Re: Logic question

Peter Fairbrother wrote:
> Ryan Reich wrote:
>
> > Peter Fairbrother wrote:
>
> >> How does math/logic handle this?
> >
> > Above I chose to interpret your question narrowly, in the sense that I
> > assumed you accepted the truth table for "A implies B":
> >
> > A | B | A => B
> > -----------------
> > T | T | T
> > T | F | F
> > F | T | T
> > F | F | T
> >
> > and were simply wondering how mathematics dealt with the case of a
> > particular tricky interaction between A and B. You might also have
> > been asking, "Why is the last line not F | F | F ?"
>
> I was asking why the last two lines are F | F | T, and not F | F | -.

In other words,you're asking why => is not defined as a "partial
Boolean function" whose domain requires its first argument to be true.
This has a funny interaction with the law of the contrapositive: A =>
B is the same as (~B) => (~A). By your rule, the second one should be
undefined unless (~B) is true; i.e. unless B is false. Thus you are
asking that A => B be undefined unless A is true and B is false, which
can hardly be said to be implication at all.

In fact, if you accept the contrapositive then you have to accept the
last line of the truth table, since (F) => (F) is by contraposition the
same as (T) => (T), which you accepted. So at least, accepting the
contrapositive means that you can only ask why (F) => (T) is true. In
words, is it the case that an untrue premise always implies a true
consequent? Think of this in terms of an actual proof: you have
already proved that B is true using some collection of axioms and
deductions, and someone tells you an untrue fact A. You can always
just insert A as the first sentence of your proof of B and then ignore
it; it is then still a (slightly less efficient) proof of B.

Here is yet another way of thinking about it. That (false) => (true)
is no less absurd than (true) => (true); for example, a statement such
as "My name begins with the letter 'R'" is true, as is the statement
"Your name begins with a 'P'". I wouldn't say that the one causes the
other (in either direction!), nor does the negation of either one imply
the negation of the other. Yet in propositional logic the implication
IS true, since both of the statements are. Propositional logic tries
to extract the idea of formal deduction by deliberately NOT asking
about the precise nature of the interaction between its variables, but
simply making deductions based on assumed truth values. And this is
the way it should be. If it were necessary to examine the most subtle
workings of causation between two statements, it might be the case that
although A appears to imply B (that is, using A we could prove B) in
fact both A and B are consequences of C, which is true for some other
reason, so that you can't say that A is _really_ the cause of B, but
simply some fact that if it were false would imply the falsehood of a
deeper cause C. This does not change the fact, however, that we were
able to deduce B from A.

In words: My name _does_ begin with an 'R' and yours with a 'P', and
although there is no direct relationship between them it is also the
case that the sun shines on Earth (at least, for a good portion of each
day) and this implies both of the former two facts, since it is the
sun's existence which allows ours: more subtly, it is through the
energy of the sun that our species evolved and our parents were born
and named us what they did. Thus the sun's shining IS the cause of our
names. If my name beginning wth 'R' did not imply that your name
begins with 'P', then the two facts would of course be independent,
but they are not actually independent, since if my name were not 'R'
presumably the sun would not shine, which would cause your name not to
begin with 'P' either.

This is a very silly argument: you might (reasonably) object that my
name isn't _necessarily_ R just because of the sun: my parents could
have named me something else. Now we're talking philosophy: is what
has happened necessarily so, or is it realistic that the same
precedents could have led to different outcomes? Well, they didn't,
and furthermore, if you're going to go there, you might also agree with
me when I say that it's possible that the universe could contain no
matter at all, in which case nothing I have said would be true anyway,
so what's all this about implication?

The fact is that propositional logic is designed to formalize (and make
rigorous) the notion of proof that mathematicians use: writing down
statements and combining them using implications to deduce other
statements. And in such a proof it is possible (as I mentioned above)
to introduce extraneous facts without affecting the validity of the
proof. Any false statement is extraneous, hence could be part of a
valid proof of anything, hence implies anything.

> > So the answer to the second perhaps-question is that implication is
> > defined that way because it is useful, which is where we started
> > anyway.
>
> Yes - but it causes much heartache - incompleteness, inconsistency,
> paradoxes, and so on.

Why do you think that incompleteness is a heartache? As for the
others, do you believe that inconsistency should never occur? And what
do you qualify as a "paradox" anyway? If it's actually contradictory
then it's an inconsistency, otherwise it's merely unexpected. And you
can probably think of examples from real life in which a very basic and
widely accepted fact has undesired consequences. This is sort of a
"baby with the bathwater" situation. I feel like your argument here is
that mathematics should be "above" all this mess, which is not a very
good argument.

> I think the flaw may be in the assumption that the sentences we are using
> are necessarily Boolean. We may start out with all Boolean sentences, but
> the new sentences they make under manipulation do not all end up that way if
> the rules for manipulating them are sufficiently complex.

This is like the difference between "or" and "xor". In English we
often use "or" to mean "xor" whereas in math they are different. I
notice that "xor" is not widely used outside of logic or computer
science, so that the non-obvious operator is in fact the more useful.

I also have to question how your notion of indeterminacy would interact
with an operator such as "or". For example, what is the status of "(A
=> B) v (A => B)"? Would you actually violate the reflexive property
of disjunction, or perhaps extend the definition of "or" to account for
this? And in this situation, how would that reflect the "obvious" fact
that one of A or B must hold in order for A v B to hold? I don't think
you can win.

People have tried multi-valued logic. I notice that it is not widely
used either; however, you might go and look at it since you seem to
feel the lack that they also felt.

--
Ryan Reich
ryan.reich@xxxxxxxxx

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