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Type of logical system
first-order logic (FOL), also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system. First-order logic uses
First-order_logic
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
Formal system of logic
In mathematics and logic, a higher-order logic (HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes
Higher-order_logic
Branch of logic
zeroth-order logic. Sometimes, it is called first-order propositional logic to contrast it with System F, but it is distinct from first-order logic. It deals
Propositional_logic
Assignment of meaning to the symbols of a formal language
and first-order logic. The sentences that are made true by a particular assignment are said to be satisfied by that assignment. In classical logic, no
Interpretation_(logic)
Translation of a text into a logical system
the translation of the English sentence "some men are bald" into first-order logic as ∃ x ( M ( x ) ∧ B ( x ) ) {\displaystyle \exists x(M(x)\land B(x))}
Logic_translation
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name
List_of_logic_symbols
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
Inference rule in logic, proof theory, and automated theorem proving
theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts
Resolution_(logic)
Method of deriving conclusions
propositional logic examines how statements formed through logical operators like "not" and "if...then..." support conclusions. First-order logic extends propositional
Rule_of_inference
Set of sentences in a formal language
mathematical 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
Theory_(mathematical_logic)
Symbol representing a property or relation in logic
In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula R ( a ,
Predicate_(logic)
Subfield of mathematics
classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic. First-order logic
Mathematical_logic
Fragment of first-order logic
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic (also called predicate calculus)
Monadic_predicate_calculus
In logic, a statement which is always true
formulas of propositional logic. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from
Tautology_(logic)
Index of articles associated with the same name
self-reference", as in first-order logic and other logic uses, where it is contrasted with "allowing some self-reference" (higher-order logic) In detail, it may
First-order
Whether a decision problem has an effective method to derive the answer
effectively determined. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. A theory (set of sentences
Decidability_(logic)
Non-contradiction of a theory
formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is
Consistency
Form of second-order logic
In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification
Monadic_second-order_logic
Branch of mathematical logic
logarithmic time. First-order logic in a signature with only the order relation corresponds to the set of star-free languages. First-order logic gains substantially
Descriptive_complexity_theory
Methods in artificial intelligence research
with first-order logic, e.g., with either Markov Logic Networks or Probabilistic Soft Logic. Other, non-probabilistic extensions to first-order logic to
Symbolic artificial intelligence
Symbolic_artificial_intelligence
Mathematical logic concept
In mathematical logic and philosophy, Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an
Skolem's_paradox
Class of formal logics
specifically propositional and first-order logic, as opposed to the other forms of classical logic. Most semantics of classical logic are bivalent, meaning all
Classical_logic
Formalism for knowledge representation
three main directions: a graphical interface for first-order logic, a diagrammatic calculus of logics, and a graph-based knowledge representation and reasoning
Conceptual_graph
Study of the scope and nature of logic
of logics in contrast to one universally true logic. These logics can be divided into classical logic, usually identified with first-order logic, extended
Philosophy_of_logic
Formal language used to construct ontologies
first-order logic or on description logic. Common Logic - and its dialects CycL DOGMA (Developing Ontology-Grounded Methods and Applications) F-Logic
Ontology_language
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
Family of formal knowledge representation
more expressive than propositional logic but less expressive than first-order logic. In contrast to the latter, the core reasoning problems for DLs are
Description_logic
Components of a mathematical or logical formula
In mathematical logic, a term is an arrangement of dependent/bound symbols that denotes a mathematical object within an expression/formula. In particular
Term_(logic)
Technique in mathematical logic
translation for propositional logic, and the Gödel–Gentzen translation and Kuroda's translation for first-order logic. The easiest double-negation translation
Double-negation_translation
Application of logical methods to philosophical problems
logic, extended logics, and deviant logics. This classification is based on the idea that classical logic, i.e. propositional logic and first-order logic
Philosophical_logic
Mathematical use of "for all" and "there exists"
second-order logic or higher-order logics. Quantifiers have been generalized beginning with the work of Andrzej Mostowski and Per Lindström. In a first-order
Quantifier_(logic)
Study of correct reasoning
commonly used system is classical logic. It consists of propositional logic and first-order logic. Propositional logic only considers logical relations
Logic
Branch of logic
Gödel's completeness theorem, and the method of ultraproducts for first-order logic (FO). These invalidities all follow from Trakhtenbrot's theorem. While
Finite_model_theory
Fundamental theorem in mathematical logic
theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem
Gödel's_completeness_theorem
Problem in computer science
hence consistent) and complete effective axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that
Halting_problem
Book on mathematical logic
Extensions of First Order Logic is a book on mathematical logic. It was written by María Manzano, and published in 1996 by the Cambridge University Press
Extensions of First Order Logic
Extensions_of_First_Order_Logic
Impossible task in computing
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 be deduced
Entscheidungsproblem
Theories in mathematical logic
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model
List_of_first-order_theories
Subfield of automated reasoning and mathematical logic
theoretically, completeness for first-order logic. Initial approaches relied on the results of Herbrand and Skolem to convert a first-order formula into successively
Automated_theorem_proving
Limitative results in mathematical logic
that first-order logic is semantically complete. But it is not syntactically complete, since there are sentences expressible in the language of first-order
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Standard system of axiomatic set theory
regularity (first proposed by John von Neumann), to Zermelo set theory yields the theory ZFC. Formally, ZFC is a one-sorted theory in first-order logic. The
Zermelo–Fraenkel_set_theory
Mapping of mathematical formulas to a particular meaning
view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics. For a given
Structure (mathematical logic)
Structure_(mathematical_logic)
Modal temporal logic with modalities referring to time
called propositional temporal logic (PTL). In terms of expressive power, LTL is a fragment of first-order logic. LTL was first proposed for the formal verification
Linear_temporal_logic
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
Tool for proving a logical formula
decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed
Method_of_analytic_tableaux
Algebraization of first-order logic
In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic
Predicate_functor_logic
Logical connective AND
In logic, mathematics and linguistics, and ( ∧ {\displaystyle \wedge } ) is the truth-functional operator of conjunction or logical conjunction. The logical
Logical_conjunction
System of formal deduction in logic
These deductive systems are most often studied for first-order logic, but are of interest for other logics as well. It is defined as a deductive system that
Hilbert_system
Template that specifies one or more axioms
second-order logic. Analogously, some first-order set-theoretic schemata can be represented by quantifying over classes or higher-order objects in an
Axiom_schema
Zero-one law holds for first-order logic (without function symbols), first-order logic extended with fixed point operators and first-order with infinite disjunctions
Zero–one_law_(logic)
Mathematical term; concerning axioms used to derive theorems
In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science
Axiomatic_system
Dutch philosopher and logician
decision procedure for propositional logic, whereas they are only semi-effective for first-order logic, since first-order logic is undecidable, as showed by Church's
Evert_Willem_Beth
Framework for a family of logic languages
Common Logic (CL) is a framework for a family of logic languages, based on first-order logic, intended to facilitate the exchange and transmission of
Common_Logic
Existence of values making formula true
respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being
Satisfiability
Axioms for the natural numbers
Peano axioms, but rather as axioms of the "underlying logic". The next three axioms are first-order statements about natural numbers expressing the fundamental
Peano_axioms
System for representing and reasoning about time
anticipations of temporal logic, and may imply an early, partially developed form of first-order temporal modal bivalent logic. Aristotle was particularly
Temporal_logic
Concept in logic
soundness of the deduction rule described in the previous section. In first-order logic, a substitution is a total mapping σ: V → T from variables to terms;
Substitution_(logic)
Axiom set used in first-order logic
specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity (i.e. is formulable as an elementary theory). As such
Tarski's_axioms
Probabilistic logic
A Markov logic network (MLN) is a probabilistic logic which applies the ideas of a Markov network to first-order logic, defining probability distributions
Markov_logic_network
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
Axiom of set theory
{\displaystyle X} contains exactly one element. This can be formalized in first-order logic as: ∀ x ( ∃ e ( e ∈ x ∧ ¬ ∃ y ( y ∈ e ) ) ∨ ∃ a ∃ b ∃ c ( a ∈ x ∧
Axiom_of_choice
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
1943 paper proposing artificial neural networks
Pr(N_{1},N_{2},\dots ,N_{p},t)} where P r {\displaystyle Pr} is a first-order logic predicate function (a function that outputs a boolean), N 1 , … ,
A Logical Calculus of the Ideas Immanent in Nervous Activity
A_Logical_Calculus_of_the_Ideas_Immanent_in_Nervous_Activity
Characteristic of some logical systems
propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example, the propositional logic statement
Completeness_(logic)
Mathematical model for deduction or proof systems
which, in order to avoid confusion, are usually called metatheorems. A logical system is a deductive system (most commonly first order logic) together
Formal_system
Basic notion of sameness in mathematics
proving all its properties; this burden is now assumed by the logic. In first-order logic without equality, two sets are defined to be equal if they contain
Equality_(mathematics)
finite structures is not decidable in first-order logic. That is, the set {φ | φ is a sentence of first-order logic that is satisfied in some finite structure}
Trakhtenbrot's_theorem
Issue in artificial intelligence and categorical algebra
with using first-order logic to express facts about a robot in the world. Representing the state of a robot with traditional first-order logic requires
Frame_problem
Area of mathematical logic
for these logics. This is made concrete by Lindström's theorem, stating roughly that first-order logic is essentially the strongest logic in which both
Model_theory
Formal system in mathematical logic
characterization, first-order logic is, up to equivalence, the only abstract logic that is countably compact and has Löwenheim number ω. Abstract algebraic logic – Aspect
Abstract_logic
throughout the 1980s, a variety of logics and extensions of first-order logic were developed both for negation as failure in logic programming and for default
History of artificial intelligence
History_of_artificial_intelligence
Statement that is taken to be true
requires the use of second-order logic. The Löwenheim–Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the
Axiom
Existence and cardinality of models of logical theories
to characterize first-order logic. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic. In its general
Löwenheim–Skolem_theorem
Logic that allows infinitely long proofs
1930s. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact
Infinitary_logic
Look up Appendix:Glossary of logic in Wiktionary, the free dictionary. This is a glossary of logic. Logic is the study of the principles of valid reasoning
Glossary_of_logic
Yes-or-no question that cannot ever be solved by a computer
all true first-order logic statements about natural numbers must be false. Undecidable problems can be related to different topics, such as logic, abstract
Undecidable_problem
Reasoning about equations with free variables
nonclassical logics are typically modeled by what are called "Boolean algebras with operators." Algebraic formalisms going beyond first-order logic in at least
Algebraic_logic
Paradox in set theory
of ZFC, with the help of Thoralf Skolem, turned out to be that of first-order logic. The paradox had already been discovered independently by the German
Russell's_paradox
Logical incompatibility between two or more propositions
tell a falsehood is impossible?". In classical logic, particularly in propositional and first-order logic, a proposition φ {\displaystyle \varphi } is a
Contradiction
Standard form of Boolean function
9.5.1 Conjunctive normal form for first-order logic. Andrews, Peter B. (2013). An Introduction to Mathematical Logic and Type Theory: To Truth Through
Conjunctive_normal_form
Fundamental result of mathematical logic
mathematical logic obtained by Jacques Herbrand (1930). It essentially allows a certain kind of reduction of first-order logic to propositional logic. Herbrand's
Herbrand's_theorem
State of being real
other. Formal logic studies deductively valid arguments. In first-order logic, which is the most-commonly used system of formal logic, existence is expressed
Existence
Academic discipline
described first-order logic (FOL) as the metric by which all AI knowledge representation formalisms should be evaluated. First-order logic is a general
Logic_in_computer_science
Approach to predicate logic
Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe
Intensional_logic
to all propositional formulas with those variables. In first-order logic and higher-order logics, a structure, (the interpretation) and the corresponding
Valuation_(logic)
Rules used for constructing, or transforming the symbols and words of a language
propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example the propositional logic statement
Syntax_(logic)
System of formal mathematical logic
mathematics comparable to first-order logic plus set theory. It is a form of higher-order logic and closely related to the logics of the HOL theorem prover
Q0_(mathematical_logic)
Field of artificial intelligence
on developing automated theorem-provers for first-order logic, motivated by the use of mathematical logic to formalise mathematics and to automate the
Knowledge representation and reasoning
Knowledge_representation_and_reasoning
Index of articles associated with the same name
(graph theory) In logic, model theory and type theory: Zeroth-order logic First-order logic Second-order logic Higher-order logic Order (journal), an academic
Order_(mathematics)
Various systems of symbolic logic
logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by
Intuitionistic_logic
first-order logic. Polyadic algebras form one of the main algebraic frameworks used in algebraic logic to study the syntax and model theory of first-order
Polyadic_algebra
Controlled language
representation structures (DRS) that use a variant of the language of first-order logic. A DRS can be further translated into other formal languages, for
Attempto_Controlled_English
Mathematical set containing no elements
of empty set can be shown redundant in at least two ways: Standard first-order logic implies, merely from the logical axioms, that something exists, and
Empty_set
Hierarchical typed logic
Categorical logic First-order logic § Many-sorted logic Carlos Caleiro, Ricardo Gonçalves (2006). "On the algebraization of many-sorted logics". Proc. 18th
Many-sorted_logic
Mathematical logic concept
a given model. The well-formed terms and propositions of ordinary first-order logic have the following syntax: Terms: t ≡ c ∣ x ∣ f ( t 1 , … , t n )
Atomic_formula
Metatheorem in mathematical logic
{\displaystyle B} . Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style
Deduction_theorem
Extension of classical first-order logic
Independence-friendly logic (IF logic; proposed by Jaakko Hintikka and Gabriel Sandu [fr] in 1989) is an extension of classical first-order logic (FOL) by means
Independence-friendly_logic
Mathematical logic concept
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent
Contraposition
Group in which each element has finite order
Mathematical logic (2. ed., 4. pr. ed.). New York [u.a.]: Springer. pp. 50. ISBN 978-0-387-94258-2. Retrieved 18 July 2012. However, in first-order logic we may
Torsion_group
travel, tourism, insurance
FIRST ORDER-LOGIC
FIRST ORDER-LOGIC
Boy/Male
Arabic, Australian, Muslim
Order
Surname or Lastname
English
English : variant of Cordier.Catalan : occupational name for a maker of cord or string, from an agent derivative of Catalan corda ‘string’, ‘cord’.
Boy/Male
Greek
Order.
Boy/Male
Indian
Order, Decree
Boy/Male
Greek
Order.
Boy/Male
Greek
Order.
Boy/Male
Muslim
Order. Discipline.
Boy/Male
Tamil
Pradarsh | பà¯à®°à®¤à®°à¯à®·
Appearance, Order
Pradarsh | பà¯à®°à®¤à®°à¯à®·
Girl/Female
Indian, Telugu
Order
Male
Swedish
Old Swedish form of Old Norse Oddr, ODDER means "point of a weapon."
Girl/Female
Australian, French, German, Greek, Italian
Order
Boy/Male
Australian, French, German, Greek
Order
Boy/Male
Muslim
Order, Decree
Surname or Lastname
English
English : topographic name for someone who lived at the edge of a village or by some other boundary, Middle English border, from Old French bordure ‘edge’.
Girl/Female
German, Greek
Order
Boy/Male
English
From the Thicket of Trees
Girl/Female
Indian, Marathi, Sindhi
Order
Girl/Female
Greek
Order.
Boy/Male
Hindu, Indian, Punjabi, Sikh
Order
Girl/Female
Indian, Traditional
Order
FIRST ORDER-LOGIC
FIRST ORDER-LOGIC
FIRST ORDER-LOGIC
FIRST ORDER-LOGIC
FIRST ORDER-LOGIC
FIRST ORDER-LOGIC
FIRST ORDER-LOGIC
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