Predicate Logic

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.

It is much harder to reason in predicate logic than in propositional calculus. In general, truth-tables are not suitable for predicate logic, as a universally quantified predicate may have an infinite domain of interest.

IDENTITIES

:

eg orall x P(x) \Leftrightarrow \exists x 
eg P(x)

eg \exists x P(x) \Leftrightarrow orall x 
eg P(x)

orall x orall y P(x,y) \Leftrightarrow orall y orall x P(x,y)

\exists x \exists y P(x,y) \Leftrightarrow \exists y \exists x P(x,y)

orall x P(x) \land orall x Q(x) \Leftrightarrow orall x (P(x) \land Q(x))

\exists x P(x) \lor \exists x Q(x) \Leftrightarrow \exists x (P(x) \lor Q(x))

INFERENCE RULES

:

\exists x orall y P(x,y) \Rightarrow orall y \exists x P(x,y)

orall x P(x) \lor orall x Q(x) \Rightarrow orall x (P(x) \lor Q(x))

\exists x (P(x) \land Q(x)) \Rightarrow \exists x P(x) \land \exists x Q(x)

\exists x P(x) \land orall x Q(x) \Rightarrow \exists x (P(x) \land Q(x))

orall x P(x) \Rightarrow P(c)

(If c is a variable, then it must not already be quantified somewhere in P(x))
:

P(c) \Rightarrow \exists x P(x)

(x must not appear free in P(c))

SEE ALSO

First Order Predicate Calculus

Second Order Predicate Calculus

Information About

Predicate Logic