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A propositional formula may also be called a propositional expression, a sentence, [1] 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:
In mathematical logic, a sentence (or closed formula) [1] of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition , something that must be true or false.
Composite formulas (all formulas besides atoms) are called molecules, [49] or molecular sentences. [34] (This is an imperfect analogy with chemistry, since a chemical molecule may sometimes have only one atom, as in monatomic gases.) [49]
A formula in first-order logic with no free variable occurrences is called a first-order sentence. These are the formulas that will have well-defined truth values under an interpretation. For example, whether a formula such as Phil(x) is true must depend on what x represents.
A tautology in first-order logic is a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable).
A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first-order language in which the variables v 1, …, v n have free occurrences, then A preceded by ∀v 1 ⋯ ∀v n is a universal closure of A.
A simple sentence consists of a single independent clause with no dependent clauses. A compound sentence consists of multiple independent clauses with no dependent clauses. These clauses are joined together using conjunctions, punctuation, or both. A complex sentence consists of one independent clause and at least one dependent clause.
A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are φ {\displaystyle \varphi } (or φ ( x ) {\displaystyle \varphi (x)} to indicate x {\displaystyle x} is the unbound variable in φ {\displaystyle \varphi } ) and ψ {\displaystyle \psi } (or ψ ( x ...
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