Set theory and the fallacies of division and composition

[bahen]

I am curious to know whether there is a set theoretic proof for the fallacies of division and composition. The former holds when one claims that something true for the whole must also be true of all or some of its parts. The latter holds when one infers that something is true of the whole from the fact that it is true of some part of the whole (or even of every proper part). The query arises out of the fact that they are informal fallacies, meaning that their flaws are about content and not structure. If this is true, then I wonder whether a formal set theoretic proof is possible.

 

[Ishuda]

I am curious to know whether there is a set theoretic proof for the fallacies of division and composition. The former holds when one claims that something true for the whole must also be true of all or some of its parts. The latter holds when one infers that something is true of the whole from the fact that it is true of some part of the whole (or even of every proper part). The query arises out of the fact that they are informal fallacies, meaning that their flaws are about content and not structure. If this is true, then I wonder whether a formal set theoretic proof is possible.

Would this work for the division part of the question?
Fallacy of division: The whole contains the whole but no proper subset of the whole contains the whole.

 

[bahen]

Would this work for the division part of the question?
Fallacy of division: The whole contains the whole but no proper subset of the whole contains the whole.

Could you show me how to describe this notationally? I am not sure how to represent whole to part relations (I have no real exposure to mereology).

 

[Ishuda]

Could you show me how to describe this notationally? I am not sure how to represent whole to part relations (I have no real exposure to mereology).

The whole contains the whole
\(\displaystyle A\, \subseteq\, A\)
but no proper subset of the whole contains the whole
\(\displaystyle \nexists\, S,\, S\, \subset\, A, \text{such that}\, A\, \subset\, S\)

 

[pka]

I am curious to know whether there is a set theoretic proof for the fallacies of division and composition. The former holds when one claims that something true for the whole must also be true of all or some of its parts. The latter holds when one infers that something is true of the whole from the fact that it is true of some part of the whole (or even of every proper part). The query arises out of the fact that they are informal fallacies, meaning that their flaws are about content and not structure. If this is true, then I wonder whether a formal set theoretic proof is possible.

I am somewhat surprised that you asked this question. Surely you have some idea of the history of logic in the twentieth century. It is not a miss to say that the problem of self-reference is twentieth logic. In c1905 Russel wrote to Frege asking a simple question: "Is a set of teaspoons a teaspoon?" Well Frege understood at once the paradox and fallacy in his system. Thus was born the famous paradox: Can the set of of all sets that are not members of themselves be a member of itself? Think of Godel, Church, and Turing's work. In some real sense all are derivatives of the paradox.

So in a real sense, the OP is an extreme simplification of about one hundred years of work in foundational logic/set theory.

 

[bahen]

Think of Godel, Church, and Turing's work. In some real sense all are derivatives of the paradox.

Thanks for your reply. Could you elaborate on the above?

 

[bahen]

The whole contains the whole
\(\displaystyle A\, \subseteq\, A\)
but no proper subset of the whole contains the whole
\(\displaystyle \nexists\, S,\, S\, \subset\, A, \text{such that}\, A\, \subset\, S\)

Thank you, Ishuda.

 

[pka]

Thanks for your reply. Could you elaborate on the above?

Enroll in a course on the history of modern logic. That is what you are asking.