9.2 Polysyllogisms - My Notes from The Reasonable Person by Mark Grannis

Polysyllogism - "An argument that links multiple syllogisms together in such a way that the conclusion of one serves as the premise for the next." 

In other words, the conclusion of one syllogism acts as a premise in the next one and they are linked together. But they can be separated into two fully complete syllogisms in themselves. 

Example:

All witches burn; 

All wood burns; 

Therefore all witches are made of wood; 

All wood floats;

Therefore all witches float. 

Can actually be separated out as: 

All witches burn; 

All wood burns; 

Therefore all witches are made of wood. 

All wood floats; 

[All witches are made of wood;] 

Therefore all witches float. 

Interpolation is used here again to supply the implied proposition. 

"...the validity of a polysyllogism depends entirely upon the validity of each of its component parts when expanded." 

"... be sure that you are evaluating each link of the chain according to the rules of validity that govern that particular kind of syllogism."