Predicate calculus

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In mathematical logic the predicate calculus, predicate logic or calculus of propositional functions is a formal system used to describe mathematical theories.

The predicate calculus is an extension of propositional calculus, which is inadequate for describing more complex mathematical structures. Grammatically speaking the predicate calculus adds a predicate-subject structure and quantifiers on top of the existing propositional calculus. A subject is a name for a member of a given group of individuals (a set) and a predicate is a relation on this group.

Inference rules

<math>\neg \bigwedge_x P(x) \Leftrightarrow \bigvee_x \neg P(x)</math>

<math>\neg \bigwedge_x \neg P(x) \Leftrightarrow \bigvee_x P(x)</math>

<math>\neg \bigvee_x P(x) \Leftrightarrow \bigwedge_x \neg P(x)</math>

<math>\neg \bigvee_x \neg P(x) \Leftrightarrow \bigwedge_x P(x)</math>

<math>\bigwedge_x \bigwedge_y P(x,y) \Leftrightarrow \bigwedge_y \bigwedge_x P(x,y)</math>

<math>\bigvee_x \bigvee_y P(x,y) \Leftrightarrow \bigvee_y \bigvee_x P(x,y)</math>

<math>\bigvee_x \bigwedge_y P(x,y) \Rightarrow \bigwedge_y \bigvee_x P(x,y)</math>

<math>\bigwedge_x P(x) \Rightarrow P(x)</math>

<math>P(x) \Rightarrow \bigvee_x P(x)</math>

See also

• First order predicate calculus
• Second order predicate calculus