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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 logics
Propositional_variable
In logic, a statement which is always true
tautology of propositional logic, and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The
Tautology_(logic)
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
Branch of logic
connectives, to make propositional formulas. Because of this, the propositional variables are called atomic formulas of a formal propositional language. While
Propositional_logic
Logical connective AND
disjunction Logical graph Negation Operation Peano–Russell notation Propositional calculus "2.2: Conjunctions and Disjunctions". Mathematics LibreTexts
Logical_conjunction
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
Type of mathematical variable
properly called metalinguistic variables. In higher-order logic, predicate variables correspond to propositional variables which can stand for well-formed
Predicate_variable
Syntactically correct logical formula
interpretations. For example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula
Well-formed_formula
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
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)
Mathematical logic concept
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
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
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)
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
Paradox in set theory
first-order logic. As José Ferreirós notes, Zermelo insisted instead that "propositional functions (conditions or predicates) used for separating off subsets
Russell's_paradox
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
Mathematical use of "there exists"
then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically
Existential_quantification
Algebraic manipulation of "true" and "false"
metavariables (variables outside the language of propositional calculus, used when talking about propositional calculus) to denote propositions. The semantics
Boolean_algebra
Logical connective
Implicational propositional calculus Laws of Form Logical graph Logical equivalence Material implication (rule of inference) Peirce's law Propositional calculus
Material_conditional
3-volume treatise on mathematics, 1910–1913
σn) that can be thought of as the classes of propositional functions of τ1,...τm obtained from propositional functions of type (τ1,...,τm,σ1,...,σn) by
Principia_Mathematica
Statement that is taken to be true
{\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are propositional variables, then A → ( B → A ) {\displaystyle A\to (B\to A)} and ( A → ¬ B
Axiom
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
Logical incompatibility between two or more propositions
impossible?". In classical logic, particularly in propositional and first-order logic, a proposition φ {\displaystyle \varphi } is a contradiction if and
Contradiction
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)
Formal semantics for non-classical logic systems
[citation needed] The language of propositional modal logic consists of a countably infinite set of propositional variables, a set of truth-functional connectives
Kripke_semantics
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
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
Branch of mathematical logic
calculi Each of these can give a complete and axiomatic formalization of propositional or predicate logic of either the classical or intuitionistic flavour
Proof_theory
Set whose elements all belong to another set
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Subset
Rules used for constructing, or transforming the symbols and words of a language
Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example the propositional logic
Syntax_(logic)
Statement that is true regardless of the truth or falsity of its constituent propositions
which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including
Logical_truth
Infinite cardinal number
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Aleph_number
Mathematical operation with two operands
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Binary_operation
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 valuation
Truth_value
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)
Symbol representing a mathematical object
Lambda calculus Observable variable Physical constant Propositional variable Sobolev, S.K. (originator). "Individual variable". Encyclopedia of Mathematics
Variable_(mathematics)
Input to a mathematical function
(computer programming) – Variable that represents an argument to a function Propositional function – Expression in propositional calculus Type signature –
Argument_of_a_function
Number of arguments required by a function
side effects). Such functions may have some hidden input, such as global variables or the whole state of the system (time, free memory, etc.). Examples of
Arity
Set of sentences in a formal language
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Theory_(mathematical_logic)
Complexity class used to classify decision problems
whether or not a certain formula in propositional logic with Boolean variables is true for some value of the variables. The decision version of the travelling
NP_(complexity)
Branch of mathematics that studies sets
12,000 theorems starting from ZFC set theory, first-order logic and propositional logic. Set theory is a major area of research in mathematics with many
Set_theory
Formal system of logic
(from a technical perspective) in such a context. Zeroth-order logic (propositional logic) First-order logic Second-order logic Type theory Higher-order
Higher-order_logic
Form of logic that allows quantification over predicates
of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that
Second-order_logic
Mathematical set containing no elements
Routledge. p. 87. George Boolos (1984), "To be is to be the value of a variable", The Journal of Philosophy 91: 430–49. Reprinted in 1998, Logic, Logic
Empty_set
Impossible task in computing
EXPTIME-complete (Theorem 2.24). The first-order logic fragment where the only variable names are x , y {\displaystyle x,y} is NEXPTIME-complete (Theorem 3.18)
Entscheidungsproblem
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
Collection of mathematical objects
objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define
Set_(mathematics)
Mathematical set of all subsets of a set
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Power_set
Set of the elements not in a given subset
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Complement_(set_theory)
Mathematical-logic system based on functions
expressing computation based on function abstraction and application using variable binding and substitution. Untyped lambda calculus, the topic of this article
Lambda_calculus
Limitative results in mathematical logic
such a system is first-order Peano arithmetic, a system in which all variables are intended to denote natural numbers. In other systems, such as set
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Mathematical function such that every output has at least one input
Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has a right inverse is equivalent to the
Surjective_function
Fundamental theorem in mathematical logic
the language of the formula (i.e. for any assignment of values to the variables of the formula). To formally state, and then prove, the completeness theorem
Gödel's_completeness_theorem
Ordered listing of items in collection
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Enumeration
System including an indeterminate value
ternary signals. This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, true}, and extends conventional
Three-valued_logic
Function returning one of only two values
expressed as a propositional formula in k {\displaystyle k} variables x 1 , . . . , x k {\displaystyle x_{1},...,x_{k}} , and two propositional formulas are
Boolean_function
Standard system of axiomatic set theory
metavariables for any wff, and x {\displaystyle x} be a metavariable for any variable. These are valid wff constructions: ¬ ϕ {\displaystyle \lnot \phi } ( ϕ
Zermelo–Fraenkel_set_theory
Collection of sets in mathematics that can be defined based on a property of its members
x(x\in A\leftrightarrow x=x)} . For a class A {\displaystyle A} and a set variable symbol x {\displaystyle x} , it is necessary to be able to expand each
Class_(set_theory)
Function in mathematical logic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Gödel_numbering
Any one of the distinct objects that make up a set in set theory
∈ 𝔇y makes this definition well-defined by ensuring that x is a bound variable in its predication of membership in y. In this case, the domain of Px,
Element_of_a_set
Formalization of the natural numbers
language of PRA consists of: A countably infinite number of variables x, y, z,.... The propositional connectives; The equality symbol =, the constant symbol
Primitive recursive arithmetic
Primitive_recursive_arithmetic
Type of logical system
a quantifier, x is a variable, and "... is a human" and "... is mortal" are predicates. This distinguishes it from propositional logic, which does not
First-order_logic
Concept in logic
propositional logic, ψ is a substitution instance of φ if and only if ψ may be obtained from φ by substituting formulas for propositional variables in
Substitution_(logic)
common understanding of formalism takes mathematics as not a body of propositions representing an abstract piece of reality but much more akin to a game
Mathematical_object
Yes-or-no question that cannot ever be solved by a computer
of a polynomial in any number of variables with integer coefficients. Since we have only one equation but n variables, infinitely many solutions exist
Undecidable_problem
Undecidability of equality of real numbers
that generated by rational numbers, the number π, the number ln 2, the variable x, the operations of addition, subtraction, multiplication, composition
Richardson's_theorem
Target set of a mathematical function
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Codomain
Process of repeating items in a self-similar way
follows: If a proposition is an axiom, it is a provable proposition. If a proposition can be derived from true reachable propositions by means of inference
Recursion
Set of elements in any of some sets
Pierpont, James (1912). Lectures On The Theory Of Functions Of Real Variables Vol II. Osmania University, Digital Library Of India. Ginn And Company
Union_(set_theory)
Mathematical model for deduction or proof systems
systems includes Indian logic of Pāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long (c. 325–250 BCE)
Formal_system
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
Mathematical proof expressed visually
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Proof_without_words
Set of elements common to all of some sets
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Intersection_(set_theory)
Components of a mathematical or logical formula
A first-order term is recursively constructed from constant symbols, variable symbols, and function symbols. An expression formed by applying a predicate
Term_(logic)
Rule defining the correct structure of expressions in formal grammar
as a propositional calculus, with the addition of quantifiers such that if we take Φ to be a formula of propositional logic and α as a variable then we
Formation_rule
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
Theorem for proving more complex theorems
fields, a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also
Lemma_(mathematics)
Set that is not a finite set
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Infinite_set
Type of theory in mathematical logic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Categorical_theory
Mathematical set containing all objects
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Universal_set
Kind of proof calculus
specified – see § Propositional inference rules (Suppes–Lemmon style). This section defines the formal syntax for a propositional logic language, contrasting
Natural_deduction
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
Subset of a function's codomain
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Range_of_a_function
Additional mathematical object
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Mathematical_structure
One-to-one correspondence
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Bijection
Form of mathematical proof
but it does so by a finite chain of deductive reasoning involving the variable n {\displaystyle n} , which can take infinitely many values. The result
Mathematical_induction
Theory of truth in the philosophy of language
(relative to an assignment of values to the variables x1, ..., xn)) if the corresponding values of variables bear the relation expressed by the predicate
Semantic_theory_of_truth
Axiomatic logical system
Burgess (2005, p. 42) (cf. also the axioms of first-order arithmetic). Variables not bound by an existential quantifier are bound by an implicit universal
Robinson_arithmetic
Set-theoretic concept
also be defined in a topos. As an example, we will prove an easy proposition. Proposition. If x ∈ U {\displaystyle x\in U} and y ⊆ x {\displaystyle y\subseteq
Grothendieck_universe
Term in mathematical logic
appearance of a predicate on specific elements is replaced by a new propositional variable. Equalities are replaced by their truth values according to their
Spectrum_of_a_sentence
Function that preserves distinctness
graphical approach for a real-valued function f {\displaystyle f} of a real variable x {\displaystyle x} is the horizontal line test. If every horizontal line
Injective_function
Term in logic and deductive reasoning
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Soundness
Subfield of mathematics
values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics
Mathematical_logic
Mathematical set that can be enumerated
Press. p. 141. ISBN 978-0-8247-7915-3. Apostol, Tom M. (June 1969), Multi-Variable Calculus and Linear Algebra with Applications, vol. 2 (2nd ed.), New York:
Countable_set
Mathematical logic concept
transformed the assertion of consistency into an arithmetic proposition. He could show that this proposition can neither be proved nor disproved within the formalism
Gentzen's_consistency_proof
Problem in computer science
about natural numbers is true or false. The reason for this is that the proposition stating that a certain program will halt given a certain input can be
Halting_problem
Computation model defining an abstract machine
state-trajectory, this is not true for the "copy" machine that can be provided with variable input "parameters". The diagram "progress of the computation" shows the
Turing_machine
Symbol representing a mathematical concept
function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply a constant symbol
Function_symbol
Index of articles associated with the same name
{\displaystyle \sigma } as the variable x. A formula is stratified if and only if it is possible to assign types to all variables appearing in the formula in
Stratification_(mathematics)
Consistency of the axioms of arithmetic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Hilbert's_second_problem
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
Boy/Male
Anglo, Australian, British, English, French, Swedish
Variable; Brave with the Spear; Spear Rule
Boy/Male
Anglo, British, English
Variable
Biblical
according to variable songs or tunes,
Surname or Lastname
English
English : topographic name for someone living on (and farming) a hide of land, Old English hī(gi)d. This was a variable measure of land, differing from place to place and time to time, and seems from the etymology to have been originally fixed as the amount necessary to support one (extended) family (Old English hīgan, hīwan ‘household’). In some cases the surname is habitational, from any of the many minor places named with this word, as for example Hyde in Greater Manchester, Bedfordshire, and Hampshire.English : variant of Ide, with inorganic initial H-. Compare Herrick.Jewish (American) : Americanized spelling of Haid.
Girl/Female
Biblical
According to variable songs or tunes.
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from Saint-Hilaire-du-Harcouët in La Manche, which gets its name from the dedication of its church to St. Hilary, or alternatively from either of the places, in La Manche and Somme, called Saint-Lô. Both of the latter are named from a 6th-century St. Lauto, bishop of Coutances; his name is of variable form in the sources and uncertain etymology.North German : habitational name for someone from Sandel.Jewish (eastern Ashkenazic) : occupational name for a cobbler or shoemaker, Yiddish sandler (from Hebrew sandelar, from Late Latin sandalarius, an agent derivative of sandalium ‘shoe’).
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from Déville in Seine-Maritime, France, probably named with Latin dei villa ‘settlement of (i.e. under the protection of) God’. This name was interpreted early on as a prepositional phrase de ville or de val and applied to dwellers in a town or valley (see Ville and Vale).English : nickname from Middle English devyle, Old English dēofol ‘devil’ (Latin diabolus, from Greek diabolos ‘slanderer’, ‘enemy’), referring to a mischievous youth or perhaps to someone who had acted the role of the Devil in a pageant or mystery play.French : variant of Ville, with the preposition de.
Surname or Lastname
English
English : from a medieval male personal name (from Latin Hilarius, a derivative of hilaris ‘cheerful’, ‘glad’, from Greek hilaros ‘propitious’, ‘joyful’). The Latin name was chosen by many early Christians to express their joy and hope of salvation, and was borne by several saints, including a 4th-century bishop of Poitiers noted for his vigorous resistance to the Arian heresy, and a 5th-century bishop of Arles. Largely due to veneration of the first of these, the name became popular in France in the forms Hilari and Hilaire, and was brought to England by the Norman conquerors.English : from the much rarer female personal name Eulalie (from Latin Eulalia, from Greek eulalos ‘eloquent’, literally well-speaking, chosen by early Christians as a reference to the gift of tongues), likewise introduced into England by the Normans. A St. Eulalia was crucified at Barcelona in the reign of the Emperor Diocletian and became the patron of that city. In England the name underwent dissimilation of the sequence -l-l- to -l-r- and the unfamiliar initial vowel was also mutilated, so that eventually the name was considered as no more than a feminine form of Hilary (of which the initial aspirate was in any case variable).
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
Surname or Lastname
English
English : variant spelling of Mitcham.
Female
Egyptian
, the sun.
Boy/Male
Australian, Finnish
Valley
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Elder; Ancestors
Boy/Male
Anglo Saxon
Name of several kings.
Girl/Female
Sikh
Light, Lustrous, Power
Girl/Female
Australian, French, Hawaiian, Hebrew
Light; Enlightens; Glowing; Encourages
Boy/Male
Scottish Celtic
At the ridge.
Boy/Male
Hindu, Indian, Kannada, Tamil, Telugu
Horizon
Girl/Female
Hindu
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
n.
The inferred proposition of a syllogism; the necessary consequence of the conditions asserted in two related propositions called premises. See Syllogism.
n.
That which is proposed; that which is offered, as for consideration, acceptance, or adoption; a proposal; as, the enemy made propositions of peace; his proposition was not accepted.
n.
Any number or quantity in a proportion; as, a mean proportional.
a.
Capable of being proportioned, or made proportional; also, proportional; proportionate.
n.
A proposition collected from the agreement of other previous propositions; any conclusion which results from reason or argument; inference.
n.
That which is offered or affirmed as the subject of the discourse; anything stated or affirmed for discussion or illustration.
n.
A statement of religious doctrine; an article of faith; creed; as, the propositions of Wyclif and Huss.
a.
Of or pertaining to a preposition; of the nature of a preposition.
n.
A disjunctive proposition.
a.
Relating to, or securing, proportion.
a.
Pertaining to, or in the nature of, a proposition; considered as a proposition; as, a propositional sense.
n.
The part of a poem in which the author states the subject or matter of it.
n.
The combining weight or equivalent of an element.
n.
A disjunctive proposition.
a.
Constituting a proportion; having the same, or a constant, ratio; as, proportional quantities; momentum is proportional to quantity of matter.
n.
A complete sentence, or part of a sentence consisting of a subject and predicate united by a copula; a thought expressed or propounded in language; a from of speech in which a predicate is affirmed or denied of a subject; as, snow is white.
n.
A statement in terms of a truth to be demonstrated, or of an operation to be performed.
a.
Following by necessary inference or rational deduction; as, a proposition consequent to other propositions.
a.
Having a due proportion, or comparative relation; being in suitable proportion or degree; as, the parts of an edifice are proportional.
n.
A subaltern proposition.