Implication (Formal Logic)

IMPLICATION, FORMALLY

Even if our aim is to use better logical implication in normal life, I think it is important to understand some more of how it is used in mathematics. Mathematical language is dry and formal, which can make it seem offputting and irrelevant. But its dryness is there for the excellent reason of making things crisp and clear rather than soggy and mushy. It also helps make things more concise, which in turn helps us build bigger and more complex arguments. It’s a bit like those vacuum bags into which you put your clothes and then suck all the air out with a vacuum cleaner, condensing a whole pile of clothes into one compact unit.

A more concise way of saying “If ... then ...” is “implies”. So instead of saying “If A then B” we can say “A implies B”. Mathematicians use the symbol (\Rightarrow) for “implies”. This implication means that whenever A is true, B absolutely has to be true. When A is false, the implication doesn’t tell us anything.

For example, “Being a US citizen implies you can legally live in the US” tells us that whenever someone is a US citizen they can legally live in the US. But when someone isn’t a US citizen this implication doesn’t tell us anything about them one way or another: they might be able to live in the US legally (for example, if they have a visa or have permanent residency) or they might not. Unfortunately this logic is lost on some people who think that anyone who isn’t a citizen is illegal. We will come back to this grave error of logic in the next chapter.

A proof is basically a whole series of implications strung together like this:

(A \Rightarrow B) (B \Rightarrow C) (C \Rightarrow D)

We can then conclude that (A \Rightarrow D). This is because if A is true then B is true by the first implication, and next if B is true then C is true by the second implication, and finally if C is true then D is true by the third implication, so the overall knock-on effect is that if A is true then (after some thought) D is true.

The critical point is “after some thought” – a chain of implications takes more concentration and command of logic to follow than a single one does, and these resources are unfortunately often lacking from arguments.

Here are some longer chains of implications that take more than a basic command of logic to follow them:

1. If you say women are inferior, that is insulting to women.
2. If you think that “feminine” is an insulting way to describe a man, you are saying that women are inferior.
3. Therefore if you think that “feminine” is an insulting way to describe a man, you are insulting women.

Here is another:

1. If you don’t stand up for minorities being harassed then you are letting bigotry flourish.
2. If you let bigotry flourish you are complicit with bigotry.
3. If you are complicit with something bad then you are almost as bad as it.
4. Therefore if you don’t stand up for minorities being harassed you are almost as bad as a bigot.

It is important to note that the conclusion is true if you don’t stand up for minorities, but the implication is true whether or not you stand up for minorities. I might not know that you are a great ally of minorities, and I might say to you “If you don’t stand up for minorities who are being harassed then you are almost as bad as a bigot yourself”. My statement is still true although you personally are not actually anywhere close to being a bigot. This is an important subtlety of implication. “If you are a US citizen or permanent resident you are required to have health insurance” is true whether or not you are in fact a citizen or resident. The implication doesn’t tell us whether or not someone needs health insurance; we only know they do if we already know they are a citizen or resident. (This point was lost on one insurance agent who told me everyone in the world needed US health insurance regardless of whether they even lived there or not.)

We can build up huge arguments from logical implications, and the argument will still have this feature: the concluding statement is only known to be true when the opening conditions are fulfilled. But the argument itself tells us that if the opening conditions are fulfilled then the concluding statment is true, and this argument is always correct. This sort of justification, in mathematics, is a proof.

I am gradually going to argue that stringing together long chains of implications gives us logical power. It is what enables us, like Kyle MacDonald, to start from something obvious and work our way to something complex and unobvious. Being able to construct and follow such complex arguments is hard but is a critical part of making good use of our human brains. I believe it is one of the things that separates us from simpler animals and tiny children, who can only deal with immediate needs and direct observations. Long chains of implications often require us to package many connected ideas into a single unit so that we can build on them more easily, like vacuum packing our clothes. What we gain in the process is new insights and deeper understanding.