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Form of logic that allows quantification over predicates
legitimate sentence of second-order logic. Here, P is a predicate variable and is semantically a set of individuals. As a result, second-order logic has greater
Second-order_logic
Aspect of mathematical logic
a second-order predicate is a predicate that takes a first-order predicate as an argument. Compare higher-order predicate. The idea of second order predication
Second-order_predicate
Type of logical system
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system used in mathematics, philosophy
First-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 predicate
First-order_predicate
Type of mathematical variable
quantified by means of (at least) second-order quantifiers. Predicate variables should be distinguished from predicate constants, which could be represented
Predicate_variable
Symbol representing a property or relation in logic
all. For instance, in the first-order formula P ( a ) {\displaystyle P(a)} , the symbol P {\displaystyle P} is a predicate that applies to the individual
Predicate_(logic)
Subject and predicate in sentences
The term predicate is used in two ways in linguistics and its subfields. The first defines a predicate as everything in a standard declarative sentence
Predicate_(grammar)
Topics referred to by the same term
derivative is the second Second-order logic, an extension of predicate logic Second-order perturbation, in perturbation theory Second-order cybernetics, the
Second-order
Form of second-order logic
elements for which the predicate is true). Monadic second-order logic is expressively equivalent to plural logic. Monadic second-order logic comes in two
Monadic_second-order_logic
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
3-volume treatise on mathematics, 1910–1913
quasi-intensional stance to a fully extensional stance also restricts predicate logic to the second order, i.e. functions of functions: "We can decide that mathematics
Principia_Mathematica
Subfield of automated reasoning and mathematical logic
Gabbay, Dov M., and Hans Jürgen Ohlbach. "Quantifier elimination in second-order predicate logic." (1992). Howlett, Joseph. "AI just solved an 80-year-old
Automated_theorem_proving
Order of syntactic constituents
In linguistics, word order (also known as linear order) is the order of the syntactic constituents of a language. Word order typology studies it from
Word_order
Formal system of logic
those of first-order logic. The term "higher-order logic" is commonly used to mean higher-order simple predicate logic. Here, "simple" indicates that the
Higher-order_logic
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
Paradox in set theory
is a formal theory, that is formulated in a first-order language with a binary non-logical predicate ∈ {\displaystyle \in } , and that includes the axiom
Russell's_paradox
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
Reformulation of Floyd-Hoare logic
proving a first-order formula. Technically, predicate transformer semantics perform a kind of symbolic execution of statements into predicates: execution runs
Predicate transformer semantics
Predicate_transformer_semantics
Assignment of meaning to the symbols of a formal language
semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of
Interpretation_(logic)
Method of deriving conclusions
Therefore, this hill has fire." Second-order logic extends first-order logic by allowing quantifiers to apply to predicates in addition to singular terms
Rule_of_inference
Mathematical system
otherwise, the second player wins.) A set is projective if and only if (as a predicate) it is expressible by a formula in the language of second-order arithmetic
Second-order_arithmetic
System of word ordering
the default word order is VSO, in which verbs and copulas are found at the beginning of a sentence. However, the object of the predicate can be emphasized
Verb–subject–object word order
Verb–subject–object_word_order
Variable that can either be true or false
as x and y attached to predicate letters such as Px and xRy, having instead individual constants a, b, ... attached to predicate letters are propositional
Propositional_variable
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
Axioms for the natural numbers
making it weaker than the second-order axiom. The reason that it is weaker is that the number of predicates in first-order language is countable, whereas
Peano_axioms
Limitative results in mathematical logic
enumeration.) The standard proof of the second incompleteness theorem assumes that the provability predicate ProvA(P) satisfies the Hilbert–Bernays provability
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
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
Framework in lambda calculus
impredicative way in second and higher order logics. In the weak higher order logics, there are variables for higher order predicates, but no quantification
Lambda_cube
Statement that is taken to be true
tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies
Axiom
Type of logical argument that applies deductive reasoning
Within some academic contexts, syllogism has been superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift
Syllogism
Form of mathematical proof
ranges over predicates rather than over individual numbers. This is a second-order quantifier, which means that this axiom is stated in second-order logic.
Mathematical_induction
Syntactically correct logical formula
propositional logic and predicate logic. A key use of formulas is in propositional logic and predicate logic such as first-order logic. In those contexts
Well-formed_formula
Mathematical-logic system based on functions
comparison predicates of natural numbers, using recursion. When Y combinator is coded directly in a strict programming language, the applicative order of evaluation
Lambda_calculus
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
In logic, a statement which is always true
unsatisfiable). The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers—a feature absent from sentences of
Tautology_(logic)
Mathematical theory of data types
to only one type. Where a subset would be used, type theory can use a predicate function or use a dependently-typed product type, where each element x
Type_theory
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
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
Basic notion of sameness in mathematics
through logic or through set theory. In logic, equality is a primitive predicate (a statement that may have free variables) with the reflexive property
Equality_(mathematics)
Typed lambda calculus
(1974). Girard proved the representation theorem: that in second-order intuitionistic predicate logic (P2), functions from the natural numbers to the natural
System_F
Measure of algorithmic complexity
description, namely "write ab 16 times", which consists of 17 characters. The second one has no obvious simple description (using the same character set) other
Kolmogorov_complexity
Psychological theory
Cause (R, S) Each set of predicates (words like hit, broke, cry, happy are first order-predicates; Cause is a second-order predicate) and arguments (often
Propositional_representation
Axiom of set theory
choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set created by choosing
Axiom_of_choice
Fundamental theorem in mathematical logic
other).[citation needed] We first fix a deductive system of first-order predicate calculus, choosing any of the well-known equivalent systems. Gödel's
Gödel's_completeness_theorem
Theorem in formal logic
Tait, 1966. A nonconstructive proof of Gentzen's Hauptsatz for second order predicate logic. In Bulletin of the American Mathematical Society, 72:980–983
Takeuti's_conjecture
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
Mathematical use of "for all"
It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation
Universal_quantification
Mathematical set containing no elements
or if Cantor merely used ≡ O {\displaystyle \equiv O} as an emptiness predicate. Zermelo accepted O {\displaystyle O} itself as a set, but considered
Empty_set
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
Mathematical use of "there exists"
In predicate logic, an existential quantification is a type of quantifier which asserts the existence of an object with a given property. It is usually
Existential_quantification
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)
Mathematical theory
only one another also cannot be interpreted in monadic second-order logic. This is because predicates such as "are shipmates", "are meeting together", "are
Plural_quantification
Smallest grammatical unit that can express a complete proposition
predicand (expressed or not) and a semantic predicate. A typical clause consists of a subject and a syntactic predicate, the latter typically a verb phrase composed
Clause
Non-contradiction of a theory
1918[citation needed] and Emil Post in 1921, while the completeness of (first order) predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for
Consistency
Representation of data types in lambda calculus
to 3, using normal order. \f.\x.f (f (f (x))) A predicate is a function that returns a Boolean value. The most fundamental predicate on Church numerals
Church_encoding
Subfield of mathematics
/ Date incompatibility (help) Kleene, Stephen Cole (1943). "Recursive Predicates and Quantifiers". Transactions of the American Mathematical Society. 53
Mathematical_logic
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
Infinite cardinal number
ε 0 {\displaystyle \varepsilon _{0}} . For example, the sequence (with order type ω ⋅ 2 {\displaystyle \omega \cdot 2} ) of all positive odd integers
Aleph_number
Branch of mathematical logic
these can give a complete and axiomatic formalization of propositional or predicate logic of either the classical or intuitionistic flavour, almost any modal
Proof_theory
Mathematical set formed from two given sets
main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way, and extract numerical
Cartesian_product
Basic framework of mathematics
with either "if S is a set then" or "if φ {\displaystyle \varphi } is a predicate then". So, Peano's axioms induce a quantification on infinite sets, and
Foundations_of_mathematics
Pair of mathematical objects
mathematics, an ordered pair, denoted (a, b), is a pair of objects in which their order is significant. If a and b are different, then (a,b) is different from (b
Ordered_pair
Logic concept
In formal theories of truth, a truth predicate is a fundamental concept based on the sentences of a formal language as interpreted logically. That is
Truth_predicate
Diagram that shows all possible logical relations between a collection of sets
higher numbers. Shown below, four intersecting spheres form the highest order Venn diagram that has the symmetry of a simplex and can be visually represented
Venn_diagram
Logical formulation of recursion
FO[PFP](X) is the set of formulas formed from X using first-order connectives and predicates, second-order variables as well as a partial fixed point operator
Fixed-point_logic
Logical quantifier
be expressed in terms of the existential and universal quantifiers of predicate logic, by defining the formula ∃ ! x ( P ( x ) ) {\displaystyle \exists
Uniqueness_quantification
Size of a possibly infinite set
of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and ω 1 {\displaystyle \omega _{1}} is the order type of
Cardinal_number
Theories in mathematical logic
handling this in first-order logic: Use first-order logic with two types. Use ordinary first-order logic, but add a new unary predicate "Set", where "Set(t)"
List_of_first-order_theories
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
Set of elements in any of some sets
extensionality to show that this set is unique. For readability, define the binary predicate Union ( X , Y ) {\displaystyle \operatorname {Union} (X,Y)} meaning
Union_(set_theory)
Whether a decision problem has an effective method to derive the answer
one other predicate symbol with two or more arguments is not decidable. Logical systems extending first-order logic, such as second-order logic and type
Decidability_(logic)
One-to-one correspondence
correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set
Bijection
Complexity class used to classify decision problems
corresponds precisely to the set of languages definable by existential second-order logic (Fagin's theorem). NP can be seen as a very simple type of interactive
NP_(complexity)
Components of a mathematical or logical formula
first-order term is recursively constructed from constant symbols, variable symbols, and function symbols. An expression formed by applying a predicate symbol
Term_(logic)
Size of a set in mathematics
set theory, equinumerosity is not a relation in the usual sense, but a predicate, defined formally as: A ∼ B ⟺ ∃ ( f : A → B ) ( ∀ b ∈ B ( ∃ ! a ∈ A
Cardinality
Non-monotonic logic created by John McCarthy
McCarthy augmented first-order logic to allow the minimization of the extension of some predicates, where the extension of a predicate is the set of tuples
Circumscription_(logic)
Reasoning for mathematical statements
inference shows that a collection of statements are pairwise equivalent. In order to prove that the statements φ 1 , … , φ n {\displaystyle \varphi _{1},\ldots
Mathematical_proof
Study of computable functions and Turing degrees
discussed many aspects of second-order arithmetic and reverse mathematics. The field of proof theory includes the study of second-order arithmetic and Peano
Computability_theory
Symbol representing a mathematical concept
only if Y = F(X). Many treatments of predicate logic don't allow functional predicates, only relational predicates. This is useful, for example, in the
Function_symbol
Programming language that uses first order logic
higher-order programming. A higher-order predicate is a predicate that takes one or more other predicates as arguments. Although support for higher-order programming
Prolog
Kind of transfinite induction
it. In first-order set theories, the common framework, the set induction principle is an axiom schema, granting an axiom for any predicate (i.e. class)
Epsilon-induction
Proof in set theory
a bijection between their underlying sets, Cantor also defines binary predicate of cardinalities | S | {\displaystyle |S|} and | T | {\displaystyle |T|}
Cantor's_diagonal_argument
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
Logical connective OR
short-circuited; that is, if the first (left) operand evaluates to true, then the second (right) operand is not evaluated. The logical disjunction operator thus
Logical_disjunction
Characteristic of some logical systems
introducing an inconsistency. Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for
Completeness_(logic)
Mapping of mathematical formulas to a particular meaning
fragments of first-order logic such as that used in universal algebra, and for second-order logic. In connection with first-order logic and model theory
Structure (mathematical logic)
Structure_(mathematical_logic)
Study of the semantics, or interpretations, of formal and natural languages
to some predefined mathematical domains: an interpretation of first-order predicate logic is given by a mapping from terms to a universe of individuals
Semantics_(logic)
Proof by Alan Turing
Computable Numbers, with an Application to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem;
Turing's_proof
Axiomatic set theories based on the principles of mathematical constructivism
closer mirroring second-order arithmetic. In set theory it also follows from full (i.e. unbounded) Separation, which says that all predicates on ∀ {\displaystyle
Constructive_set_theory
Set of the elements not in a given subset
logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus Set theory
Complement_(set_theory)
School of thought in philosophy of mathematics
survived: [In the second edition] The axiom of reducibility is dropped, and it is stated explicitly that all primitive predicates belong to the lowest
Logicism
Branch of mathematical logic
theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by
Reverse_mathematics
Process of repeating items in a self-similar way
apply to intransitive verbs, transitive verbs, or ditransitive verbs. In order to provide a single denotation for it that is suitably flexible, and is
Recursion
Set whose elements all belong to another set
partial order is—up to an order isomorphism—the Cartesian product of k = | S | {\displaystyle k=|S|} (the cardinality of S) copies of the partial order on
Subset
Symbol representing a mathematical object
variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple replacement. Viète's convention was to
Variable_(mathematics)
Proposition in mathematical logic
the concept of "size" of the set Q {\displaystyle \mathbb {Q} } with the order or topological structure placed on it. In fact, it turns out the rational
Continuum_hypothesis
Computer programming function
predicate even to every element of the list of integers 1, 2, …, 10 in that order and creating a new list of those elements for which the predicate returns
Filter (higher-order function)
Filter_(higher-order_function)
Type of infinite number in set theory
which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is
Inaccessible_cardinal
Type of logic diagram
simple proposition containing two terms, subject (S) and predicate (P), in which the predicate is either asserted or denied of the subject. Every categorical
Square_of_opposition
Argument whose conclusion must be true if its premises are
Mars. (False) The problem with the argument is that it is not sound. In order for a deductive argument to be sound, the argument must be valid and all
Validity_(logic)
SECOND ORDER-PREDICATE
SECOND ORDER-PREDICATE
Boy/Male
Greek
Order.
Girl/Female
Australian, French, German, Greek, Italian
Order
Male
Swedish
Old Swedish form of Old Norse Oddr, ODDER means "point of a weapon."
Boy/Male
Indian
Order, Decree
Boy/Male
Australian, French, German, Greek
Order
Male
English
Variant spelling of Middle English Estmond, ESMOND means "gracious protector."Â
Boy/Male
Hindu, Indian, Punjabi, Sikh
Order
Girl/Female
Indian, Marathi, Sindhi
Order
Boy/Male
Tamil
Pradarsh | பà¯à®°à®¤à®°à¯à®·
Appearance, Order
Pradarsh | பà¯à®°à®¤à®°à¯à®·
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.
Girl/Female
German, Greek
Order
Boy/Male
Greek
Order.
Boy/Male
Arabic, Australian, Muslim
Order
Female
English
Anglicized form of Scottish Gaelic Seònaid, SEONA means "God is gracious."
Girl/Female
Greek
Order.
Male
Irish
Older form of Irish Gaelic Seachlainn, SEACHNALL means "second."
Girl/Female
Indian, Traditional
Order
Girl/Female
Indian, Telugu
Order
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’.
SECOND ORDER-PREDICATE
SECOND ORDER-PREDICATE
Surname or Lastname
German
German : variant of Lepper with excrescent -t.English : variant of Leopard.
Girl/Female
Norse
Divine strength.
Male
Egyptian
, the overseer of the altar of Osiris (?).
Boy/Male
American, Australian, British, Celtic, Chinese, Christian, English, French, Hebrew, Irish, Norse, Scandinavian
Beloved; Dear One; Bright Finn; Brilliant Finn; Black One; Variant of David
Boy/Male
Tamil
Who has faith in God
Boy/Male
Arabic
Worshippers; Adorers
Boy/Male
African, Australian, Nigerian
Father's Heart
Boy/Male
Arabic, Indian, Kurdish, Muslim
Wise; Far-sighted
Boy/Male
Indian
Beautiful, Perfect, One of the ninety nine qualities of God
Girl/Female
Hebrew Latin
Bitter.
SECOND ORDER-PREDICATE
SECOND ORDER-PREDICATE
SECOND ORDER-PREDICATE
SECOND ORDER-PREDICATE
SECOND ORDER-PREDICATE
a.
Immediately following the first; next to the first in order of place or time; hence, occuring again; another; other.
n.
One who seconds or supports what another attempts, affirms, moves, or proposes; as, the seconder of an enterprise or of a motion.
v. i.
To give orders; to issue commands.
n.
The second part in a concerted piece; -- often popularly applied to the alto.
n.
Rank; degree; thus, the order of a curve or surface is the same as the degree of its equation.
a.
Being of the same kind as another that has preceded; another, like a protype; as, a second Cato; a second Troy; a second deluge.
a.
Of the rank or degree below the best highest; inferior; second-rate; as, a second-class house; a second-class passage.
a.
To follow or attend for the purpose of assisting; to support; to back; to act as the second of; to assist; to forward; to encourage.
adv.
In the second place.
a.
The sixtieth part of a minute of time or of a minute of space, that is, the second regular subdivision of the degree; as, sound moves about 1,140 English feet in a second; five minutes and ten seconds north of this place.
n.
A body of persons having some common honorary distinction or rule of obligation; esp., a body of religious persons or aggregate of convents living under a common rule; as, the Order of the Bath; the Franciscan order.
n.
The second part in a concerted piece.
n.
To give an order to; to command; as, to order troops to advance.
a.
Of the second size, rank, quality, or value; as, a second-rate ship; second-rate cloth; a second-rate champion.
v. t.
To order a second time.
imp. & p. p.
of Second
adv.
Secondly; in the second place.
n.
To give an order for; to secure by an order; as, to order a carriage; to order groceries.
n.
Right arrangement; a normal, correct, or fit condition; as, the house is in order; the machinery is out of order.
a.
Having the power of second-sight.