

You Can Prove a Negative [pdf] - miralabs
http://departments.bloomu.edu/philosophy/pages/content/hales/articlepdf/proveanegative.pdf

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ThePhysicist
In physics (and other natural sciences) you can ONLY prove negatives in fact,
so to a physicist this statement is not surprising at all.

Example: If we observe an apple falling to the ground with the same
acceleration many times we will try to generalize this observation and create
a theory that explains this phenomenon and similar ones (such as Newton's laws
and his theory of gravity). Unfortunately we can never prove such a theory in
the mathematical sense since we can never be sure that the next time we
perform a given experiment it will still yield the same result (it could be
that the laws of physics change over time or work differently in other parts
of the universe). However, we can easily falsify a given theory if we can
produce even one single observation where the theory does not fit reality
(which is precisely why general relativity was needed to replace the classical
theory of gravity, which could not explain all experimental data [actually the
new theory came even before the data in that case, which is amazing in
itself]). So, in a sense, proving that something is not true is the only thing
we can do with absolute certainty.

That said, theories that are proven wrong by experiment and replaced by other
theories are still valid in their domain of applicability, so to say as an
approxomation of the larger theory.

~~~
scoofy
You're reference for negative here is wrong. When you prove that something is
not the case, then you do so by demonstrating an there exist something
contrary to your hypothesis. This is positive knowledge, not negative
knowledge.

This is a very convoluted philosophical concept, but it's essential Karl
Popper's argument. We don't, and can't prove negatives, but what we mean by
that, as i discuss below, is that we cannot apply the universal quantifier
with any certainty.

~~~
Retra
What exactly do you mean by "positive knowledge" or "negative knowledge?"

~~~
scoofy
Positive knowledge is: there is an instance of x (where not existing is not a
predicate of x, as existence is not a predicate)

Negative knowledge is: there is not an instance of x (where not existing is
not a predicate of x)

~~~
Retra
So you're not actually talking about knowledge, just the presence of the word
'not' in a sentence?

~~~
scoofy
Of course i'm talking about knowledge. Empirical knowledge of a proposition
being true or false.

~~~
Retra
Ok, so you can know if a proposition is false, or you can know if it is true,
and these two things fundamentally behave differently if the proposition
contains the word 'not' in it?

~~~
scoofy
Yes, because a proposition without it is a statement about a thing: "I exist."
Whereas, with not in it is a statement about everything that exists: "Black
swans do not exist." You need knowledge of one thing in the former, you need
knowledge of everything in the latter.

It's not just the word "not" it's an odd number of nots (double negatives and
all that).

~~~
Retra
"Black swans do not exist" isn't wholly a statement about everything that
exists. It's a statement about the _possibility_ of the property of "being a
black swan" to exist.

It is a statement about an abstraction, not existence. Like saying "the cup is
above the table" is a statement about the existence of the "above-ness with
respect to the table" of the cup. One needn't survey all of reality to know if
it is possible for some abstraction to exist. It need only be consistent with
the rest of your language, so that's all you need to survey.

~~~
ectoplasm
I would describe what you're talking about as "black swans cannot exist". That
is an even stronger statement, because it also implies that black swans do not
currently exist.

------
wfo
I think the article dances around what people really mean when they say "you
can't prove a negative". The writer touches on it but somehow skirts the real
issue. When we say "you can't prove a negative" you're really saying that in
order to prove a statement of the form "not exists x P(x)", it's equivalent to
proving "for all x not P(x)" and so you're really making a statement about
EVERYTHING that has or does or could ever exist somehow. These proofs are
either analytic and trivial (i.e. there doesn't exist a rectangular circle) or
not proofs.

He suggests you get around this by providing an argument about unicorns and
then goes on to talk about how premises don't always need to be justified, but
the problem is proving the nonexistence of a thing -- and he provides a formal
deductive example of a proof of nonexistence of a thing by instead
substituting as a premise nonexistence of another thing (evidence). Sure, if
you can prove no evidence for unicorns exists and that for unicorns to exist
they MUST have left evidence you can prove they don't exist. But the first one
is a reduction to the original problem -- in order to show something doesn't
exist (be it unicorns or evidence for unicorns) you need complete and total
knowledge about existence, which no human up to this point has had enough
hubris to claim, and the second requires a level of certainty that doesn't
really exist.

If we're discussing the existence of a thing that doesn't leave evidence (god)
especially it's very fair to say you can't prove a negative and I think his
attack on this tactic is a non-starter.

There has never been even the tiniest bit of evidence against the existence of
god. There is certainly evidence against a god with specific qualities; a
benevolent god, a god that wants this to happen, etc. But if you don't require
god to have any specific qualities there isn't a shred.

~~~
cristianpascu
I think the problem of proving the (non) existence of something is as simple
as understanding that (most of the time) it can not be deductive. I'd say that
when it comes to existence, it is always inductive. What kind of proof can
anyone give me that she/he exists? Or the rest of the world? There is no such
thing. Even in mathematics, if one doubts basic entities as numbers and their
properties, the whole establishment of mathematics starts to crack, at least
for that particualr one human being.

All of our reasoning about the world is inference based, hence probabilistic.
That's in the nature of science, religion and everything else in between.
There is strong evidence for the existence of God, but it's inherently
inductive. Not to mention various conceputal complication which follow from in
different kind of reasoning about God (in whatever view).

~~~
Dylan16807
> Even in mathematics, if one doubts basic entities as numbers and their
> properties, the whole establishment of mathematics starts to crack, at least
> for that particualr one human being.

That's not how it works. Numbers _are_ their properties. You are free to use
any properties you want when you commit mathematics, and you will get
different sorts of behavior from your numbers depending on your choices. There
is nothing to doubt about a set of properties being itself. No 'establishment'
tells you how numbers work. Use whatever you like.

~~~
danielam
Well, if mathematics is to say anything about the world, then the foundations
must be sound (and drawn from experience which brings us back to empiricism).
Sometimes the problem isn't necessarily with the concepts in our heads but the
formalizations we choose to proceed from. Set theory is a great example. Take
Russell's paradox, for instance, and the different responses to it (e.g., ZFC,
Lesniewski's mereology).

I'm reminded of Duhem's rather humorous "German Science" where he
characterizes German mathematics as being disproportionately oriented toward
rigorously deducing nonsense from arbitrary axioms as opposed to the French
tendency to rely on intuition to arrive at sensible axioms but neglecting
systematic rigor afterwards.

~~~
Dylan16807
Physics says things about the world.

Math is a tool to help you make logical conclusions.

------
awptimus
This is an unnecessary. "You can't prove a negative" isn't a statement about
logic, it's a statement statistical evidence (incorporating logic). You need
to observe the entire population in order to reduce a probability to zero.

~~~
TelmoMenezes
And likewise you can't prove a positive. In fact, you can't prove anything
with statistics, and science is not about proof. Mathematics is about proof.
Science is doubt. All theories are forever on probation -- or until they are
falsified by new experiments.

~~~
awptimus
Not true. Given a hypothesis "there exists" the first observed instance is
proof of the positive. The inverse however, again, requires you to sample the
entire population and collect all data.

~~~
duaneb
Depending on what you're observing, it's only statistical probability you
actually observed anything. There is no such thing as an inductive proof;
there is no way you can prove anything exists to another person.

I suggest Hume and the writings of the logical positivists. This will wipe the
idea of being able to "prove" with science or observation right out of your
brain. No matter how much information you have, you don't have enough to
predict the future. If our planet collides with some huge energy burst, all
the people who said "I know for a fact that the earth will still be moving
around the sun at the approximately same rate and mass" will have been PROVEN
false. But there is no such trick to prove anything about tomorrow.

~~~
awptimus
Other people don't factor into this. Proving to them is not necessary. Once an
event occurs, it occurred. Your knowledge of what that event or the total
state of that event might be uncertain, but the event's occurrence is no
longer probabilistic.

~~~
duaneb
Yea, but you are probably talking about a completely different event than the
other person, or whatever happened in your memory, or maybe just a past
version of yourself. The world is not naturally quantized into discrete
events, causes, and effects, and it is in this disagreement over simple
assumptions that "this event happened" that cause people to disagree about
"certain" things. Look at how utterly useless witness testimony can be, no
matter how hard they swear up and down, if the memory is suspect at all.

So, if you want to lie to yourself, I guess you could "prove" something to
yourself. But you know you can't trust your observations for about a billion
different reasons.

Why is this important? Because you can always know if a bridge has failed, but
you never know if it's not going to fail. Never. No matter how certain you
are, unless you can form a deductive proof showing that the structure will
NEVER fail (which I suspect is impossible for a computer, let alone a human),
you will always have doubt. If you don't recognize the doubt, you are lying to
yourself.

Furthermore, once you realize how.... loosely science binds together, you
realize that the "laws" we think we "know" are broadly accurate but fall to
pieces in the details.

So no, proving a negative is not possible. It's not a proof unless you can
show the state of the universe, which is again impossible without consensus,
which is itself probabilistic and flawed. If you are 100% certain that an
event has happened in any way that you could deductively prove something, that
gamma ray burst could easily mess with your plans.

A non-deterministic universe is one of increasing, but always <100%,
certainty.

------
animefan
Similarly in frequentist statistics "you can only disprove a hypothesis" is
false.

If you think you can disprove x > y, (where x and y are parameters like the
mean of some quantity for two different populations) then it follows that you
can prove x <= y.

On the other hand you cannot ever prove x = y since there will always be
values of x and y that are so close that your test has no power. but you can
still put bounds on how close x and y must be in this case

~~~
darkmighty
It really depends on the statistical model of the quantities you're working
with. If somehow your model dictates that x and y must be discrete, you can
get certainty of measurement (i.e. prove x=y) even with some noise.

In this case I think both sides are right, so the argument should be used in
context. You can only prove things about the real world (and not an axiomatic
system for example) assuming there are no unknowns. In some (perhaps most)
cases those unknowns are not there, but in some important cases unexpected
behavior may occur in spite of "overwhelming evidence", because the
assumptions made by the conclusions failed. It would be a much simpler
discussion to state "When you want to prove something, make your assumptions
clear, and test them well. Everything that hopes for rigor is proved within a
well defined set of assumptions, so it's not like nothing can be proved
either.".

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scoofy
Induction allows us to use the existential quantifier, deductions allows us to
add the universal quantifier. If we want to say something about the world, we
are bound to empiricism, thus our deductive claims rest upon our inductive
evidence. This is a framework. Any use of the universal quantifier (necessary
for a negative claim), is unsupported by our empirical data. This is
essentially Popper/Hume.

~~~
Retra
That's one model of the process. Another is that the distinction between
negative and positive claims is entirely artificial, and that there are really
only "claims" and the truth of them is determined by their effectiveness at
constructing reliable models of reality. And thus any demonstration of self-
contradiction is a disprove of a claim, whether negative or not.

~~~
scoofy
Negative and positive claims are not artificial or equivalent, one is making a
claim about positive existence, the other is making claims about lack of
existence.

Of course we can manufacture the universal quantifier insofar as it's a
mapping of the existential quantifier. That is to say, proving something is
"not nonexistent" is not a statement that requires the universal quantifier,
it's a claim that has simply mapped the equivalency of the existential
quantifier, using the universal one.

This is still only a statement about positive knowledge.

------
jsprogrammer
I think, "not being able to prove a negative" is more related to the idea that
you cannot prove/disprove (ie. Assign an absolute truth value to) the non-
existence of something. For example, you cannot prove that a God does not
exist.

~~~
ginolomelino
The paper very specifically addresses that problem at length.

~~~
themodelplumber
I wouldn't say it addresses the problem at length. At best it goes one level
deeper than "you can't prove a negative" to say "you _can_ actually offer some
evidence or probability for said negative" but even then it hands off the
threshold of [what's been proven vs. not proven at that point] to
epistemology.

After that, the author concludes that you should never dismiss an inductive
argument just because you think any probability it provides is immaterial.
After all, we rely on induction all the time in day-to-day life! Well, duh. So
let's hear it for induction, everybody. At least it gives us some probability.

I dunno. It's just not as intriguing a line of thinking as I hoped it would be
based on the title of the paper. Despite all the excitement about proving a
negative, the author goes completely silent on the issue of how much
probability proves what!

