FOM: Universal Generalisation (No Aribitary Objects)

[Dean Buckner]

UG is one of four rules used in logic to move between statements using
constants
and statements using variables.  What does this rule mean?

I looked at a number of academic websites, all of which give broadly similar
accounts of this rule.  They say that we must reason with an "arbitrarily
selected" member A of a domain, using only facts or features "true of all
members", being careful not to "import" any special fact or feature "unique"
or particular to the chosen member.  If we can show (or "demonstrate" or
"prove") that the A is also B, then the rule allows us to deduce that all
A's are B's.

This is superficially plausible, but does it really make any sense?

Let's grant that the chosen A is B.  How do we know that this fact is not
something special or unique to this particular A , which we have "imported"
by mistake?  Either we have already established that being a B is a fact
"generally true of all A" - in which case we have no need of the arbitrary
object - or we have not, in which case we can't be sure that being a B is
not a special feature of this A.

It doesn't help to argue that we must "demonstrate"  (rather than just
discover or find out) that the A is a B.  We select an A, making sure, as
the textbooks say, that it has only properties common to all A's.  We cannot
establish at this point whether it is B, for if we have, we must have
checked whether being B is a "property common to all A's", and thus
established what was to be proved.  Now we go on to "prove" or "show" that
the A is a B.  But does that mean we didn't look closely enough in the first
place, when we were first screening the A's?  If "demonstration" is simply a
closer scrutiny, then we haven't taken taken the "great care" insisted on by
the textbooks, to ensure A is "arbitrarily selected".  If on the other hand,
"demonstration" means establishing a truth of a universal character (such as
all A are B - nothing less will do) then it is the demonstration that does
all the work, and not the arbitrary object.

This of course does not mean the proofs given by the textbooks are invalid.
But if you look at them closely, they all appeal to some covert or explicit
rule of general inference.  Imagine, for example, that we could not deduce
that all A are are C, from the assumptions that all A are B and all B are C.
If not, all these proofs fail.

In conclusion, the "arbitrary object" method of establishing a general
conclusion implictly relies on assumptions that are general, rather than an
inference respecting a particular (though "arbitrary") object.

BTW some definitions say UG is the inference from "any A is B" to "all A are
B".  This seems better.  The problem is from "some (arbitrary) A is B" to
...

Dean Buckner