Logical Statements

There are two more important symbols in logic.

$ \exists $ translates to "there exists," and $ \forall $ translates to "for all."

These are used to state how much of a group the next statement will effect. For example,

There exists a school named Kibo.

There exists some schools that teach computer science.

For all students at Kibo, the students study computer science.

For all classes at Kibo, the class is related to computer science.

"There exists" refers to the fact that at least one of the group follows the next statement. There could be more than one member, but for the statement to be truthful, only one member needs to fall under the statement.

"For all" requires that all members of the group follow the statement. If even one is false, the entire "for all" statement is false.

Check your understanding: All four statements before were true. Can you pick out which statements below would be false?

  • For all schools, a computer science degree is offered.
  • There exists a student who studies at Kibo.
  • There exists a founder of Kibo named John.
  • For all students, the students study computer science.
Check your answer
  • For all schools, a computer science degree is offered.

False, there are trade schools, high schools, etc.

  • There exists a student who studies at Kibo.

True, you are an example!

  • There exists a CEO of Kibo named John.

False, the founders were Ope and Rob (with the help of Keno).

  • For all students, the students study computer science.

False, not every student everywhere studies computer science. Who would fly our planes or cook our food?

Notice, we can turn a false statement into a true statement by negating it. To negate one of these statements, we change "there exists" into "for all" (or the other way around), and then negate the statement that follows. For example,

For all schools, a computer science degree is offered.

Can be negated as

There exists a school where a computer science degree is not offered.

One of these statements is true, and the other is false.

Check your understanding: Try negating all false statements from the previous list. Are all of your negations now true?

Check your answer
  • For all schools, a computer science degree is offered.

There exists a school which does not offer a computer science degree.

This is true, as mentioned before: trade schools and high schools exist.

  • There exists a CEO of Kibo named John.

Of all CEOs of Kibo, none are named John.

True, the founders were Ope and Rob (with the help of Keno).

  • For all students, the students study computer science.

There exists a student who does not study computer science.

True, there are some students out there with other interests.