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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.
The notion of a predicate in traditional grammar traces back to Aristotelian logic. [2] A predicate is seen as a property that a subject has or is characterized by. A predicate is therefore an expression that can be true of something. [3] Thus, the expression "is moving" is true of anything that is moving.
[2] Example. In a given propositional logic, a formula can be defined as follows: Every propositional variable is a formula. 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.)
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information.
A predicative expression (or just predicative) is part of a clause predicate, and is an expression that typically follows a copula or linking verb, e.g. be, seem, appear, or that appears as a second complement of a certain type of verb, e.g. call, make, name, etc. [1] The most frequently acknowledged types of predicative expressions are predicative adjectives (also predicate adjectives) and ...
For example, propositional logic is decidable, because the truth-table method can be used to determine whether an arbitrary propositional formula is logically valid. First-order logic is not decidable in general; in particular, the set of logical validities in any signature that includes equality and at least one other predicate with two or ...
The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything true in all models is provable".