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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 ...
Russell and Whitehead found it impossible to develop mathematics while maintaining the difference between predicative and non-predicative functions, so they introduced the axiom of reducibility, saying that for every non-predicative function there is a predicative function taking the same values.
Given a formula X, the negation ¬X is a formula. Given two formulas X and Y, and a binary connective b (such as the logical conjunction ∧), the expression (X b Y) is a formula. (Note the parentheses.) Through this construction, all of the formulas of propositional logic can be built up from propositional variables as a basic unit.
A definition of something as the unique object with a given property descriptive function A function taking values that need not be truth values, in other words what is not called just a function. diversity The inequality relation domain The domain of a relation R is the class of x such that xRy for some y. elementary proposition
Definition 1: Atomic propositional variables are formulas. Definition 2: If is a propositional connective, and A, B, C, … is a sequence of m, possibly but not necessarily atomic, possibly but not necessarily distinct, formulas, then the result of applying to A, B, C, … is a formula.
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 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 that within the sentence there is a variable (x) that is not defined or specified (thus being a free variable), which leaves the statement undetermined.
A formula is logically valid (or simply valid) if it is true in every interpretation. [22] These formulas play a role similar to tautologies in propositional logic. A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.