2

I'm reading a mathematical logic book and the book introduces some old logic stuff such as Aristotellian logic, and according to it there are 24 valid categorical syllogisms one of which is described as AAI-3, in essence it's an argument of the form:

All humans are mortal

All humans are earthlings

$\therefore$ Some earthlings are mortal.

This seems ok at first but what happens when the first class mentioned, in this case "human" is empty? Then in theory I could write down an argument of the same form with true premises that has a false conclusion, example:

All humans born in 1203 who are still alive are 500 meters tall (which is vacuously true)

All humans born in 1203 who are still alive are mortal

$\therefore$ Some mortals are 500 meters tall (which I believe is false).

Can someone explain this to me?

zlaaemi
  • 1,037

2 Answers2

2

You are right: if there are no humans, then this syllogism doesn’t work. This is why this syllogism is said to be conditionally valid: it is valid once you make the assumption of Existential Categorical Import, which is that for every category there is at least one member.

It turns out that out of those 24 ‘valid’ syllogisms there are 9 that are conditionally valid, this being one of them . Only 15 are unconditionally valid.

Shaun
  • 44,997
Bram28
  • 100,612
  • 6
  • 70
  • 118
1

From what I read from here, regarding the “Nine conditionally valid syllogisms” (in which $AAI-3$ is included):

Nine Conditionally Valid (for only Aristotle, not Boole; these assume that a term in the conclusion exists)

Since your proposition is of type

All M are S.

All M are P.

Some S are P.

The conclusion has, as terms, S and P, which is “mortals” and “500 meters tall”. Since the set of 500 meters tall people doesn’t have elements, thus the term P doesn’t exist, the syllogism isn’t valid.

selenio34
  • 158