Scopus (logic)

In the logic is meant by the region , the range or the scope of the negation ( engl. Microscope "area" of lat. Scopus "target") of a quantifier the shortest formula, immediately following this quantifier.

The term is used, for example, to define the concepts of freedom and the constraint of variables .

The following applies:

An occurrence of a variable is free in a formula B if it does not occur in the range of a quantifier.
An occurrence of a variable is bound by an occurrence of a quantifier if the variable occurs in area B of the quantifier and if the variable in B is free.

Explanation and examples

As an example, consider the two statements

A:

{\ displaystyle \ forall x (\ exists xG (x) \ wedge (F (x) \ supset H (x)))}

and

B:

{\ displaystyle \ forall x (\ exists x (G (x) \ wedge F (x)) \ supset H (x))}

The scope of the existential quantifier in statement A consists only of the formula , in statement B it consists of , the scope of the universal quantifier in both cases covers the entire formula. ${\ displaystyle \ exists x}$ ${\ displaystyle G (x)}$ ${\ displaystyle G (x) \ wedge F (x)}$ ${\ displaystyle \ forall x}$

We can now show that the x in in A is bound by the universal quantifier and in B by the existential quantifier. In the first case, the scope of the universal quantifier is the formula: ${\ displaystyle F (x)}$

{\ displaystyle \ exists xG (x) \ wedge (F (x) \ supset H (x))}

As already said, the x from here is not in the scope of the existential quantifier. So the occurrence is free. A free occurrence of a variable in the scope of a quantifier is bound by this quantifier, so the variable is bound by the universal quantifier. ${\ displaystyle F (x)}$

In B the scope of the existential quantifier is the formula:

{\ displaystyle G (x) \ wedge F (x)}

Both variables are free in this formula, so they are bound by the existential quantifier.

The Scopus difference also corresponds to a difference in the meaning of the two formulas: To clarify this, we interpret G (x) as “x is God”, F (x) “x is just” and H (x) as “x is happy". Then statement A is to be read as

There is one God and all righteous are happy

Statement B on the other hand as

If there is a God who is righteous, everyone is happy.

Scopus (logic)

Explanation and examples

See also