IN THE LOVELY STOP MOTION film Chicken Run, the smooth-talking American rooster Rocky does a deal with the sly salesmen rats saying he will pay them “all the eggs he lays that month”. The rats are sly but not very knowledgeable about chickens, so they don’t realize that roosters don’t lay eggs, and that the total number of eggs Rocky lays that month is therefore going to be zero. Rocky is being entirely truthful in telling them that he will give them “all” those eggs, because “all” happens to be zero. Ginger, the heroic leader of the hens, is outraged and thinks Rocky has been dishonest. Rocky has certainly misled the rats but has not been strictly dishonest in terms of logic; he has just been dishonest in terms of suggestion or emotion.
A reverse emotional–logical situation occurs when someone has an emotional outburst and wails “Men are sexist pigs!”. Do they mean all men are sexist pigs? That seems rather extreme. Do they mean most men are sexist pigs? Still a bit melodramatic. Perhaps we can agree that some men are sexist pigs? This is now a true statement but it’s become rather tame–no wonder the wailer preferred the dramatic outburst.
Probably, in practice, they mean “I have encountered enough men who were sexist pigs today that I am fed up with it.” This takes longer to say but is more precise. It also sounds a bit pedantic, but perhaps it is actually illuminating: it expresses the fact that actually what’s going on is your emotional response of being fed up. However, to express that fed-up-ness it may be tempting to say “All men are sexist pigs!” but this might provoke someone to argue that not every man is a sexist pig, say, perhaps, Justin Trudeau. In this case you’ve expressed true emotions but with inaccurate logic, and in doing so you’ve tempted certain types of people to argue with your logic instead of soothe your emotions.
A toxic form of this kind of outburst is when someone accuses their partner of something like “You never do the washing up!” or “You always leave a mess in the kitchen!” The logical refutations of these statements are rather simple to prove:
• Statement: You never do the washing up!
Negation: I did the washing up one time.
• Statement: You always leave a mess in the kitchen.
Negation: There was one time I did not leave a mess in the kitchen.
Of course, the sweeping statement isn’t meant literally. More precisely, it probably means something like “I feel like you do vastly less than your fair share of the washing up, so little that it feels negligible, so I’m very frustrated and feel overworked and underappreciated.” Or “I feel like you leave a mess in the kitchen so often that cleaning it up becomes a big burden on me and I am tired of it.”
It would probably be not just more accurate but also more productive to say those things instead of the exasperated sweeping statements.
People are prone to making sweeping statements. See, I just did it myself. What did I mean? Did I mean that all people are prone to making sweeping statements? Do I mean some people are? This is certainly true, but hardly a very emphatic statement. Do I mean that most people are? I think everyone I know is, but I’ve only met a tiny proportion of all people, so really I should just say “Everyone I know is prone to making sweeping statements”. I have now refined my statement and made it less ambiguous, hence more defensible using logic. The way I did it was by refining its scope. Refining your scope means being precise about what world of objects you are focusing on.
“Mozart is more boring than Brahms” is a sweeping statement that many people would disagree with. Whereas “In my opinion Mozart is more boring than Brahms” is now just a statement about my taste, so nobody can logically disagree with it. It would be even more precise to say “In my opinion almost all Mozart is more boring than almost all Brahms.” I can probably think of one example of a piece of Mozart that I think is less boring than one example of a piece of Brahms: for example, I am not a huge fan of Brahms’s second symphony but I like Mozart’s Masonic Funeral Music. Whereas I can make a sweeping statement “The macarons in Paris are much better than the macarons in Chicago” by which I mean “In my experience every macaron I have bought in Paris was better than every macaron I have bought in Chicago.” Harsh, but true. Also impossible for someone other than me to refute logically.
“Almost all” is a form of qualifier that has a whole related family, like “most”, “some”. If you qualify a statement with one of those, together with perhaps “In my experience”, you can almost never be wrong. (See?) “Perhaps” also works, along with “probably”, “maybe” and “might”. Carefully qualifed statements with these words are very precise in their correctness, but do not make good headlines, so unfortunately the media tends to overstate everything. “New research shows that sugar causes cancer!” screams a headline, but if you look at the research what it shows is that there is evidence to suggest that there might be some kind of a connection between sugar consumption and cancer. We could also add “seems”, as in “It seems like there might be some kind of a connection between sugar and cancer.”
Looking for the truth in someone’s statement can be much more productive than pedantically demonstrating that they are wrong. I think this is an instance of the principle of charity, where you try to think the best of everyone all the time. Finding truth in someone’s sweeping statement by applying the right qualifiers can lead to greater understanding of what people are trying to say and where disagreements are coming from.
For example, in a typical argument about homeopathy, someone says that there is no evidence that homeopathy works, and someone else insists that homeopathic remedies make them feel better. The probable truth behind these statements is that there is no scientific research showing that homeopathy works any better than placebo, and the person insisting that homeopathy works for them is probably benefiting from the placebo effect. The placebo effect certainly has been demonstrated to work better than nothing. So the anti-homeopathy person is comparing homeopathy with placebo, whereas the pro-homeopathy person is comparing homeopathy with nothing. “No better than placebo” does not contradict “better than nothing”, and so there isn’t a logical disagreement here, probably just an emotional one about whether it’s worth paying for something that is “just” placebo.
Basic logic doesn’t handle these nuances much better than we do in emotional outbursts. We’ve discussed gray areas and the way that basic logic forces us to push all the gray to one side or the other. When it comes to qualifying statements, there are two logically unambiguous ways to do it:
1. The statement is true of everything in your world. Perhaps “all mathematicians are awkward”.
2. The statement is true of at least one thing in your world. Perhaps “there is at least one mathematician who is friendly”. (I hope I count as this.)
Everyone in the US is obese.
Someone in the US is obese.
The basic statement concerns being obese. “In the US” narrows the scope from the entire world to just the US, and then we say whether we’re talking about everyone in that scope, or just someone, at least one person. If two people are obese it is still true that “someone is obese”.
As usual, the way we turn this into formal language is a bit tricky, because we need something that is more rigid than our fluid, flexible, spoken language. Formally these two types of statement would be rendered using “for all” and “there exists” like this:
For all people X in the US, X is obese.
There exists a person X in the US such that X is obese.
This sounds terribly pedantic in normal language, but makes things easier to manipulate in formal mathematics. “For all” and “there exists” are called quantifiers in mathematics; they quantify the scope of our statement.
In the case of Rocky promising all his eggs to the rats, the “for all” clause was fulfilled because “all” happened to be zero. This can often seem like a bit of a cheat although the logic strictly holds. This is called vacuous truth, or a condition being vacuously satisfied. Consider this statement:
All the elephants in the room have two heads.
This sounds (and is) a ridiculous statement, but it is certainly true of the room I’m in at the moment. Unless you’re reading this at the zoo, it’s probably true of the room you’re in as well. There are no elephants in the room, and all zero of them have two heads. This is related to the fact that a falsehood implies anything, logically. You might meet someone who very implausibly claims to be a billionnaire, and you might exclaim “If you’re a billionnaire then I’m the Queen of Sheba!” In effect, this means you are completely sure the person in question is not a billionnaire. If a falsehood is true then truth and falsehood have become the same thing, which means everything is true, but also everything is false. It’s not a very useful situation to be in.
We can now revisit our emotional ouburst, “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 favorite 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:
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 logicial 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.
Sweeping statements are very close to stereotypes, and so are dangerous if you aren’t open to the possibility of counterexamples, or if you don’t respond to the situation at hand for what it is, rather than for what the sweeping statement says it is. I often complain about portrayals of mathematicians in popular culture, because they are too often awkward male people who are not very good at social interaction and are possibly insane. Someone recently said to me, “Yes, but mathematicians are like that!” This sounded awfully like “All mathematicians are awkward” and I didn’t like that at all, because I am a mathematician and I believe I am not awkward. Therefore my existence refutes the statement:
For all X in the set of mathematicians, X is awkward.
If someone says to me “All mathematicians are awkward” after meeting me it sounds to me that they are implying either that I’m not a mathematician, or that I am awkward, because those are the only ways to reconcile my existence with the implication:
Being a mathematician implies being awkward.
Either they think I am a mathematician, therefore they must think I’m awkward, in which case I take offence. Or else they don’t think I’m awkward, in which case they must think I’m not a mathematician, in which case I also take offence. There is one more possibility, which is that they think I’m not a mathematician but still awkward: double offence.
This sounds like an overanalysis, but that is like the difference between pedantry and precision. What’s the difference between analysis and overanalysis? Sometimes people say to me “You’re overthinking!” and I often want to reply “No, you’re underthinking!” I think the difference is illumination: I don’t call it overanalysis if it has helped me with something. In this case I find it helpful to know exactly why it is so frustrating when people make those sweeping statements to me about mathematicians.
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 have to do something much harder–you have to check every single female science student, and make sure that nobody has been hit on by their supervisor. Unfortunately this won’t be possible.
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:
1. 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.
2. 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
If you are very precise about how you quantify your statements you can ensure that you are never wrong about anything. This is one of the reasons I as a mathematician might be quite annoying to argue with: I am careful to use enough quantifiers that it’s almost impossible for me to be wrong. We’ve seen a few ways of doing this with phrases like
In my opinion…
In my experience…
Maybe we could add some more, like
Maybe…
Sometimes…
Apparently…
It seems to me…
I recently did an interview in which I lamented that some math lessons leave little lasting effect on people except math phobia–the students don’t remember much actual math, and mostly only remember fear. In that case, teaching them math has been a waste of time and money, and worse, has actually had a negative effect. Thus we might have achieved a better result by teaching them no math at all, because that would have been a zero effect, which is better than negative. Unfortunately various tweets went out declaring that I said “Teaching math is a waste of time and money” and “We would be better off not teaching math at all.” In fact, what I said is that in some cases teaching math is something of a waste of time and money, and in those cases, we might be better off not teaching it at all. Qualifiers abound.
My wonderful, wise, precise and illuminating PhD supervisor, Martin Hyland, is well known among his students for prefacing statements with “There is a sense in which”. “There is a sense in which Mozart is more boring than Brahms” is another way of correcting my sweeping statement “Mozart is more boring than Brahms”. “There is a sense in which teaching math can be a waste of time and money.” The phrase has a wonderful way of focusing one’s attention on exactly what is the sense–or what are the possible senses–in which something is true. It reminds us all that mathematics is not just about finding the right answer, but is about finding the sense in which things might or might not be true.
I believe that a useful way to be a rational person is to look for the sense in which things are true rather than simply deciding if they are true or false. Someone might say something that is untrue in strictly logical terms, but perhaps they were really trying to say something else, perhaps something with strong emotional content that we should listen to if we are intelligent humans rather than intelligent emotionless robots.
1 Basic quantifiers only quantify over sets, that is, you can only say “For all objects in a certain set”. You can’t quantify over sets of objects, as that is a higher-order level of expressivity and would cause problems of self-reference. The difference is a bit technical, but the idea is one that we’ll come back to when discussing paradoxes in Chapter 9.