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Type of mathematical variable
In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not
Predicate_variable
Symbol representing a property or relation in logic
truth. Free variables and bound variables Hypostatic abstraction Multigrade predicate Opaque predicate Philosophical predication Predicate functor logic
Predicate_(logic)
Type of logical system
called predicate logic, predicate calculus, or quantificational logic, is a type of formal system. First-order logic uses quantified variables over non-logical
First-order_logic
Variable that can either be true or false
{\displaystyle \gamma } . Propositional variables with no object variables such as x and y attached to predicate letters such as Px and xRy, having instead
Propositional_variable
Mathematical use of "for all"
domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable. It is usually denoted by
Universal_quantification
Mathematical use of "there exists"
by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"), read
Existential_quantification
Form of logic that allows quantification over predicates
notable that while we have variables for predicates in second-order logic, we don't have variables for properties of predicates. We cannot say, for example
Second-order_logic
Mathematical theory
defined as Predicate symbols F {\displaystyle F} , G {\displaystyle G} , etc. (with appropriate arities, which are left implicit) Singular variable symbols
Plural_quantification
Being present, not nothing
scope of an existential quantifier is true of at least one value of a predicate variable. Eli Hirsch, Quantifier Variance and Realism: Essays in Metaontology
Something_(concept)
Algebraization of first-order logic
i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors (or predicate modifiers) that operate on
Predicate_functor_logic
Syntactically correct logical formula
In mathematical logic, propositional logic, and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence
Well-formed_formula
Symbol representing a mathematical concept
symbols.) This schema states (in one form), for any functional predicate F in one variable: ∀ A , ∃ B , ∀ C , C ∈ A → F ( C ) ∈ B . {\displaystyle \forall
Function_symbol
Database model
corresponds to a predicate variable; the contents of a table to a relation; key constraints, other constraints, and SQL queries correspond to predicates. However
Relational_model
Symbol representing a mathematical object
that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable. The values a variable can take
Variable_(mathematics)
Overview of and topical guide to logic
Monadic predicate calculus Predicate (mathematical logic) Predicate logic Predicate variable Quantification Second-order predicate Sentence (mathematical
Outline_of_logic
In logic, a statement which is always true
sentences of predicate logic that can be reduced to propositional tautologies by substitution. Propositional logic begins with propositional variables, atomic
Tautology_(logic)
Reformulation of Floyd-Hoare logic
Predicate transformer semantics were introduced by Edsger Dijkstra in his seminal paper "Guarded commands, nondeterminacy and formal derivation of programs"
Predicate transformer semantics
Predicate_transformer_semantics
Mathematical logic concept
formal expression that denotes an atomic formula. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being
Atomic_formula
3-volume treatise on mathematics, 1910–1913
"⊃"), "&" (and), "∨" (or), "¬" (not), "∀" (for all), "∃" (there exists); predicate symbol: "=" (equals); function symbols: "+" (arithmetic addition), "∙"
Principia_Mathematica
Theory of truth in the philosophy of language
used in his incompleteness theorems. Roughly, this states that a truth-predicate satisfying Convention T for the sentences of a given language cannot be
Semantic_theory_of_truth
Fragment of first-order logic
logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic (also called predicate calculus) in which
Monadic_predicate_calculus
Assignment of meaning to the symbols of a formal language
variables. Then, terms can be combined into an atomic formula using a predicate symbol (relation symbol) from the signature or the special predicate symbol
Interpretation_(logic)
Index of articles associated with the same name
mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical
Stratification_(mathematics)
syntactic predicate specifies the syntactic validity of applying a production in a formal grammar and is analogous to a semantic predicate that specifies
Syntactic_predicate
Term that does not contain any variables
operators, and P {\displaystyle P} the set of predicate symbols. A ground term is a term that contains no variables. Ground terms may be defined by logical
Ground_expression
System of formal deduction in logic
ponens, for propositional logics – or two – with generalisation, to handle predicate logics, as well – and several infinite axiom schemas. Hilbert systems
Hilbert_system
Term in logic
be predicate letters; let a, b, c be individual constants; let x, y, z be variables. These wffs are atomic sentences; they contain no free variables or
Atomic_sentence
Form of second-order logic
predicate variables only. In the variant considered in automata theory and the Büchi–Elgot–Trakhtenbrot theorem, all predicates, constant or variable
Monadic_second-order_logic
Logical statement with variables, predicates, and quantifiers over objects
first-order predicate is a predicate that takes only individual(s) constants or variables as argument(s). Compare second-order predicate and higher-order
First-order_predicate
Mathematical use of "for all" and "there exists"
"quantifier rank". If D is a domain of x and P(x) is a predicate dependent on object variable x, then the universal proposition can be expressed as ∀
Quantifier_(logic)
Object or module in concurrent programming
precondition that our predicate // must be true. // This code might make cv's condition false, and/or make other condition variables' // predicates true. // Call
Monitor_(synchronization)
Mathematical-logic system
expressing computation based on function abstraction and application using variable binding and substitution. Untyped lambda calculus, the topic of this article
Lambda_calculus
of a pointer variable is dependent on the underlying computer architecture. postcondition In computer programming, a condition or predicate that must always
Glossary_of_computer_science
Axiomatic set theories based on the principles of mathematical constructivism
or predicate variable in axiom schemas and P {\displaystyle P} or Q {\displaystyle Q} is used for particular such predicates. The word "predicate" is
Constructive_set_theory
Use of braces for specifying sets
set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the separator, and
Set-builder_notation
Non-monotonic logic created by John McCarthy
propositional variable with a predicate of zero arity (i.e., a predicate with no arguments). Therefore, minimization is done on predicates in the first-order
Circumscription_(logic)
Rule defining the correct structure of expressions in formal grammar
and α as a variable then we can take ( ∀ {\displaystyle \forall } α)Φ and ( ∃ {\displaystyle \exists } α)Φ each to be formulas of our predicate calculus
Formation_rule
Set of rules defining correctly structured Prolog programs
anonymous variable and means "any term". Unlike other variables, the underscore does not represent the same value everywhere it occurs within a predicate definition
Prolog_syntax_and_semantics
Number of arguments required by a function
that accepts a variable number of arguments is called variadic. In logic and philosophy, predicates or relations accepting a variable number of arguments
Arity
Variable used for specification
be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicate variables, and when defining
Parameter
all". If E is a logical predicate, ∀ x E {\displaystyle \forall x\;E} means that E is true for all possible values of the variable x. 2. Often used in plain
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Basic notion of sameness in mathematics
through set theory. In logic, equality is a primitive predicate (a statement that may have free variables) with the reflexive property (called the law of identity)
Equality_(mathematics)
Measure of the structural complexity of a software program
compound predicates like those found in high-level languages like IF cond1 AND cond2 THEN ... should be counted in terms of predicate variables involved
Cyclomatic_complexity
Mathematical logic concept
to the predicate of the inferred proposition, it is permissible that it could be the original subject or its contradictory, and the predicate term of
Contraposition
Approach to logic
with the advent of new logic, remaining dominant until the advent of predicate logic in the late nineteenth century. However, even if eclipsed by newer
Term_logic
Statement that is taken to be true
sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical
Axiom
Set of tuples consisting of values indexed by attributes
extension of some n-adic predicate: all and only those n-tuples whose values, substituted for corresponding free variables in the predicate, yield propositions
Relation_(database)
Study of correct reasoning
Q(John))} ". In this case, the existential quantifier is applied to the predicate variable " Q {\displaystyle Q} ". The added expressive power is especially
Logic
Problem in computer science
we can read a definite answer, 'Yes' or 'No,' to the question, 'Is the predicate value true?'." 1952 (1952): Kleene includes a discussion of the unsolvability
Halting_problem
Formal system of logic
term "higher-order logic" is commonly used to mean higher-order simple predicate logic. Here, "simple" indicates that the underlying type theory is the
Higher-order_logic
Concept in mathematics or computer science
science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for free variable and bound
Free variables and bound variables
Free_variables_and_bound_variables
Mathematical function characterizing set membership
Laplacian of the indicator Dirac delta Extension (predicate logic) Free variables and bound variables Heaviside step function Identity function Iverson
Indicator_function
Variable that stores data about other variables or program structure
In logic, a metavariable (also metalinguistic variable or syntactical variable) is a symbol or symbol string which belongs to a metalanguage and stands
Metavariable
Type of logical argument that applies deductive reasoning
some academic contexts, syllogism has been superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift
Syllogism
Impossible task in computing
15), thus undecidable. The monadic predicate calculus is the fragment where each formula contains only 1-ary predicates and no function symbols. Its S a
Entscheidungsproblem
Function computable with bounded loops
primitive recursive in ψ. #C: A predicate P obtained by substituting functions χ1,..., χm for the respective variables of a predicate Q is primitive recursive
Primitive_recursive_function
Kind of transfinite induction
stated in terms of a negated predicate ¬ S {\displaystyle \neg S} is then just as strong as one in terms of a predicate variable P {\displaystyle P} , as
Epsilon-induction
In mathematical logic, a well-formed formula with no free variables
a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing
Sentence_(mathematical_logic)
Logical formulation of recursion
formulas formed from X using first-order connectives and predicates, second-order variables as well as a partial fixed point operator PFP {\displaystyle
Fixed-point_logic
Formula that contains at least one free variable
An open formula is a formula that contains at least one free variable. An open formula does not have a truth value assigned to it, in contrast with a closed
Open_formula
Graphical method to simplify Boolean expressions
(2004) [2003-11-05]. "Karnaugh Maps". Switching Theory: Insight Through Predicate Logic. Berlin, Heidelberg, New York: Springer-Verlag. pp. 57–76. ISBN 3-540-40343-4
Karnaugh_map
Paradox in set theory
y\forall x(x\in y\iff \varphi (x))} for any predicate φ {\displaystyle \varphi } with x as a free variable inside φ {\displaystyle \varphi } . Substitute
Russell's_paradox
Logical formalism using combinators instead of variables
of quantified variables in logic, essentially by eliminating them. Another way of eliminating quantified variables is Quine's predicate functor logic
Combinatory_logic
{\displaystyle m} are fixed constants and x {\displaystyle x} is a variable. A predicate is regular if and only if it can be defined in the language of congruence
Regular_numerical_predicate
Logic formula
discourse) and (ii) a predicate (a verb or possibly verb-clause that asserts a quality or attribute of the object(s)). The predicate calculus then generalizes
Propositional_formula
Topological model
When testing two geometries against a scheme, the result is a spatial predicate named by the scheme. The model was developed by Clementini and others
DE-9IM
Limitative results in mathematical logic
to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Programming language that uses first order logic
and higher-order programming. A higher-order predicate is a predicate that takes one or more other predicates as arguments. Although support for higher-order
Prolog
Constructed language
verbs, adjectives and adverbs. A predicate may act as any of these, depending on its position in a sentence. Each predicate has its own argument structure
Loglan
Model that describes the programmable interface of a computer processor
instruction sets include a predicate field in every instruction. Having predicates on instructions is called predication, and can include conditional-branches
Instruction_set_architecture
Logic for proving computer program correctness
program state. The assertions are logical statements—predicates with variables, where the variables define the state of the program. Algebraic semantics
Axiomatic_semantics
ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃
Mathematical_object
French psychoanalyst and writer (1901–1981)
linguistics and anthropology to his own work, which he augmented with predicate logic and topology. Taking this new direction, and introducing controversial
Jacques_Lacan
Words joined with underscores
identifiers. Prolog, for both atoms (predicate names, function names, and constants) and variables Python, for variable names, function names, method names
Snake_case
Input to a mathematical function
provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x , y ) = x 2 + y 2 {\displaystyle
Argument_of_a_function
contains at most the shown free variables, where X is again a unary (set) variable, and Y is a fresh binary predicate variable. We write x ∈ J A μ {\displaystyle
Theories of iterated inductive definitions
Theories_of_iterated_inductive_definitions
Algebraic manipulation of "true" and "false"
values of the variables are the truth values true and false, usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers
Boolean_algebra
Subfield of automated reasoning and mathematical logic
both a complete propositional calculus and what is essentially modern predicate logic. His Foundations of Arithmetic, published in 1884, expressed (parts
Automated_theorem_proving
Standard system of axiomatic set theory
common. The signature has a single predicate symbol, usually denoted ∈ {\displaystyle \in } , which is a predicate symbol of arity 2 (a binary relation
Zermelo–Fraenkel_set_theory
Expression in propositional calculus
or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that
Propositional_function
Axioms for the natural numbers
induction axiom is sometimes stated in the following form: If φ is a unary predicate such that: φ(0) is true, and for every natural number n, φ(n) being true
Peano_axioms
Theorem for proving more complex theorems
ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃
Lemma_(mathematics)
Programming paradigm based on formal logic
function fibonacci(N) = M, and the predicate N is Expression is Prolog notation for the predicate that instantiates the variable N to the value of Expression
Logic_programming
Area of mathematical logic
formula in one variable. Quantifier-free formulas in one variable express Boolean combinations of polynomial equations in one variable, and since a nontrivial
Model_theory
Theorem that arithmetical truth cannot be defined in arithmetic
metalanguage capable of expressing the semantics of some object language (e.g. a predicate is definable in Zermelo–Fraenkel set theory for whether formulae in the
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Components of a mathematical or logical formula
constructed from constant symbols, variable symbols, and function symbols. An expression formed by applying a predicate symbol to an appropriate number of
Term_(logic)
Sequence of words formed by specific rules
contains infinitely many elements x0, x1, x2, … that play the role of variables. See e.g. Reghizzi, Stefano Crespi (2009). Formal Languages and Compilation
Formal_language
Axiom of set theory
ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃
Axiom_of_choice
Concept in logic
certain variables into a derivation. A propositional formula is a tautology if it is true under every valuation (or interpretation) of its predicate symbols
Substitution_(logic)
Formal semantics and 1998 book
denotes the universal closure of all variables in the alphabet. The most basic UTP theory is the alphabetised predicate calculus, which has no alphabet restrictions
Unifying Theories of Programming
Unifying_Theories_of_Programming
Thesis on the nature of computability
Thesis I. Every effectively calculable function (effectively decidable predicate) is general recursive [Kleene's italics] Since a precise mathematical
Church–Turing_thesis
Knowledge representation framework
known as "properties", "predicates", "features", "dimensions", "characteristics", "fields", "headers" or "independent variables" depending on the context)
Attribute–value_system
Measure of algorithmic complexity
ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃
Kolmogorov_complexity
Form of mathematical proof
(}P(n){\bigr )}{\Bigr )},} where P(·) is a variable for predicates involving one natural number and k and n are variables for natural numbers. In words, the base
Mathematical_induction
Symbolic description of a mathematical object
syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions. Other symbols include punctuation marks and
Expression_(mathematics)
Whether a decision problem has an effective method to derive the answer
validities in any signature that includes equality and at least one other predicate symbol with two or more arguments is not decidable. Logical systems extending
Decidability_(logic)
Logical principle
ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃
Law_of_excluded_middle
Mathematical model for deduction or proof systems
ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃
Formal_system
Collection of mathematical objects
objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Sets cannot be mathematically defined
Set_(mathematics)
Logical connective OR
∨ c) → (b ∨ c)) Truth-preserving: The interpretation under which all variables are assigned a truth value of 'true', produces a truth value of 'true'
Logical_disjunction
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