Universal and Existential Quantifiers: The Logic of 'All' and 'Some'
Universal statements like 'All men are sexist pigs' require refutation by finding a single counterexample (existential statement). A joke about three logicians illustrates how individuals can deduce the truth of a universal claim through sequential self-reporting.
THE IDEA
Not Everyone Is Horrible
We can now revisit our emotional outburst, “All men are sexist pigs!” This is technically a “for all” statement:
For all X in the set of men, X is a sexist pig.
so to refute it you have to show that there exists someone in the set of men who is not a sexist pig. So this is the negation of the above statement:
There exists X in the set of men, such that X is not a sexist pig.
This means you just have to find one man who is not a sexist pig. My friend Greg is definitely not a sexist pig. (Admittedly I can’t really prove that without introducing you to him.)
One of my favourite mathematical jokes goes like this:
Three logicians walk into a bar. The bartender says “Would everyone like a beer?” The first logician says “I don’t know.” The second logician says “I don’t know.” The third logician says “Yes.”
The point is that the bartender has asked a “for all” question, and the three logicians, being logicians, know how to verify and refute it properly. Either:
• A: All three logicians want a beer, or • not A: There exists a logician who does not want a beer.
The first logician answers that they don’t know, which means they definitely want a beer: otherwise they would know that there exists a logician who does not want a beer.
Likewise the second logician must definitely want a beer, otherwise they in turn would know that there exists a logician who does not want a beer. The third logician can then answer on behalf of “everyone” because they also want a beer.
Like in many mathematical jokes, there is an element of truth in this that I find slightly endearing. I have spent enough time around mathematicians to know that this sort of precision is likely to spill over into their normal lives, where normal people would just answer “Yes” if they want a beer, even though that’s not the technically correct answer to the question. Is it pedantic? It is in fact illuminating to the logicians, so perhaps it is still just narrowly on the side of precision.
When to use it
Debunking stereotypes or overgeneralizations in conversations or social media debates.
Evaluating scientific claims: a single well-documented counterexample can falsify a universal hypothesis.
Clarifying vague statements in meetings: asking 'Does this apply to all cases or just some?' to avoid logical errors.
Improving critical thinking by recognizing when a universal proposition requires only one exception to be disproven.
FROM THEORY TO EVERYDAY LIFE
In practice
Refuting a Sexist Generalization
The claim 'All men are sexist pigs' is a universal statement. To refute it, you only need to find one man who is not a sexist pig, such as 'my friend Greg.' A single counterexample disproves a universal claim.
The Three Logicians Joke
In a bar, the bartender asks 'Would everyone like a beer?' Each logician answers in turn. The first two say 'I don't know,' implying they each want a beer—otherwise they'd know a counterexample exists. The third knows that since the first two want beer and she also wants one, the universal statement 'everyone wants a beer' is true, so she answers 'Yes.'
A MOMENT TO REFLECT
Put it to work.
Think of a sweeping generalization you've heard recently. What specific counterexample could you provide to challenge it?
How might the logicians' method of verifying a universal statement apply to a team decision-making scenario?
When have you personally mistaken an existential claim (e.g., 'some people are unreliable') for a universal one (e.g., 'everyone is unreliable')?
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)