Negation of Quantified Statements
Learn about the negation of logical statements involving quantifiers and the role of DeMorgan’s laws in negating quantified statements.
Negation of universal quantifier
Assume the statement , for some domain . This statement asserts that every element of has the property . By negation, we mean that it is not the case. If it is not the case that every element of has the property , then there must be at least one element in that does not have the property . That is,
Similarly, the negation has the same effect for more than one universal quantifier. Again, with a generic domain , assume the following statement:
This statement asserts that for every selection of three elements and , from , the predicate is true. The negation of this statement means that it is not the case. This means there must be at least one selection of three elements and , such that is not true. We can state this fact as follows:
Let’s look at a few examples.
Examples
Let’s look at a few examples:
Assume the domain to be , which is the set of all mammals. That is,
is a mammal.
Make a predicate as follows:
- : has four legs.
Quantify this predicate to make the following statement:
- : Every mammal has four legs.
The negation of this statement is as follows:
- : It is not the case that every mammal has four legs.
The logical equivalent of this statement is as follows:
- : There exists a mammal that does not have four legs.
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