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An open formula is a formula that contains at least one free variable. [citation needed] An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like true or false. An open formula can be transformed into a closed formula by applying a ...
A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier ∀ {\displaystyle \forall } in the first order formula ∀ x P ( x ) {\displaystyle \forall xP(x)} expresses that everything in the domain satisfies the property denoted by P ...
Formulas in logic are typically built up recursively from some propositional variables, some number of logical connectives, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic, and are often denoted using capital roman letters such as P {\displaystyle P} , Q {\displaystyle Q} and R {\displaystyle ...
Propositional constants represent some particular proposition, [56] while propositional variables range over the set of all atomic propositions. [56] Schemata, or schematic letters , however, range over all formulas.
Two important types of logical constants are logical connectives and quantifiers. The equality predicate (usually written '=') is also treated as a logical constant in many systems of logic . One of the fundamental questions in the philosophy of logic is "What is a logical constant?"; [ 1 ] that is, what special feature of certain constants ...
Generally, automated theorem provers focus on supporting full first-order logic with quantifiers, whereas SMT solvers focus more on supporting various theories (interpreted predicate symbols). ATPs excel at problems with lots of quantifiers, whereas SMT solvers do well on large problems without quantifiers. [23]
A second-order propositional logic is a propositional logic extended with quantification over propositions. A special case are the logics that allow second-order Boolean propositions, where quantifiers may range either just over the Boolean truth values, or over the Boolean-valued truth functions.