Re: Modus Ponens and Trivial Truth


rhys44503@xxxxxxxxx wrote:
I have a problem. If you take two true but completely unrelated
premises and pop them into a modus ponens, how can you call the result
valid--i.e. If the moon orbits the earth, then I am wearing white
carpenter's pants. The moon orbits the earth. Hence, I am wearing
white carpenter's pants--and take my word for it, I am.

Example and more description below.


Here are the truth functions of modus ponens:

((P > Q ) & P) > Q
1 1 1 1 1 [1] 1
1 0 0 0 1 [1] 0
0 1 1 0 0 [1] 1
0 1 0 0 0 [1] 0
*
Underneath the main (conclusion) operator, all lines of the truth table
are true, hence the argument is valid.

But there is a problem once you start filling in the variables. The
usual example is If one is a man, then one is mortal. Socrates is a
man, hence Socrates is mortal. That works. P is true, Q is true, and
the conclusion, by the magic of modus ponens comes out true. But what
about this: If the moon orbits the earth, then I am wearing white
carpenter's pants. Again, the first premise, P, is true. And take my
word for it that the second premise, Q, is also true. Given the
foregoing, the conclusion is valid. But why? It doesn't seem like the
moon orbiting the earth should have any bearing on what I am wearing
today, does it?

There are only two ways I have to deal with this, and I hope someone
can help.

The only problem is thinking that it's a problem. People think of P=>Q
as meaning that sometimes P is true and when it is there is some reason
for Q to be true, and at other times P is false and that reason
typically isn't there "for the same reason" i.e. Q could still be true
but the impetus caused by P is not there by P being there (althugh
something else oculd cause it.)

When the set of situations in which P is true is a trivial set
(universal set or empty set), then they freak out because they don't
see that "reason" occurring when P and not occurring when ~P. But if
you merely think about the list of possible scenarios in which P is
true, and the list of scenarios in which ~P is true (P is false), and
see that being in "all of them" includes when it is an empty list, then
there is no problem.

The problem (in not realizing the above) has caused normal logic to be
doubted by Logicians (a misnomer in this case) and brought about
supposed "paradoxes" such as the Paradox of the Raven, which occurs
when there are no ravens in sight. They have even gone so far as to
say that P=>Q is not the same as ~Q=>~P!!!!

Maybe if they used a Computationally Based Logic, they wouldn't be so
confused and say such stupid things as that. Have you ever used one?

(They have also invented new "types of logic" in frustration or simply
to justify grant requests.)

C-B

First is simply to say that propositional logic doesn't
account for modalities--whether the moon necessarily or possibly
orbiting the earth has any impact on my choice of pants. Granted,
modal logic, temporal logic, fuzzy logic, and some applications of
predicate logic capture all of that. But as to basic bone-headed
propositional logic, the conclusion seems odd, because it leaves open
the possibility of a modus ponens sentence returning an invalid
result--which it shouldn't be able to do.

So I think I have a second answer that works better. Because basic
propositional logic doesn't account for time, modality, probability,
etc. Given that, propositional logic describes a world in which all
true propositions are necessarily related to each other (or necessarily
not related to each other.) For instance, in the world that
propositional logic can describe--every time a butterfly flaps its
wings, there either must or must not be a hurricane.

That's about all I have to describe it, but I'd love to hear what
anyone else has to say.

Thx.
--
/s/ Rhys B. Cartwright-Jones /s/
Rhys B. Cartwright-Jones

Attorney & Counselor at Law
46 Chagrin Pl., P.M.B. 168
Chagrin Falls, OH 44022-3022
rhys44503@xxxxxxxxx

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