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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
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
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
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
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
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)
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
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
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)
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
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
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
Symbol representing a mathematical object
Lambda calculus Observable variable Physical constant Propositional variable Sobolev, S.K. (originator). "Individual variable". Encyclopedia of Mathematics
Variable_(mathematics)
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)
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
Statement that is taken to be true
an infinite number of axioms. For example, if A, B, and C are propositional variables, then A→(B→A) and (A→¬B)→(C→(A→¬B)) are both instances of axiom
Axiom
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
Mathematical operation with two operands
are binary operations on sets. There are 16 binary operations on binary variables, such as 'AND', 'OR', and 'XOR'. If a broad definition of "binary operation"
Binary_operation
Logical formulation of recursion
P {\displaystyle P} is a second-order variable, x → {\displaystyle {\vec {x}}} a tuple of first-order variables, t → {\displaystyle {\vec {t}}} a tuple
Fixed-point_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
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
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
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
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
Infinite cardinal number
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Aleph_number
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
Logical connective
Implicational propositional calculus Laws of Form Logical graph Logical equivalence Material implication (rule of inference) Peirce's law Propositional calculus
Material_conditional
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
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
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
Problem in computer science
point of view". 1931 (1931): Gödel publishes "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I". 19 April 1935 (1935-04-19):
Halting_problem
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
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
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
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
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
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
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
Set of all things that may be the input of a mathematical function
f is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, and the set of real numbers on
Domain_of_a_function
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
Establishment of a theorem using inference from the axioms
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Formal_proof
Theorem for proving more complex theorems
fields, a lemma (pl.: lemmas or lemmata) is a generally minor proven proposition used to prove a larger statement. For that reason, it is also known as
Lemma_(mathematics)
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)
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
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
Standard form of a boolean function
used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for DNF: DNF → {\displaystyle
Disjunctive_normal_form
In mathematical logic, a well-formed formula with no free variables
free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is
Sentence_(mathematical_logic)
Collection of mathematical objects
objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Sets cannot be mathematically defined
Set_(mathematics)
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
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
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
Concept in logic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Logical_equivalence
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
Symbols requiring interpretation
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Non-logical_symbol
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
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)
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Saturated_set
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
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
Additional mathematical object
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Mathematical_structure
Symbol representing a property or relation in logic
contains variables, sometimes referred to as predicate variables, and may be true or false depending on those variables’ value or values. In propositional logic
Predicate_(logic)
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
Function in mathematical logic
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Gödel_numbering
Standard system of axiomatic set theory
metavariables for any wff, and x {\displaystyle x} be a metavariable for any variable. These are all the valid wff constructions: ¬ ϕ {\displaystyle \lnot \phi
Zermelo–Fraenkel_set_theory
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 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
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)
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
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
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
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 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
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
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)
Logical quantifier
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Uniqueness_quantification
Infinite set that is not countable
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Uncountable_set
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
Identities and relationships involving sets
weakened to the rule (A∁)∁=A, then this is exactly the algebra of propositional linear logic[clarification needed]. Each of the identities stated above
Algebra_of_sets
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
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
Logical connective AND
Logical connective OR Logical graph – Type of diagrammatic notation for propositional logicPages displaying short descriptions of redirect targets Negation –
Logical_conjunction
Data whose unit can take on only two possible states
structure for collection of bits, with a semantic of a collection of propositional variables. Boolean algebra operations are known as "bitwise operations" in
Binary_data
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)
Yes/no problem in computer science
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Decision_problem
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
Function, homomorphism, or morphism
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Map_(mathematics)
Logical principle
diagrammatic notation for propositional logicPages displaying short descriptions of redirect targets: a graphical syntax for propositional logic Mathematical
Law_of_excluded_middle
Mathematical table used in logic
of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate
Truth_table
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
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
Mathematical set formed from two given sets
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Cartesian_product
Set theory concept
The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be
Large_cardinal
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
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