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See searches and references containing PROPOSITIONAL FUNCTION!PROPOSITIONAL FUNCTION
Expression in propositional calculus
In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except
Propositional_function
About mathematical functions
proposition; this proposition is called a "value" of the propositional function. In our example there are four values of the propositional function,
History of the function concept
History_of_the_function_concept
3-volume treatise on mathematics, 1910–1913
matrix is (at least for propositional functions), a truth table, i.e., all truth-values of a propositional or predicate function. Sheffer stroke: Is the
Principia_Mathematica
Input to a mathematical function
argument to a function Propositional function – Expression in propositional calculus Type signature – Defines the inputs and outputs for a function, subroutine
Argument_of_a_function
Branch of logic
Propositional logic is a branch of classical logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic,
Propositional_logic
Formulaic summary of Buddhist doctrines
important teaching in Buddhism where they have both a symbolic and a propositional function. As symbols, they represent the awakening and liberation of the
Four_Noble_Truths
Mathematical use of "there exists"
the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically
Existential_quantification
Variable that can either be true or false
false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order
Propositional_variable
Mathematical use of "for all"
{\displaystyle \lnot } denotes negation. For example, if P(x) is the propositional function "x is married", then, for the set X of all living human beings,
Universal_quantification
Logic formula
propositional logic, a propositional formula is a type of syntactic formula which is well formed. If the values of all variables in a propositional formula
Propositional_formula
School of thought in philosophy of mathematics
the proposition, his argument being that, indeed, the arguments x do not belong to the propositional function aka "class" created by the function. The
Logicism
Algebraic manipulation of "true" and "false"
language of propositional calculus, used when talking about propositional calculus) to denote propositions. The semantics of propositional logic rely on
Boolean_algebra
Function in logic
operator Propositional calculus Truth-functional propositional logic Roy T. Cook (2009). A Dictionary of Philosophical Logic, p. 294: Truth Function. Edinburgh
Truth_function
Function returning one of only two values
2^{k}} entries. Every k {\displaystyle k} -ary Boolean function can be expressed as a propositional formula in k {\displaystyle k} variables x 1 , . . .
Boolean_function
Bearer of truth values
of its sensory nature, or as a propositional process whose contents can be true or false. Psychological propositionalism is the view that all intentional
Proposition
Every set is smaller than its power set
that there are more propositional functions than objects. "For suppose a correlation of all objects and some propositional functions to have been affected
Cantor's_theorem
Topics referred to by the same term
formal logic: Predicate (logic) Propositional function Finitary relation, or n-ary predicate Boolean-valued function Syntactic predicate, in formal grammars
Predicate
Process of grouping elements into fuzzy sets
sets whose membership functions are defined by the truth value of a fuzzy propositional function. A fuzzy propositional function is analogous to an expression
Fuzzy_classification
Communication about how information is meant to be interpreted
principle, that no propositional function can be defined prior to specifying the function's scope of application. In other words, before a function can be defined
Meta-communication
In logic, a statement which is always true
valuation is a function that assigns each propositional variable to either T (for truth) or F (for falsity). So by using the propositional variables A and
Tautology_(logic)
Assignment of meaning to the symbols of a formal language
for propositional logic consists of formulas built up from propositional symbols (also called sentential symbols, sentential variables, propositional variables)
Interpretation_(logic)
Philosophical concept
The type of propositional function that Ramsey is referring to here is a function that takes a proposition as input and gives a proposition as output.
Redundancy_theory_of_truth
Paradox in set theory
instead that "propositional functions (conditions or predicates) used for separating off subsets, as well as the replacement functions, can be 'entirely
Russell's_paradox
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
Mathematical table used in logic
logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions
Truth_table
In propositional calculus and proof complexity a propositional proof system (pps), also called a Cook–Reckhow propositional proof system, is a system for
Propositional_proof_system
Overview of and topical guide to logic
consequence Negation normal form Open sentence Propositional calculus Propositional formula Propositional variable Rule of inference Strict conditional
Outline_of_logic
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
System of mathematical set theory
axiom schemas by formalizing the concept of "definite propositional function" with his functions, whose construction requires only finitely many axioms
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Syntactically correct logical formula
Two key uses of formulas are in propositional logic and predicate logic. A key use of formulas is in propositional logic and predicate logic such as
Well-formed_formula
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Branch of philosophy
a semantic fact (i.e., the proposition that is represented by "The horse is red"). In other words, a propositional function is like an algorithm. The meaning
Philosophy_of_language
Symbol representing a mathematical concept
systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse, though
Function_symbol
Type of formal logic
concurrent programs. Flavors of temporal logic include propositional dynamic logic (PDL), (propositional) linear temporal logic (LTL), computation tree logic
Modal_logic
Type of logical system
it from propositional logic, which does not use quantifiers or relations; in this sense, first-order logic is an extension of propositional logic. A
First-order_logic
System of mathematical set theory
III. Axiom of separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is defined for all elements of a set M, M possesses a subset
Zermelo_set_theory
proposition is allowed to have quantification over individuals but not over things of higher type. function This often means a propositional function
Glossary of Principia Mathematica
Glossary_of_Principia_Mathematica
Characteristic of some logical systems
Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example, the propositional logic
Completeness_(logic)
Logical connective AND
Logical connective OR Logical graph – Type of diagrammatic notation for propositional logicPages displaying short descriptions of redirect targets Negation –
Logical_conjunction
Logical operation
that P → ⊥ {\displaystyle P\rightarrow \bot } . As a result, in the propositional case, a sentence is classically provable if its double negation is intuitionistically
Negation
Process of repeating items in a self-similar way
where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values),
Recursion
Existence of values making formula true
the positive propositional calculus, the questions of validity and satisfiability may be unrelated. In the case of the positive propositional calculus, the
Satisfiability
Logical incompatibility between two or more propositions
impossible?". In classical logic, particularly in propositional and first-order logic, a proposition φ is a contradiction if and only if φ⊢⊥. Since for
Contradiction
Concept of an existence; not nothing
exists," "there is at least one," or "for some." It expresses that a propositional function can be satisfied by at least one member of a domain of discourse
Something_(concept)
Rule defining the correct structure of expressions in formal grammar
more other expressions. Propositional and predicate calculi are examples of formal systems. The formation rules of a propositional calculus may, for instance
Formation_rule
Target set of a mathematical function
mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in
Codomain
Symbol connecting formulas in logic
combine or negate arithmetic expressions. For instance, in the syntax of propositional logic, the binary connective ∨ {\displaystyle \lor } (meaning "or")
Logical_connective
1921 philosophical work by Ludwig Wittgenstein
of atomic propositions. Wittgenstein drew from Henry M. Sheffer's logical theorem making that statement in the context of the propositional calculus.
Tractatus Logico-Philosophicus
Tractatus_Logico-Philosophicus
uniform equivalents of propositional proof systems. The connection is particularly useful for constructions of short propositional proofs. It is often easier
Bounded_arithmetic
Axiom in Russell's ramified theory of types
states that any truth function (i.e. propositional function) can be expressed by a formally equivalent predicative truth function. It made its first appearance
Axiom_of_reducibility
Whether a decision problem has an effective method to derive the answer
For example, propositional logic is decidable, because the truth-table method can be used to determine whether an arbitrary propositional formula is logically
Decidability_(logic)
A propositional directed acyclic graph (PDAG) is a data structure that is used to represent a Boolean function. A Boolean function can be represented as
Propositional directed acyclic graph
Propositional_directed_acyclic_graph
Type of mathematical variable
such letters represent propositional functions, such that the domain of the arguments is mapped to a range of different propositions, and when such variables
Predicate_variable
System of formal deduction in logic
extend the propositional system to axiomatise classical predicate logic. Likewise, these three rules extend system for intuitionistic propositional logic (with
Hilbert_system
Symbol representing a mathematical object
of parabolas. Lambda calculus Observable variable Physical constant Propositional variable Sobolev, S.K. (originator). "Individual variable". Encyclopedia
Variable_(mathematics)
Study of computable functions and Turing degrees
computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study
Computability_theory
Mathematical-logic system
as λ-calculus) is a formal system for expressing computation based on function abstraction and application using variable binding and substitution. Untyped
Lambda_calculus
Hilbert-style deductive systems for propositional logics. Classical propositional calculus is the standard propositional logic. Its intended semantics is
List of axiomatic systems in logic
List_of_axiomatic_systems_in_logic
algebra and propositional logic. Algebra of sets Boolean algebra (structure) Boolean algebra Field of sets Logical connective Propositional calculus Ampheck
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Computation model defining an abstract machine
The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Mineola, NY: Dover Publ. ISBN 978-0486432281
Turing_machine
Number of arguments required by a function
science, arity (/ˈærɪti/ ) is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank,
Arity
Value indicating the relation of a proposition to truth
¬p ∨ ¬q ¬(p ∨ q) ⇔ ¬p ∧ ¬q Propositional variables become variables in the Boolean domain. Assigning values for propositional variables is referred to as
Truth_value
Method of deriving conclusions
Propositional logic is not concerned with the concrete meaning of propositions other than their truth values. Key rules of inference in propositional
Rule_of_inference
Class of formal logics
apparent that classical propositional calculus admits other semantics. In Boolean-valued semantics (for classical propositional logic), the truth values
Classical_logic
In mathematics, a statement that has been proven
This should not be confused with "proposition" as used in propositional logic. In classical geometry the term "proposition" was used differently: in Euclid's
Theorem
Function that outputs either true or false
domain Boolean logic Propositional calculus Truth table Logic minimization Indicator function Predicate Proposition Boolean function Brown, Frank Markham
Boolean-valued_function
Non-contradiction of a theory
Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency
Consistency
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Subfield of automated reasoning and mathematical logic
constructed proofs from a small set of propositional axioms and three deduction rules: modus ponens, (propositional) variable substitution, and the replacement
Automated_theorem_proving
Quickly growing function
Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not
Ackermann_function
Version of classical propositional calculus that uses only one connective
In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus that uses only one connective, called
Implicational propositional calculus
Implicational_propositional_calculus
Subset of a function's codomain
the range of a function may refer to either of two closely related concepts: the codomain of the function, or the image of the function. In some cases
Range_of_a_function
Logical connective OR
Retrieved 25 Dec 2023. "A Brief Introduction to the Intuitionistic Propositional Calculus" (PDF). California Institute of Technology. Retrieved 2026-05-19
Logical_disjunction
Form of logic that allows quantification over predicates
an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic
Second-order_logic
Logic principle
properties). There are various extensionality principles in mathematics. Propositional extensionality of predicates P , Q {\displaystyle P,Q} : if P ⟺ Q {\displaystyle
Extensionality
Problem of determining if a Boolean formula could be made true
computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT)
Boolean satisfiability problem
Boolean_satisfiability_problem
Theorem for proving more complex theorems
Often, a theorem is broken into multiple cases (for example, a quadratic function may have no real roots, one double root, or two distinct roots), and each
Lemma_(mathematics)
Yes/no problem in computer science
function problem can be turned into a decision problem; the decision problem is just the graph of the associated function. (The graph of a function f
Decision_problem
governing the logic of predicates Propositional calculus, specifies the rules of inference governing the logic of propositions Modal μ-calculus, a common temporal
List_of_formal_systems
Symbolic description of a mathematical object
mathematical notation. Symbols can denote numbers, variables, operations, and functions. Other symbols include punctuation marks and brackets, used for grouping
Expression_(mathematics)
Property of a mathematical operation
rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions
Associative_property
Area of mathematical logic
symbol, or a function or relation symbol with a specified arity. Note that in some literature, constant symbols are considered as function symbols with
Model_theory
Logical principle
diagrammatic notation for propositional logicPages displaying short descriptions of redirect targets: a graphical syntax for propositional logic Mathematical
Law_of_excluded_middle
One-to-one correspondence
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the
Bijection
Size of a possibly infinite set
A . {\displaystyle \#A.} Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one
Cardinal_number
Mathematical logic concept
formulas depends on the logic under consideration; for propositional logic, for example, a propositional variable is often more briefly referred to as an "atomic
Atomic_formula
Mathematical logic concept
truth-functional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia
Contraposition
Mathematical theory of data types
Curry–Howard Correspondence, the identity type is a type introduced to mirror propositional equivalence, as opposed to the judgmental (syntactic) equivalence that
Type_theory
Function, homomorphism, or morphism
In mathematics, a map or mapping is synonymous to a function, though various branches add further nuances to differentiate them. The terms may have originated
Map_(mathematics)
1905 philosophy essay by Bertrand Russell
of a propositional function. This is basically a modified version of Frege's idea of unsaturated concepts. Hence, "C(x) stands for a proposition in which
On_Denoting
Argument whose conclusion must be true if its premises are
it is true under every possible interpretation of the language. In propositional logic, they are tautologies. A statement can be called valid, i.e. logical
Validity_(logic)
Fundamental theorem in mathematical logic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Gödel's_completeness_theorem
propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas
Valuation_(logic)
Diagram that shows all possible logical relations between a collection of sets
Representation of Propositions and Reasonings" in the Philosophical Magazine and Journal of Science, about the different ways to represent propositions by diagrams
Venn_diagram
Concept in first-order logic
also sometimes referred as effectively propositional (EPR) since it can be effectively translated into propositional logic formulas by a process of grounding
Bernays–Schönfinkel_class
Reasoning for mathematical statements
must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture
Mathematical_proof
Mathematical set containing no elements
exists precisely one function f {\displaystyle f} from ∅ {\displaystyle \varnothing } to A , {\displaystyle A,} the empty function. As a result, the empty
Empty_set
Concept in mathematical logic
called a universal gate (or a universal set of gates). In a context of propositional logic, functionally complete sets of connectives are also called (expressively)
Functional_completeness
Sequence of words formed by specific rules
sequence of well-formed formulas (which may be interpreted as sentences, or propositions) each of which is an axiom or follows from the preceding formulas in
Formal_language
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PROPOSITIONAL FUNCTION
PROPOSITIONAL FUNCTION
PROPOSITIONAL FUNCTION
PROPOSITIONAL FUNCTION
PROPOSITIONAL FUNCTION
PROPOSITIONAL FUNCTION
PROPOSITIONAL FUNCTION
PROPOSITIONAL FUNCTION
PROPOSITIONAL FUNCTION
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