Predicate Logic : Standard Predicates

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Standard Predicates

There are two, "standard" predicates, i.e. functions that take something in and return a boolean value.

One of them is the ForAll(x,y), where y is a predicate and x is one of y's arguments. The other one is the ExistsAtLeastOne(x,y), where y is a predicate and x is one of y's arguments. Like all predicates as functions that return boolean values, the two standard functions can be combined by using propositional logic. The "standard" predicates have a "standard" notation:

ForAll(x,y) is written as $\forall x y$ or as $\forall x(y)$

ExistsAtLeastOne(x,y) is written as $\exists x y$ or as $\exists x(y)$

Sometimes some mathematicians also use a third "standard function", the ExistsExactlyOne(x,y), where y is a predicate and x is one of y's arguments. The ExistsExactlyOne(x,y) has standard notations of $\exists!x y$ and $\exists!x(y)$ .


Semantics of the Standard Predicates

The ForAll(x,y) returns true only, if for every x in the set of all possible x values the predicate y returns true.

The ExistsAtLeastOne(x,y) returns true only, if for at least one x in the set of all possible x values the predicate y returns true.

As the standard functions accept other predicates as its arguments, they can be "fed to eachother" in various ways.