Don't use Superset reasoning

**Introduction**

This can be a complicated topic for some people to grasp, but I will try and explain it with simple examples so that anyone can understand.

The hardest part is understanding what sets are and their relationships. Then, we can talk about the logic concerning them.

A set is simply a collection of items. I'm going to use a simple example, "Fruit". The word fruit is a collection of different things, like apples, bananas and pears. So far, so easy. The next thing to understand is what a "subset" and "superset" is. A subset or superset is another collection of items and it has a specific relationship with another set. So sticking with our example, let's talk about the set of apples. We have macintosh, red delicious, granny smith etc. What relationship does the set of apples have with the set of fruit? Well, everything that is in the set of apples is also inside the set of fruit. Macintoshes are fruit, red delicious are fruit and granny smiths are fruit. But it doesn't work the other way around. We can't say that everything in the set of fruit is in the set of apples. Banana's belongs in fruit, but it doesn't belong in apples.

So the set of apples and the set of fruit share a relationship. We would say it as the following: The set of apples is a subset of the set of fruit. (Another way of saying, everything in apples is also in fruit). The other way to say it is that the set of fruit is a superset of the set of apples. (And this is saying the same thing, everything in apples is in fruit.)

Let us introduce the set of "Food". Everything in fruit is in food. Everything in apples is also in food. (Maybe you can come up with counter-examples but lets just keep it simple!)

Now, we might look at it this way:

FOOD

FRUIT

APPLES

When we ask about the relationships that "fruit" has, we would say that: Apples are a subset of Fruit and Food is a superset of Fruit.

Understanding these definitions is the hardest part, once you have that done, the rest of this will be simple.

**Reasoning with Sets**

If I told you "I like all fruit", what could we logically say follows from this statement? Does it follow that I like bananas? Yes. Does it follow that I like all apples? Yes. This is essentially *subset reasoning.*

However, if I said "I like all fruit" can we logically say.. I like steak? Does it follow that I like cabbage? Hopefully you said no. I \*might\* like those things in addition, but it doesn't logically follow from "I like all fruit." If you come to a conclusion like "Oh, you like all fruit, well fruit is a food, and so is cabbage, so you must like cabbage!" This is *superset* *reasoning*, and hopefully you realize this would be nonsense.

Yet, this sort of reasoning does happen. People make these sorts of errors all the time. I constantly see this sort of reasoning being applied in ethical conversations.

Let us first abstract what we’ve discussed above.

Subset reasoning: You X all A’s. Therefore, you X all subset of A’s. [This is valid].

X can equal “like, hate, believe in, get angry at, enjoy the taste of, etc.” and A is a set. A can be “fruit” and a subset of A could be something like apples. But you can use other examples. A could be mammals and a subset of mammals is cats.

Superset reasoning: You X all A’s. B is a superset of A. C is a subset of B. Therefore, you like all C’s. [This is not valid].

Hopefully the above isn’t too complicated with all the variables, let’s just quickly go through it to make sure.

X can equal “like, hate, believe in, get angry at, enjoy the taste of, etc.” again. A is a set. Let’s pretend it’s fruit. B is a superset of A. So, something like “food” or “things made of matter” or “things that French impressionists seem to really like painting for some reason”. But let’s stick with food. Well, what is a different subset of food other than fruit? Maybe… vegetables? So because someone said they like all fruit, we conclude they like all vegetables. Clearly that’s not right.

Here is an illustration of each form of reasoning:

Now, you might have read this and thought this was all obvious, and no one really thinks like this, do they? Well, sadly, all the time.

Here’s a common conversation within the vegan debate I have seen many times:

M = Meat Eater

V = Vegan

M: “I’m okay with eating meat.”

V: “Are you okay with dog fighting?”

M: “No.”

V: “But they are both using animals for pleasure.”

M: “Yeah, but I don’t approve of dog fighting.”

V: “Well, logically you should, because they are both using animals for pleasure.”

The Vegan in this example is using the bad superset reasoning we have discussed above, his argument is not valid. Whether the meat-eater in this example knows why he approves of one and disproves of another is irrelevant, it is still an illogical inference to suggest that he \*must\* accept both. The above example conversation might even seem intuitively correct to you, which is why I’m writing this article. It follows the same form of bad superset reasoning as the fruit example.

Here we can represent it with a picture as well:

I hope that once you notice how these superset/subset reasonings go, you will notice a lot of people employing this bad reasoning and know how to explain why it doesn’t work.

And, just in case you want this in a jargon logical form, you can use this predicate logic:

∀x(Fx->Lx), ∀x(Ax->Fx) ∴ ∀x(Ax->Lx) [Valid subset reasoning].

∀x(Fx->Lx), ∀x(Fx->Ox), ∀x(Vx->Ox) ∴ ∀x(Vx->Lx) [Invalid superset reasoning]