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In 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 are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, [1] or a sentential formula.
In this way, the truth value of the compound sentence is defined as a certain truth function of the truth values of the simpler sentences. The connectives are usually taken to be logical constants, meaning that the meaning of the connectives is always the same, independent of what interpretations are given to the other symbols in a formula.
Sentences are then built up out of atomic sentences by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory.
Propositional logic, as currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence, or whether a sentence logically follows from some other sentence or group of sentences.
A sentence is a wff in which any variables are bound. An atomic sentence is an atomic formula containing no variables. It follows that an atomic sentence contains no logical connectives, variables, or quantifiers. A sentence consisting of one or more sentences and a logical connective is a compound (or molecular) sentence.
In some logical calculi (notably, in classical logic), certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy is the classical equivalence between ¬ p ∨ q {\displaystyle \neg p\vee q} and p → q {\displaystyle p\to q} .
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.
In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. For example, a formula that states, "the ball is green or the ball is not green," is always true, regardless of what a ball is ...