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A predicate is a statement or mathematical assertion that contains variables, sometimes referred to as predicate variables, and may be true or false depending on those variables’ value or values. In propositional logic, atomic formulas are sometimes regarded as zero-place predicates. [1] In a sense, these are nullary (i.e. 0-arity) predicates.
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.)
The following example in first-order logic (=) is a sentence. This sentence means that for every y, there is an x such that =. This sentence is true for positive real numbers, false for real numbers, and true for complex numbers. However, the formula
Propositional logic deals with statements, which are defined as declarative sentences having truth value. [29] [1] Examples of statements might include: Wikipedia is a free online encyclopedia that anyone can edit. London is the capital of England. All Wikipedia editors speak at least three languages.
A predicate evaluates to true or false for an entity or entities in the domain of discourse. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences themselves are viewed as the individuals of study, and might be denoted, for example, by variables such as p and q.
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.
Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples: The column-14 operator (OR), shows Addition rule: when p=T (the hypothesis selects the first two lines of the table), we see (at column-14) that p∨q=T.
If "predicate variables" are only allowed to be bound to predicate letters of zero arity (which have no arguments), where such letters represent propositions, then such variables are propositional variables, and any predicate logic which allows second-order quantifiers to be used to bind such propositional variables is a second-order predicate ...