In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula.
A propositional formula is constructed from simple propositions, such as "five is greater than three" or propositional variables such as p and q, using connectives or logical operators such as NOT, AND, OR, or IMPLIES; for example:
(p AND NOT q) IMPLIES (p OR q).
In mathematics, a propositional formula is often more briefly referred to as a "proposition", but, more precisely, a propositional formula is not a proposition but a formal expression that denotes a proposition, a formal object under discussion, just like an expression such as "x + y" is not a value, but denotes a value. In some contexts, maintaining the distinction may be of importance.
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In propositional logic, a propositionalformula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables...
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes...
function. Propositional variables are the basic building-blocks of propositionalformulas, used in propositional logic and higher-order logics. Formulas in logic...
needed] In relation to the mind, propositions are discussed primarily as they fit into propositional attitudes. Propositional attitudes are simply attitudes...
logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains...
nested within an OR In classical logic each propositionalformula can be converted to an equivalent formula that is in CNF. This transformation is based...
calculus. This is similar to a way of axiomatizing classical propositional logic. In propositional logic, the inference rule is modus ponens MP: from ϕ → ψ...
of propositional logic correspond to equational theorems of Boolean algebra. Syntactically, every Boolean term corresponds to a propositionalformula of...
as a propositionalformula in k {\displaystyle k} variables x 1 , . . . , x k {\displaystyle x_{1},...,x_{k}} , and two propositionalformulas are logically...
D))} , since an OR is nested within an AND In classical logic each propositionalformula can be converted to DNF ... The conversion involves using logical...
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...
constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective ∨ {\displaystyle \lor...
then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically...
Skeptical entailment deciding whether a propositional default theory skeptically entails a propositionalformula is Π 2 P {\displaystyle \Pi _{2}^{P}} -complete;...
formal language for classical propositional logic can be expressed using just negation (¬), implication (→) and propositional symbols. A well-known axiomatization...
the positive propositional calculus, the questions of validity and satisfiability may be unrelated. In the case of the positive propositional calculus, the...
only implies an existential quantification of the propositional function. By convention, the formula ∀ x ∈ ∅ P ( x ) {\displaystyle \forall {x}{\in }\emptyset...
to be a propositionalformula we can also take ¬ {\displaystyle \neg } Φ to be a formula; if we take Φ and Ψ to be a propositionalformulas we can also...
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...
established in (Carnielli 1991). A formula in propositional logic consists of letters, which stand for propositions, and connectives for conjunction, disjunction...
theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural...
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