Non-Existence (Quantifier Negation)
Negating a universal claim produces an existential claim: at least one case fails the condition. Negating an existential claim produces a universal claim: no case meets the condition. Proving either negation requires evidence or reasoning appropriate to its scope.
THE IDEA
NON-EXISTENCE
Imagine I say “Every female science student has been hit on by their supervisor.” I actually heard someone say this at a panel event for women in science. Suppose you want to point out that this isn’t true. What do you need to do? You just need to find one female science student who was not hit on by their supervisor, for example, me.
Whereas if I say “Some female science students have been hit on by their supervisor” and you want to say this isn’t true, you must establish that no female science student has been hit on by their supervisor. Checking every student would be one way to try to establish that, but it is not feasible for this example. In other settings, a general proof can establish a universal statement without checking cases individually.
In the first case you are trying to negate a “for all” statement, and in the second case you are negating a “there exists” statement.
We have the following negations:
- World: All female science students in time.
Original statement: For all female science students X, X has been hit on by their supervisor.
Negation: There exists a female science student X such that X was not hit on by their supervisor.
- World: All female science students in time.
Original statement: There exists a female science student X such that X has been hit on by their supervisor.
Negation: For all female science students X, X has not been hit on by their supervisor.
Just like “and” and “or”, these two quantifiers go hand in hand as related by the negations: when you negate a statement involving one you get a statement involving the other, just like with “and” and “or”.
If we add quantifiers to our logical language we now have what is called predicate logic, or first-order logic. The word “predicate” is to distinguish from “propositional”, which is what we had without the quantifiers. “First-order” is to distinguish it from higher-order versions of logic, which are more complicated in the way that the quantifiers work.1
When to use it
Debunking overgeneralizations in everyday arguments or media.
Designing scientific experiments to test hypotheses (e.g., falsifying a theory by finding a counterexample).
Evaluating the strength of evidence in legal cases (e.g., proving someone 'always' does something vs. proving someone 'sometimes' does it).
Understanding logical fallacies like 'hasty generalization' (inferring a universal from a few instances).
FROM THEORY TO EVERYDAY LIFE
In practice
Negating a Universal Claim
To disprove "Every female science student has been hit on by their supervisor," one counterexample is enough: a female science student to whom that has not happened.
Negating an Existential Claim
To disprove "Some female science student has been hit on by their supervisor," you must establish that it has happened to none of them. The negation is universal, though universal claims do not always require checking cases one by one.
A MOMENT TO REFLECT
Put it to work.
Can you think of a recent news claim that is a universal statement? How would you prove it false?
When making an argument, why is it often easier to disprove a universal claim than an existential one?
How does the difficulty of negating existential statements relate to the problem of induction in science?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)