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Symbol representing a property or relation in logic
degree of truth. Free variables and bound variables Hypostatic abstraction Multigrade predicate Opaque predicate Philosophical predication Predicate functor
Predicate_(logic)
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
Type of logical system
linguistics, and computer science, first-order logic (FOL), also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system
First-order_logic
Family of philosophical theories
in common the claim that assertions of predicate truth of a statement do not attribute a property called "truth" to such a statement. Gottlob Frege was
Deflationary_theory_of_truth
Theory of truth within pragmatism
ideas about truth are often confused with the quite distinct notions of "logic and inquiry", "judging what is true", and "truth predicates". In one classical
Pragmatic_theory_of_truth
Theory of truth in the philosophy of language
Gödel used in his incompleteness theorems. Roughly, this states that a truth-predicate satisfying Convention T for the sentences of a given language cannot
Semantic_theory_of_truth
words add an additional parameter to the construction of an accurate truth predicate. Among the philosophers who grappled with this problem is Alfred Tarski
Theories_of_truth
American philosopher and logician (1940–2022)
contain the truth predicate, and defining a truth predicate over just that segment: this action adds new sentences to the language, and truth is in turn
Saul_Kripke
Conformity to reality
possible. A truth predicate is a linguistic device that ascribes truth to a sentence, like the expression "... is true". A priori truths are typically
Truth
Assignment of meaning to the symbols of a formal language
truth values true and false. Because the first-order interpretations described here are defined in set theory, they do not associate each predicate symbol
Interpretation_(logic)
Function that outputs either true or false
or syntactic expression. In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for
Boolean-valued_function
Limitative results in mathematical logic
analysis of the truth of the liar sentence. It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
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
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
the truth predicate. Some sentences are stable in their evaluations, such as the truth-teller sentence, The truth-teller is true. Assuming the truth-teller
Revision_theory
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)
Method of deriving conclusions
analyzing how the internal structure of propositions, like names and predicates, influences reasoning. Other logical systems explore inferential patterns
Rule_of_inference
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
Theorem that arithmetical truth cannot be defined in arithmetic
does not define truth for the stronger system: this formula T r u e ( n ) {\displaystyle \mathrm {True} (n)} only defines a truth predicate for formulas
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Branch of logic
or valid, for example by means of truth tables. One notable difference between propositional calculus and predicate calculus is that satisfiability of
Propositional_logic
Form of logic that allows quantification over predicates
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, that
Second-order_logic
Symbol connecting formulas in logic
standard systems of classical logic, these connectives are interpreted as truth functions, though they receive a variety of alternative interpretations
Logical_connective
Mathematical table used in logic
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which
Truth_table
Paradox in set theory
following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its
Russell's_paradox
Alternative to Tarskian semantics
between truth-value semantics and the standard semantics for predicate logic is that there are no domains for truth-value semantics. Only the truth clauses
Truth-value_semantics
Collection of mathematical objects
mathematical induction, which is called transfinite induction. Given a property (predicate) P ( n ) {\displaystyle P(n)} depending on a natural number, mathematical
Set_(mathematics)
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
Statement that is taken to be true
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 in the
Axiom
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
Testing device for logical soundness
occurrences of the truth predicate in natural language. In particular, Schema T treats only "freestanding" uses of the predicate—cases when it is applied
T-schema
Characteristic of some logical systems
to it without introducing an inconsistency. Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically
Completeness_(logic)
Value indicating the relation of a proposition to truth
logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic
Truth_value
In mathematics, a statement that has been proven
as expressing some truth, but in contrast to the notion of a scientific law, which is experimental, the justification of the truth of a theorem is purely
Theorem
Number of arguments required by a function
logarithm operator, the addition operator, and the division operator. Logical predicates such as OR, XOR, AND, IMP are typically used as binary operators with
Arity
3-volume treatise on mathematics, 1910–1913
(at least for propositional functions), a truth table, i.e., all truth-values of a propositional or predicate function. Sheffer stroke: Is the contemporary
Principia_Mathematica
Term in logic
has developed artificial languages, for example sentential calculus and predicate calculus, partly with the purpose of revealing the underlying logic of
Atomic_sentence
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
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
Type of infinite structure
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
O-minimal_theory
Sequence of words formed by specific rules
boolean algebra, which is a formal way of describing logical operations using truth values and set operators. In his work An Investigation of The Laws of Thought
Formal_language
Branch of mathematics that studies sets
doi:10.2307/2274520 Evangeliou, Christos (1985), "Aristotle's Doctrine of Predicables and Porphyry's Isagoge", Journal of the History of Philosophy, 23 (1):
Set_theory
Subfield of mathematics
/ Date incompatibility (help) Kleene, Stephen Cole (1943). "Recursive Predicates and Quantifiers". Transactions of the American Mathematical Society. 53
Mathematical_logic
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
Approach to logic
terms, one of which, the "predicate", is "affirmed" or "denied" of the other, the "subject", and which is capable of truth or falsity. The syllogism is
Term_logic
Measure of algorithmic complexity
Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability
Kolmogorov_complexity
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
Process of repeating items in a self-similar way
Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability
Recursion
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
Algebraic manipulation of "true" and "false"
elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted by 1 and 0, whereas in elementary
Boolean_algebra
truth. In his 1992 book Truth and Objectivity, Wright argued that any predicate which satisfied certain platitudes about truth qualified as a truth predicate
Pluralist_theories_of_truth
Infinite cardinal number
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Aleph_number
Mathematical-logic system
FALSE is equivalent to FALSE. A predicate is a function that returns a Boolean value. The most fundamental predicate is ISZERO, which returns TRUE if
Lambda_calculus
Additional mathematical object
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Mathematical_structure
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
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
individuals to truth values, essentially a generalization of a predicate. predicate functor logic A logical system that combines elements of predicate logic with
Glossary_of_logic
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
Statement that is true regardless of the truth or falsity of its constituent propositions
Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity
Logical_truth
Study of the semantics, or interpretations, of formal and natural languages
first-order predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values "true"
Semantics_(logic)
Study of correct reasoning
P\land Q} . Unlike predicate logic where terms and predicates are the smallest units, propositional logic takes full propositions with truth values as its
Logic
Set of all things that may be the input of a mathematical function
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Domain_of_a_function
Computation model defining an abstract machine
Andrew (1983). "The Spirit of Truth". Alan Turing: The Enigma. New York: Simon and Schuster. Cf. Chapter "The Spirit of Truth" for a history leading to,
Turing_machine
Logical operation
notions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity
Negation
Attempt to persuade or to determine the truth of a conclusion
logical steps, arguments are intended to determine or show the degree of truth or acceptability of a logical conclusion. The process of crafting or delivering
Argument
Theorem in mathematical logic
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Lindström's_theorem
Form of mathematical proof
requires an axiom schema containing a separate axiom for each possible predicate. The article Peano axioms contains further discussion of this issue. The
Mathematical_induction
Diagram that shows all possible logical relations between a collection of sets
inside the circle that represents the set F. Venn diagrams correspond to truth tables for the propositions x ∈ A {\displaystyle x\in A} , x ∈ B {\displaystyle
Venn_diagram
Class of formal logics
Orman Quine believed that a formal system that allows quantification over predicates (higher-order logic) didn't meet the requirements to be a logic, saying
Classical_logic
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
Target set of a mathematical function
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Codomain
One-to-one correspondence
Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability
Bijection
Set of tuples in mathematical logic that satisfy a predicate
The extension of a predicate – a truth-valued function – is the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples
Extension_(predicate_logic)
Collection of sets in mathematics that can be defined based on a property of its members
{\displaystyle \phi (x)} holds; thus, the class can be described as the set of all predicates equivalent to ϕ {\displaystyle \phi } (which includes ϕ {\displaystyle
Class_(set_theory)
Theorem for proving more complex theorems
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Lemma_(mathematics)
Any one of the distinct objects that make up a set in set theory
predication of x called membership that is equivalent to the statement ‘x is a member of y if and only if, for all objects x, the general predication
Element_of_a_set
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
Establishment of a theorem using inference from the axioms
interpretation of a formal system is the assignment of meanings to the symbols, and truth values to the sentences of a formal system. The study of interpretations
Formal_proof
Study of computable functions and Turing degrees
f such that each n is in A if and only if f(n) is in B. Truth-table reducibility: A is truth-table reducible to B if A is Turing reducible to B via an
Computability_theory
abstract objects. Logicism asserts that all mathematical truths can be reduced to logical truths, and all objects forming the subject matter of those branches
Mathematical_object
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)
Reasoning for mathematical statements
language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout
Mathematical_proof
Size of a possibly infinite set
Schindler, Ralf-Dieter (2014). Set theory : exploring independence and truth. Universitext. Cham: Springer-Verlag. doi:10.1007/978-3-319-06725-4.
Cardinal_number
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
Mathematical set formed from two given sets
Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability
Cartesian_product
Function, homomorphism, or morphism
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Map_(mathematics)
View that there are statements that are both true and false
advocating constructive (methodological) deflationism regarding the truth predicate. The term was coined by Graham Priest and Richard Sylvan (then Routley)
Dialetheism
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)
Set of the elements not in a given subset
{\displaystyle X,} and columns elements of Y . {\displaystyle Y.} The truth of a R b {\displaystyle aRb} corresponds to 1 in row a , {\displaystyle
Complement_(set_theory)
Yes-or-no question that cannot ever be solved by a computer
does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be
Undecidable_problem
Logical connective
natural-language conditionals are truth functional in the sense that the truth value of "If P, then Q" is determined solely by the truth values of P and Q. Thus
Material_conditional
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
Logical connective AND
mathematics and linguistics, and ( ∧ {\displaystyle \wedge } ) is the truth-functional operator of conjunction or logical conjunction. The logical connective
Logical_conjunction
Theorem in set theory
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Kőnig's_theorem_(set_theory)
Mathematical use of "for all" and "there exists"
let X be the set of all Peter's friends, P(x) the predicate "x likes to dance", and Q(x) the predicate "x likes to go to the beach". Then the above sentence
Quantifier_(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
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
Complexity class used to classify decision problems
Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability
NP_(complexity)
Yes/no problem in computer science
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Decision_problem
Logical incompatibility between two or more propositions
its usual "truth values" of "truth" and "falsity". They observed that: The property of being a tautology has been defined in notions of truth and falsity
Contradiction
Mathematical concept for comparing objects
formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic
Equivalence_relation
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