Logic Statement False Implies True

Date: 02/06/2008 at 23:49:17
From: Abe
Subject: False implies True?

I am well familiar with the linguistic arguments which clarifies this 
somehow confusing concept.  Is there a deeper philosophical argument 
that touches on the underlying logic of this concept {logic axiom}?

The most disturbing thing about this logic axiom is that it 
eliminates {defeats} the logical contingency of the conclusion on 
the premise, which somehow goes against the very essence of logic! 
By layman definition, logic is something that allows "naturally" the 
consequence to flow from a premise.  When the same conclusion happens 
no matter what the premise is, the connectedness of the logic in 
between the conclusion and its premise loses significance, just like 
the definition of a function is defeated when one certain value from 
the domain point {maps} into two or more different values from the 
range. 

If the moon is made of cheese, then I will go to the movie next week
can rather best describe a sarcastic {insane} mode of thinking than a
flow of "natural" logic especially when it can be said equally that if
the moon is NOT made of cheese, I still go to the movie!  The dilemma
of this concept, though I use it myself to prove some propositions
like the empty set is a subset of every set, is that it kills the
"natural" connectedness inherent in logic.
 
Unfortunately, the very definition of logic itself is so intuitive 
and vague in the same way the set or sanity is defined, otherwise 
undefined!  Even though I trained myself to live with this concept 
and I use it in my formal proofs, I try to avoid using it as much as 
possible.  It is like employing proof by contradiction.  I would 
rather prove directly.

Date: 02/07/2008 at 22:59:18
From: Doctor Peterson
Subject: Re: False implies True?

Hi, Abe.

There are several different brands of "logic", in philosophy and in
math, and I am only familiar with some of them, so you may need to
tell me the context of your question.  Are you talking about a formal
system of logic, or just about the way it is used in ordinary
mathematical proofs?

In the symbolic logic I am familiar with, what is commonly read as
"implies" (A->B) is not really an implication.  It should be read
simply as "if A then B", and is taken to be true when the truth values
of A and B are such that they do not contradict the claim that B is
true whenever A is true.  It is important NOT to read into it any
claim of a cause-and-effect connection, or anything of that sort.

I think the basic underlying reasons for this are twofold:

First, we want A->B to be defined for all values of A and B; we don't
accept "not enough information" or something like that.  So we have to
consider it either true or false, and the question becomes, which
makes sense in the contexts in which we will be using it?

Second, the main context is judging the validity of an argument.  When
we put an argument into symbolic form, we want it to be always true if
it is a valid argument.  It turns out that this works IF we define
A->B as we do.  Essentially, this means that we say something is true
when there is no evidence that it is false; it is "innocent until
proven guilty".

Your example of the empty set is a slightly different issue, though
clearly related.  Again, we choose to say that a set is a subset of
another when "every element of the former is an element of the
latter", and take that to mean "there is no element of the former that
is NOT an element of the latter", because that yields the results we
need for further theorems.  This is the same "innocent until proven
guilty" idea.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/12/2008 at 15:29:16
From: Abe
Subject: Thank you (False implies True?)

Thanks plenty Dr. Math for your interesting answer about "P implies
Q." I found it rather interesting that there is indeed some
philosophical underlying mode of thinking built into the definition 
of "P implies Q," that is, if there is no evidence that something is
false, then we must assume that it is true.  This we may cast as the
"default rule of the truth." 

I understand that we have a substantial liberty over our definitions,
although there is always something logical to them, and that is why we
do not have to prove definitions.  So if we chose to define the truth
table of "P implies Q" as such, there is no harm in that as long as we
proceed consistently with any argument we base on that definition. 

Obviously "if P is true and Q is true" then the implication is true
and that is very clear.  Similarly, if P is false and Q is false, then
the implication is true as well, and that is clear, too. The third
case is when we claim that if P is true yet Q is False, then our logic
of implication is flawed because by doing that we just nullified the
case that if P is true then Q must be true, something we just upheld a
moment ago! 

Now comes the interesting case which P is false and Q is true, yet we 
must assume that the implication is True ONLY for lack of better
knowledge or evidence pointing to the other direction.  This is to me
a philosophy or a mode of thinking which I accept as a sound one
though it has some serious implications beyond math.  

Thanks a million for the explanation.

Date: 02/12/2008 at 23:35:52
From: Doctor Peterson
Subject: Re: Thank you (False implies True?)

Hi, Abe.

Well stated, except for a few details.

This is only a definition made for a specific purpose in math, so you
can't really take it beyond that context and make it a philosophical
principle.  I think the same reasoning does apply in other areas, but
you'd have to think through whether it's appropriate on a case by case
basis.  In particular, we don't always HAVE to "assume" something is
true when there's no evidence for it; we just can't assume that it is
false.

Also, when P is false and Q is false, you really have no evidence to
justify the statement that P implies Q.  I'd like to pursue that a
little more deeply.

Let's take an example.  I have a piece of paper here that is coated
with a chemical that changes color.  I claim that if the paper is wet,
it is red.  That is,

  WET -> RED

(Note that I didn't say "wet implies red"; the word "implies", as I
said before, is not really appropriate for this connective, as you'll
see in a moment.)

Now let's consider what you might see when I show you the paper,
taking the four cases in your order.

1. It's wet, and it's red.  That agrees with my statement, so you say
my statement is true.  (You do not have enough evidence to conclude
that my statement is ALWAYS true; you've just seen one case.  Maybe
tomorrow it will be cooler, and you'll find that the paper is only red
if it's wet AND warm.  That's why you can't say it's true that wet
implies red, only that it is true in this instance that "if it's wet,
then it's red."  Do you see the difference?)

2. It's dry, and it's blue.  You don't know that it would be red if it
were wet; there's no evidence one way or the other.  So, simply by
convention, you say that my statement is true, meaning that the
evidence is consistent with that conclusion.  But you can't say that
you've proved that wetness IMPLIES redness; all you can say is that it
might.

3. It's wet, and it's blue.  That disproves my statement; we have a
case where it is wet but NOT red.  My statement is definitely false.
(This case IS enough to disprove the stronger claim that wet implies
red; you have a counterexample.)

4. It's dry, and it's red.  Hmmm ... maybe it's ALWAYS red, and my
statement was technically true but misleading; or maybe it's red for
some other reason than wetness.  Or maybe it actually turns blue when
it gets wet, and I just lied.  Again, you really don't know!  The
evidence at hand deals only with the case where it's dry, and my
statement is about what would be true if it were wet.  So you have to
say that it's true, because you haven't disproved it, just like in case 2.

So your cases 2 and 4 are both "true" for the same reason, not for
different reasons.  The evidence in both cases is consistent with my
statement, so we call it true.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 02/13/2008 at 02:18:10
From: Abe
Subject: Thank you (False implies True?)

Great example Dr. Peterson. If I may modify my understanding by the
following:

Assuming for the sake of argument that:
1.  P ---> Q, wet implies red.

Without contradicting our first statement, we could also claim that 
2. ~ P ---> Q, NOT wet implies red because the case that wet
correlates with red might not be exhaustive (reserved thinking!)

Further, without contradicting our first statement, we may also claim
that 
3. ~P ---> ~Q, NOT wet implies  NOT red, although this is a useless
(vacuous) statement since it deals with none of our original variables
(wet and red.)

Furthermore, 
4. P ---> ~ Q, wet implies NOT red,  clearly contradicts our first
statement which we assumed to be true.

In sum, P implies Q is nothing more than a claim or a proposition.  We
may uphold the rest of the logic table for P implies Q since the logic
equivalence (truth value) for the remaining three cases does NOT
contradict our claim about P implies Q, although not useful statements
in some cases.  Thanks again for the great example.

Date: 02/13/2008 at 08:52:30
From: Doctor Peterson
Subject: Re: Thank you (False implies True?)

Hi, Abe.

Yes.  Well said.

Well, I'll make one small change.  Your 3 is not really useless or 
vacuous (the latter word has a specific meaning in logic which does 
not apply here); it's just irrelevant to the original statement, 
which is probably what you meant.  Your 2 and 3 are two alternative 
possibilities for what happens when P is false, either of which 
would be alike compatible with P->Q, because the latter doesn't say 
anything about what happens when P is false. 

Thanks for the opportunity to think through this, because I've been 
looking for better ways to explain it to students, and I think I've 
got it.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/