term
work An incomplete formal system, see Completeness (logic) Gödel's incompleteness theorems, a specification of logic "Incomplete" (Bad Religion song), 1994
Incomplete
System of logic in mathematics and philosophy
all MV-algebras (general completeness) A {\displaystyle A} is valid in all linearly ordered MV-algebras (linear completeness) A {\displaystyle A} is valid
Łukasiewicz_logic
Study of correct reasoning
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical
Logic
Logic constructed only from NAND gates
structures and chip deposition geometries that produce NAND logic elements Functional completeness NOR logic – like NAND gates, NOR gates are also universal gates
NAND_logic
Binary operation that is true if and only if both operands are false
propositional logic are: Bitwise NOR Boolean algebra Boolean domain Boolean function Functional completeness NOR gate Propositional logic Sole sufficient
Logical_NOR
Form of logic that allows quantification over predicates
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic
Second-order_logic
Logic that allows infinitely long proofs
compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logics. Therefore for infinitary logics, notions of
Infinitary_logic
Inference rule in logic, proof theory, and automated theorem proving
refutation completeness. The clause produced by a resolution rule is sometimes called a resolvent. The resolution rule in propositional logic is a single
Resolution_(logic)
Whether a decision problem has an effective method to derive the answer
logic where Gödel's completeness theorem establishes the equivalence of semantic and syntactic consequence. In other settings, such as linear logic,
Decidability_(logic)
Various systems of symbolic logic
was proved complete by Bob Constable, but with a different notion of completeness than classically. Unproved statements in intuitionistic logic are not given
Intuitionistic_logic
American scientist (1839–1914)
contributions to logic, such as theories of relations and quantification. C. I. Lewis wrote, "The contributions of C. S. Peirce to symbolic logic are more numerous
Charles_Sanders_Peirce
AND and OR logic with diodes and resistors
transistors in diode–transistor logic) is additionally required to provide logical inversion (NOT) for functional completeness and amplification for voltage
Diode_logic
Non-contradiction of a theory
theory formulated in a particular deductive logic, the logic is called complete.[citation needed] The completeness of the propositional calculus was proved
Consistency
Ability of a computing system to simulate Turing machines
able to recognize or decode other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule
Turing_completeness
Rules used for constructing, or transforming the symbols and words of a language
Handbook of Mathematical Logic. Elsevier Science. p. 236. ISBN 9780080933641. Retrieved 2014-10-15. "syntactic completeness from FOLDOC". swif.uniba.it
Syntax_(logic)
Approach to logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to
Term_logic
Family of modal logics that extend provability logic
The completeness of TOL with respect to its arithmetical interpretation was proven by Giorgi Japaridze. Giorgi Japaridze and Dick de Jongh, The Logic of
Interpretability_logic
Propositional calculus in which there are more than two truth values
proven that way. Functional completeness is a term used to describe a special property of finite logics and algebras. A logic's set of connectives is said
Many-valued_logic
Mathematical logician and philosopher
Logik (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: "Are the axioms of a formal
Kurt_Gödel
Concept in mathematical logic
sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness. Complete theories are closed under a number of conditions
Complete_theory
System of formal deduction in logic
Type Deductive System for Sentential Logic, Completeness and Compactness" (PDF). Farmer, W. M. "Propositional logic" (PDF). It describes (among others)
Hilbert_system
Formal semantics for non-classical logic systems
its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose
Kripke_semantics
In logic, a statement which is always true
In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms
Tautology_(logic)
Type of formal logic
Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fields it is used as a tool for
Modal_logic
Type of diagrammatic notation for propositional logic
expressions, created by Charles Sanders Peirce, who wrote on graphical logic as early as 1882, and continued to develop the method until his death in
Existential_graph
Statement that is taken to be true
interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system. Note that "completeness" has a different
Axiom
Term in logic and deductive reasoning
of mathematical logic. The soundness property provides the initial reason for counting a logical system as desirable. The completeness property means that
Soundness
Branch of logic
Propositional logic is a branch of classical logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes
Propositional_logic
Method of deriving conclusions
of deriving conclusions from premises. They are integral parts of formal logic, serving as the logical structure of valid arguments. If an argument with
Rule_of_inference
Study of the scope and nature of logic
like consistency and completeness. Various characterizations of the nature of logic are found in the academic literature. Logic is often seen as the study
Philosophy_of_logic
Limitative results in mathematical logic
with semantic completeness, which means that the set of axioms proves all the semantic tautologies of the given language. In his completeness theorem (not
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
System of resource-aware logic
Linear logic is a substructural logic proposed by French logician Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the
Linear_logic
System including an indeterminate value
three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which
Three-valued_logic
Learning logic programs from data
logical entailment: Completeness: B ∪ H ⊨ E + Consistency: B ∪ H ∪ E − ⊭ false {\displaystyle {\begin{array}{llll}{\text{Completeness:}}&B\cup H&\models
Inductive_logic_programming
Mathematical model for deduction or proof systems
sufficiently powerful to express basic arithmetic cannot prove its own completeness. This effectively showed that Hilbert's program was impossible as stated
Formal_system
Class of formal logics
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had
Classical_logic
Axiom used in logic and philosophy
In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional
Peirce's_law
Programming paradigm based on formal logic
Logic programming is a programming, database, and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical
Logic_programming
Subfield of automated reasoning and mathematical logic
more systematic algorithms achieved, at least theoretically, completeness for first-order logic. Initial approaches relied on the results of Herbrand and
Automated_theorem_proving
Impossible task in computing
impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can
Entscheidungsproblem
Basic framework of mathematics
further axioms to add to set theory. Gödel's completeness theorem establishes an equivalence in first-order logic between the formal provability of a formula
Foundations_of_mathematics
Algebraic Geometry= Geometric Logic". Proc. Logic Colloquium Bristol 1973. Amsterdam: North-Holland. pp. 135–156. "Deligne completeness theorem in nLab". v t
Deligne's completeness theorem
Deligne's_completeness_theorem
Peirce's understanding of logic as formal semiotic. By "logic" he meant philosophical logic. He eventually divided (philosophical) logic, or formal semiotics
Semiotic theory of Charles Sanders Peirce
Semiotic_theory_of_Charles_Sanders_Peirce
Symbol representing a property or relation in logic
In logic, a predicate is a non-logical symbol that represents a property or a relation, though, formally, does not need to represent anything at all.
Predicate_(logic)
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India
History_of_logic
Philosophical concept
pragmatism), and other three-way distinctions in Peirce's work. In Aristotle's logic, categories are adjuncts to reasoning that are designed to resolve equivocations:
Categories_(Peirce)
Set of sentences in a formal language
logic, the most important case, it follows from the completeness theorem that the two meanings coincide. In other logics, such as second-order logic,
Theory_(mathematical_logic)
"Topological completeness of provability logic GLP". Annals of Pure and Applied Logic 164 (2013), pp. 1201–1223. G. Boolos, "The analytical completeness of Japaridze's
Japaridze's_polymodal_logic
American philosopher and logician (1940–2022)
its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose
Saul_Kripke
Application of logical methods to philosophical problems
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often
Philosophical_logic
Formal system of logic
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers
Higher-order_logic
American rapper and singer (born 1990)
Robert Bryson Hall II (born January 22, 1990), known professionally as Logic, is an American rapper, singer, songwriter, and record producer from Gaithersburg
Logic_(rapper)
Area of mathematical logic
higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not in general hold for these logics. This is made
Model_theory
Logical formalism using combinators instead of variables
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell
Combinatory_logic
Branch of pragmatic philosophy
view from other pragmatisms by its commitments to the spirit of strict logic, the immutability of truth, the reality of infinity, and the difference
Pragmaticism
American mathematician
proof of the completeness of the theory of types, which he was able to adapt to also give a new proof of the completeness of first-order logic. These results
Leon_Henkin
Mathematical use of "for all" and "there exists"
is the notation of Kurt Gödel's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic
Quantifier_(logic)
Digital audio workstation
Notator Logic, or Logic, by German software developer C-Lab which later went by Emagic. Apple acquired Emagic in 2002 and rebranded Logic to Logic Pro, adding
Logic_Pro
Logical incompatibility between two or more propositions
In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect
Contradiction
1956 computer program written by Allen Newell, Herbert A. Simon and Cliff Shaw
Logic Theorist is a computer program completed in 1956 by Allen Newell, Herbert A. Simon, and Cliff Shaw. It was the first program deliberately engineered
Logic_Theorist
Norwegian mathematician
though they prove the existence of uncountable sets. The completeness of first-order logic is a corollary of results Skolem proved in the early 1920s
Thoralf_Skolem
School of thought in philosophy of mathematics
is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and
Logicism
Logical operation
functional completeness. In 1913, Sheffer described non-disjunction using ∣ {\displaystyle \mid } and showed its functional completeness. Sheffer also
Sheffer_stroke
Trial and error problem solvers with a metaheuristic or stochastic optimization character
biological organisms are fundamentally incomplete and undecidable (completeness (logic)), implying that “there is more than a crude metaphor behind the
Evolutionary_computation
published as an article in 1930, titled "The completeness of the axioms of the functional calculus of logic" (in German)) is not easy to read today; it
Original proof of Gödel's completeness theorem
Original_proof_of_Gödel's_completeness_theorem
Second-order logic Soundness theorem Gödel's completeness theorem Original proof of Gödel's completeness theorem Compactness theorem Löwenheim–Skolem
List of mathematical logic topics
List_of_mathematical_logic_topics
Computer programming language
combinatory logic (BCL) is a computer programming language that uses binary terms 0 and 1 to create a complete formulation of combinatory logic using only
Binary_combinatory_logic
Study of the properties of logical systems
of diagonalization. Major completeness or incompleteness results include: Completeness of truth-functional propositional logic (Paul Bernays 1918), (Emil
Metalogic
Paradox in set theory
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician
Russell's_paradox
Digital circuit without clock cycles
Asynchronous circuit (clockless or self-timed circuit) is a sequential digital logic circuit that does not use a global clock circuit or signal generator to
Asynchronous_circuit
Type of decision problem in computer science
formulas as the basis for PSPACE-completeness proofs of many other problems in this area, involve nondeterministic constraint logic, in which the states are orientations
PSPACE-complete
System for reasoning about vagueness
Łukasziewicz fuzzy logic. A generalization of the classical Gödel completeness theorem is provable in EVŁ. Similar to the way predicate logic is created from
Fuzzy_logic
Work by Georg Wilhelm Friedrich Hegel
Science of Logic (German: Wissenschaft der Logik), first published between 1812 and 1816, is the work in which Georg Wilhelm Friedrich Hegel outlined
Science_of_Logic
Input value for which an existential statement of a function is true
2002, Computability and Logic: Fourth Edition, Cambridge University Press, ISBN 0-521-00758-5. Leon Henkin, 1949, "The completeness of the first-order functional
Witness_(mathematics)
Existence of values making formula true
in fact is equivalent to consistency for first-order logic, a result known as Gödel's completeness theorem. The negation of satisfiability is unsatisfiability
Satisfiability
Algebraic manipulation of "true" and "false"
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the
Boolean_algebra
Approach to the semantics of logic that locates meaning in inferential role
Gheorghiu's first-order result completes this line by establishing completeness for full first-order classical logic by native proof-theoretic means
Proof-theoretic_semantics
Argument whose conclusion must be true if its premises are
In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true
Validity_(logic)
Number representing a continuous quantity
structures have a notion of completeness; the description in § Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather
Real_number
The term completeness as applied to knowledge bases refers to two different concepts. In formal logic, a knowledge base KB is complete if there is no
Completeness (knowledge bases)
Completeness_(knowledge_bases)
1974 studio album by Steely Dan
Pretzel Logic is the third studio album by American rock band Steely Dan, released on February 20, 1974, by ABC Records. It was recorded at the Village
Pretzel_Logic
Family of logics for natural-language and counterfactual conditionals
are often related by soundness and completeness theorems to the underlying semantic frameworks. Conditional logics are also closely linked to nonmonotonic
Conditional_logic
Approach to formal semantics
and full completeness for multiplicative linear logic. Journal of Symbolic Logic 59 (1994): 543-574. A. Blass, A game semantics for linear logic. Annals
Game_semantics
ISBN 0-19-853779-4 P. Cintula and G. Metcalfe, Structural completeness in fuzzy logics, Notre Dame Journal of Formal Logic 50 (2009), no. 2, pp. 153–182. doi:10.1215/00294527-2009-004
Admissible_rule
numbers 0 and 1. The logic MTL is complete with respect to standard MTL-algebras; this fact is expressed by the standard completeness theorem (Jenei & Montagna
Monoidal_t-norm_logic
Ancient philosophy
traditionally divided into three interconnected disciplines: logic, physics, and ethics. Stoic logic focuses on highly intentional reasoning through propositions
Stoicism
In model theory, a subfield of mathematical logic, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula
Atomic model (mathematical logic)
Atomic_model_(mathematical_logic)
3-volume treatise on mathematics, 1910–1913
extension of it is complete and consistent. In 1930 Gödel's completeness theorem showed that first-order predicate logic itself was complete in a much weaker
Principia_Mathematica
Making other gates using just NOR gates
logic — Like NOR gates, NAND gates are also universal gates. Functional completeness Storr, Wayne (2013-08-21). "Logic NOR Gate Tutorial with Logic NOR
NOR_logic
Theorem in mathematical logic
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model
Compactness_theorem
Certain kind of modal formula
Correspondence and completeness in the first- and second-order semantics for modal logic. In Proceedings of the Third Scandinavian Logic Symposium. North-Holland
Sahlqvist_formula
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