Re: Logic question

Peter Fairbrother wrote:
|Keith Ramsay wrote:
[...]
|> We expect the Riemann hypothesis to be true, but it hasn't
|> been proven yet. Now, in the terms you prefer, would you
|> say *now* that his work is "meaningless"? I hope not; I
|> don't think he'd like that very well!
|
|No. I do know what the Reimann hypothesis is, in fact I asked the
question
|because in cryptology (I am a cryptologist) we often have "proofs"
that
|begin "If the RSA conjecture is true .. "

Sure. Now imagine that the hypothesis in question, say the
Riemann hypothesis, is discovered to be false after all.
Do we then go back and retroactively declare that my
acquaintance's proofs are no longer valid? That the
conclusion is no longer true but meaningless?

This runs counter to what we ordinarily have in mind when
we talk about whether a proof is valid and whether its
conclusion is true. If we said that A->B stopped being true
once we discovered that A is false, then the only way to
be sure that A->B was true and would remain always true
would be to prove that both A and B are true.

|Now we don't whether either of those is true, but we do not know for
sure
|that they are false. I have no problem with assigning them an
undecided
|status, and assuming that they do in fact have a true xor false
property, we
|just don't know what it is.
|
|However the statement in question, "if A then B" does not imo have
such a
|true/false property when A is known to be false.

If "if A then B" can be true when we don't know whether A
is true, but then stops being true if we discover A is false,
then then whether "if A then B" is true becomes a subjective
issue, depending on our state of mind.

[...]
|> Your position seems to imply that we have to wait until we
|> know whether the Riemann hypothesis is true to decide whether
|> his work is meaningful. I think this is at least a poor use
|> of terminology. Alternatively, we could say that if the
|> Riemann hypothesis turns out to be false, then these results
|> will be "false", but that also seems like a really bad way
|> to look at it.
|
|I did not intend to imply that at all.

But are you willing to accept that such a result can be true
without our knowing whether the Riemann hypothesis is true?
If it is true, now, then it shouldn't magically stop being
true just because we discovered something like the falsity
of the Riemann hypothesis.

|> Under the standard way people use these terms, once the
|> referees have verified that his paper is "okay", we reckon
|> that his results are meaningful and true, and always will
|> be, regardless of whether we prove or disprove the Riemann
|> hypothesis. If we were to prove that the Riemann hypothesis
|> is false, then these results of his will become
|> *uninteresting*, that's all.
|
|Yes - I don't have a problem with that either. In fact they will
probably
|still be interesting, and they may well still be be true.
|
|What I have a problem with is when the referees (or even the author)
decides
|a paper that depends on Reimann is false, and when we do find that
Reimann
|is false the paper suddenly becomes correct.

But this doesn't happen. As somebody else pointed out, if
the reasoning is incorrect, it's still incorrect if the
Riemann hypothesis turns out to be false.

|> Many of the logics where ~A->(A->B) is not always valid are
|> called "relevance logics". I guess the idea is something
|> vaguely like taking implications A->B to be true only when
|> A and B are relevant to each other. I don't think anybody
|> has figured out how to do mathematics in a relevance logic
|> in a convenient way. I think as long as nobody knows how to
|> make it work nicely, we're just going to keep using other
|> logics where this rule is valid.
|>
|> Let me expand a little on some observations Torkel Franzen
|> made for you. In relevance logics typically the rule allowing
|> you to deduce B from "A or B" and "not A" isn't valid.
|
|I am not disallowing any such rules. I would regard a logic that did
that as
|rather useless at best.

Well... I don't know if I'd entirely agree, but it would be
quite a change.

|> This
|> is a pain in the neck for doing mathematics, because in
|> proofs people are constantly dividing things into cases and
|> then eliminating ones they know don't occur. Without the
|> rule allowing you to deduce B from "A or B" and "not A",
|> you're not allowed to eliminate a case just because it's
|> false.
|>
|> It's tricky to have the rule allowing you to deduce B from
|> "A or B" and "not A" without also having the rule allowing
|> you to deduce B from "A and not A". Usually the following
|> reasoning is valid:
|>
|> Given: A and not A.
|
|Whoah there. That statement doesn't have the property of being either
true
|or false. Anything derived from will share that same lack.

You then add:
|What I meant was the statement "given A and not A" does not have a
|true/false value - you can't be given a false. But that's confusing.

Well, "A and not A" is plainly false. But there's no rule
against making deductions starting from false premises. In
fact, we do it all the time in mathematics. That's the
usual way of showing that the premise is false, in fact.


|Put another way, the statement "A and not A" is false - and from a
false
|given, iirc, you can prove anything.

Yes, and that's essentially what I was just demonstrating.
>>From "A and not A" we can infer any arbitrary conclusion B,
using just the two rules,

(1) from "X or Y" and "not X" deduce Y,
(2) from X deduce "X or Y".

Since these are rules we don't want to give up, we usually
allow anything to be proven from a contradiction.

Moreover, it's a general principle that if it's possible to
deduce B from A, then "if A then B" holds true. This is one
more reason why "if A then B" is taken as true when A is
contradictory.

|Perhaps that was a typo on your part?

No, not at all.

|I would be interested in knowing why
|"it's tricky". There doesn't seem to be any difficulty, but I may have
|missed something.

I said it was tricky to have a certain set of rules of: ones
where rule (1) is allowed but you aren't allowed to deduce
an arbitrary B from "A and not A". I don't know what set of
rules you have in mind that would work that way.

If your rule is "stop if you know that the premise is false",
then it's usually impossible to know whether I'm following the
rules without reading my mind. I'm not talking about that
kind of rule that depends on who I am and how much I know.
I'm talking about the kind of rule that we can tell I'm
following just by checking my work.

|> 1. A (by the rule allowing us to deduce X from "X and Y"
|> applied to the assumption).
|> 2. A or B (by the rule allowing us to deduce "X or Y"
|> from X applied to #1)
|> 3. not A (from the given)
|> 4. B (from #2 and #3 by the special rule)
|>
|> Some kind of trickery is needed to block such reasoning, and
|> there doesn't seem to be any value to blocking it.
|>
|> People tend to reject certain kinds of statements which are
|> categorized by philosophers as violating an "implicature".
|> An implicature is a kind of conversational rule. For example
|> it would be abnormal for me to say, "I have no unmarried
|> sisters" since in fact I don't have any sisters at all. My
|> saying I have no unmarried sisters leads the listener to
|> assume that I have a married sister, since otherwise I'm
|> introducing this irrelevant issue of marriage. But one also
|> wants to say that "Keith has no unmarried sisters" is true.
|> If we were to divide people up into those who have unmarried
|> sisters and those who don't, I would have to be put into the
|> "not" bin of course.
|
|Is that the same as the intuitionists that Rob Johnson mentioned?

No. It's just garden variety philosophy of language.

|> I think an implicature is responsible for why people gag on
|> sentences of the form A->B when A is false.
|
|Nope, I'm not gagging on that - just that given ~A, _any_ sentence of
the
|form A->B is _always_ considered to be true.

But this only is a problem, surely, if there are some
*specific* cases when you don't want A->B to be considered
true. I think all such problems arise from something like an
implicature.

Implication and disjunction are parallel cases here.

Ordinarily A->B is treated as equivalent to "either A is
false or B is true". For my acquaintance to say, "either
the Riemann hypothesis is false or ... is true" would be an
equivalent way of stating his results.

How often do people say "either X or Y" when they know that
X is true? Sometimes they do, certainly, but once everybody
knows that X is true, saying "either X or Y" seems silly.
But it's definitely true.

Saying "if A then B", i.e. "either A is false or B is true",
becomes redundant once you know that A is false or
you know that B is true. If I said, "if the Riemann hypothesis
is true, then integers satisfy unique factorization", people
would say I was being silly, because we know that integers
satisfy unique factorization anyway. But is such a statement
treated as true? Yes.

|Now if we have a sentence A->B that we know is false, perhaps because
B is
|always false, when we find that A is false the sentence suddenly
becomes
|true.

No, this doesn't happen. The only time that A->B is counted
as false is if B is false AND A is true. How else are you
going to know that A->B is false?

|> If we *knew*
|> that A was false, then asserting A->B would be pointlessly
|> mentioning B. Once you know A is false there's not too much
|> point (in mathematics, say) in talking about what would be
|> true if A were true. But as in the case of my acquaintance,
|> in mathematics we deal much of the time with implications
|> A->B where it's unknown whether A is true. If we prove that
|> A->B and later prove that A is false, we still say that the
|> proof of A->B was correct, just not as interesting. There
|> doesn't seem to be any point in distinguishing between the
|> cases where A is proven false first and the cases where
|> A->B is proven first. That's just a historical accident. So we
|> consider A->B always to hold when A is false. This also
|> makes A->B have a well-defined truth value whenever A and
|> B have them.
|>
|> Another big reason for treating A->B as true when A is false
|> has to do with variables. Probably most of the time when an
|> implication A->B appears in mathematics, there's a free
|> variable in A and B, like "if x>2 then x^2>4". Here we're
|> not dealing with simple sentences A and B, but with formulas
|> that are sometimes true and sometimes false, depending on
|> the value of x. It would be pretty inconvenient if we
|> weren't allowed to say that such an implication holds
|> regardless of what value x has.
|
|But it is pretty inconvenient to say that "if x>2 then x^2<4" is true
when
|x<2, when it is false if x>2.

I don't see that it is. It's another example of the kind of
thing people wouldn't ordinarily say, sort of like if you
saw someone write in a paper, "let p+1-1 be prime". You
probably would wonder why they wrote "p+1-1" and not "p".
"We have discovered 2^{...}-5+4 is prime". Huh? But one
doesn't want to treat such things as *false*, because that
would mess up the rules when we apply them to cases where
it isn't silly.

There are a lot of definitions in mathematics that work out
to something silly-looking in special cases. Perhaps if
you've taken complex analysis you remember the notion of
"meromorphic function". One definition says that it's a
function of the form f/g where f and g are holomorphic.
Often f and g are assumed not to have any common zeros.
Sometimes you want to state conditions that hold for all
of the zeros of g, i.e. for every z such that g(z)=0.

One example of a meromorphic function is z/e^z. In this
case, statements like,

"if g(z)=0 then ..."

become

"if e^z=0 then...".

It would be somewhat bizarre if a person were just to start
saying things like

"if e^z=0 then..."

but for the sake of keeping our terminology orderly, we want
such special cases to be regarded as TRUE. All of the values
of z such that e^z=0 are on the real line: TRUE. All of the
values of z such that e^z=0 are in the right half plane: TRUE.

I hope this has made the rationale for categorizing these
things as true but not necessarily useful to say, as opposed
to "meaningless" clearer.

Keith Ramsay

.

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