[FOM] The empty set

[Stephen Pollard]

A brief reply to Prof. Tennant:

A post of mine that appeared a few minutes after yours may help to  
clarify what I was thinking. To follow up on that post: Kummer did  
existentially quantify over his ideal divisors. So, presumably, he  
would accept the inference from "gcd(3,alpha-4) divides 3 and  
alpha-4" to "Something divides 3 and alpha-4." It is a non-trivial  
question whether he was justified in doing so prior to Dedekind's  
intervention. Some folks think he was. Perhaps it's not so clear,  
then, that the empty set inference is fallacious.

As Prof. Tait points out, the empty set argument is not monumentally  
interesting in itself. More interesting is the question of when  
existential generalizations such as Kummer's are mathematically  
justified.

Stephen Pollard
Professor of Philosophy
Division of Social Science
Truman State University
spollard at truman.edu