23.1 4 Aristotelian Forms
One of the first philosophers to systematically study logic was Aristotle.
He analyzed several important quantificational claims that are very common in logic. Consequently, they’ve become known as the Aristotelian forms.
Let’s see if you can figure out how to translate them into FOL.
The first Aristotelian form is “All Ps are Q.”
The translation for All Ps are Q might not have been intuitive, because it’s not obvious that there is a conditional in it.
But if you think about it, it makes sense: the claim in English looks like a wide-scope universal. But it doesn’t mean that everything is a P; nor does it mean that everything is Q.
Rather, what it requires is that, for any object x, if x is a dog, then it is running.
All Ps are Q is probably the most important quantificational structure, because it appears in reasoning all the time.
All humans are mortal.
Socrates is human.
Socrates is moral.
Which we translate:
The second Aristotelian form is “Some Ps are Q.”
In order to figure this out, you have to think carefully about the English: What does Some Ps are Q require?
It requires that something is P, and that it is also Q. That is why there is an existential around a conjunction.
If you were tempted by the existential around the arrow, think about this: arrow is really like a type of disjunction. Using the Boolean Definition of Conditional (BDC), we get:
Now it is clear that that does not say that Some Ps are Q.
Ex(~P(x)vQ(x)) is true if there are no Ps at all, because then something would be ~P(x).
In fact, existential around arrow is never the right way to translate anything from English to FOL, so I call it the Abominable Form.
It’s not that it doesn’t make sense, from a logical point of view. Rather, it is just an odd form which we never naturally use in reasoning.
Now for the third Aristotelian form: “No Ps are Q.”
This is probably the most confusing one of the four forms, because there are two equivalent and equally good ways of translating it.
First, note that we don’t have a symbol for “No”.
Instead, we have to figure out how to express it ∀ or ∃. Either one can do it, which is why there are two good answers.
But you have to think carefully to figure out how each one works.
Using the universal: if No Ps are Q, then whenever something is a P, it better be ~Q.
Using the existential: if No Ps are Q, then there cannot exist something that is P and Q.
Lastly, the fourth Aristotelian form is this: Some Ps are not Q.
All Ps are Q: Ax(P(x)->Q(x))
Some Ps are Q: Ex(P(x)&Q(x))
No Ps are Q: Ax(P(x)->~Q(x)) or ~Ex(P(x)&Q(x))
Some Ps are not Q: Ex(P(x)&~Q(x))
Some Ps are not Q is an existence claim: there is something that is P, and it is also not Q.
Here’s a summary:
- All Ps are Q: Ax(P(x)->Q(x))
- Some Ps are Q: Ex(P(x)&Q(x))
- No Ps are Q: Ax(P(x)->~Q(x)) or ~Ex(P(x)&Q(x))
- Some Ps are not Q: Ex(P(x)&~Q(x))