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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)
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
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
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
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)
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
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
Axiomatic system of formal deduction in logic
particular formulas containing propositional variables and add a primitive rule of uniform substitution. For propositional logic this rule has the form
Hilbert_system
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)
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
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
Symbol representing a mathematical object
Lambda calculus Observable variable Physical constant Propositional variable Sobolev, S.K. (originator). "Individual variable". Encyclopedia of Mathematics
Variable_(mathematics)
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
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
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)
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
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
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
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
Type of infinite structure
{\displaystyle M} is o-minimal if and only if every formula with one free variable and parameters in M {\displaystyle M} is equivalent to a quantifier-free
O-minimal_theory
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
Basic framework of mathematics
and the basis of propositional calculus. Independently, in the 1870's, Charles Sanders Peirce and Gottlob Frege extended propositional calculus by introducing
Foundations_of_mathematics
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)
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
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
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
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
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
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 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
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
Logical connective AND
Logical connective OR Logical graph – Type of diagrammatic notation for propositional logicPages displaying short descriptions of redirect targets Negation –
Logical_conjunction
Logic principle
properties). There are various extensionality principles in mathematics. Propositional extensionality of predicates P , Q {\displaystyle P,Q} : if P ⟺ Q {\displaystyle
Extensionality
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
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)
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)
Logical connective
Implicational propositional calculus Laws of Form Logical graph Logical equivalence Material implication (rule of inference) Peirce's law Propositional calculus
Material_conditional
Function in mathematical logic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Gödel_numbering
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
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
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
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
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
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
Infinite cardinal number
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Aleph_number
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
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
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
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
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
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
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
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
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)
Category of mathematical proof
particle. There is also Bell's theorem: no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. While
Proof_of_impossibility
Every set is smaller than its power set
shows 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
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
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
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
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
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)
Study of the semantics, or interpretations, of formal and natural languages
traditionally is not interested in the sentence as uttered but in the proposition, an idealised sentence suitable for logical manipulation.[citation needed]
Semantics_(logic)
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 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)
Area of mathematical logic
formula in one variable. Quantifier-free formulas in one variable express Boolean combinations of polynomial equations in one variable, and since a nontrivial
Model_theory
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
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
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
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
Function, homomorphism, or morphism
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Map_(mathematics)
Topics referred to by the same term
used in many sciences Propositional variable, taking the value true or false in mathematical logic Random variable, a variable in statistics whose value
Variable
Type of logical argument that applies deductive reasoning
First, in the realm of foundations, Boole reduced Aristotle's four propositional forms to one form, the form of equations, which by itself was a revolutionary
Syllogism
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
Additional mathematical object
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Mathematical_structure
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
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)
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)
Mathematical-logic system
expressing computation based on function abstraction and application using variable binding and substitution. Untyped lambda calculus, the topic of this article
Lambda_calculus
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
Apparent contradiction in metamathematics
incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The paradox was also
Richard's_paradox
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
Undecidability of equality of real numbers
rational numbers, the number π, the number ln 2 {\displaystyle \ln 2} , the variable x {\displaystyle x} , the operations of addition, subtraction, multiplication
Richardson's_theorem
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
Mathematical concept
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Transfinite_induction
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
Proof by Alan Turing
in (i) logic (ii) the paper of Kurt Gödel: "On Formally Undecidable Propositions of Principia Mathematica and Related Systems". For assistance with Gödel's
Turing's_proof
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)
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
Theorem equivalent to the Axiom of Choice
implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest. The goal
Tarski's_theorem_about_choice
Abstract mathematics problem
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Ross–Littlewood_paradox
Theorem in mathematical logic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Lindström's_theorem
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
Theorem in set theory
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Kőnig's_theorem_(set_theory)
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
Concept in logic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Logical_equivalence
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
Class of mathematical set whose elements are all subsets
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Transitive_set
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
Sequence of words formed by specific rules
contains infinitely many elements x0, x1, x2, … that play the role of variables. See e.g. Reghizzi, Stefano Crespi (2009). Formal Languages and Compilation
Formal_language
PROPOSITIONAL VARIABLE
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PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE
PROPOSITIONAL VARIABLE