
One Man's Modus Ponens - Smaug123
https://www.gwern.net/Modus
======
rocqua
I recall hearing this adage from a philisopher of mathematics a few years ago,
and it has really stuck with me. To me, this seems like the source of much
unproductive disagreement.

It highlights on one hand the value of Baysian theory (for it can break the
stalemate). On the other hand, it highlights an uncomfortable truth for
science. Two reasonable people with different experiences (and hence different
priors) can interpret the same evidence in wholly different ways. Hence, there
is not a clear-cut criterion for 'conclusive' evidence.

~~~
nabla9
How does Bayesian thinking hep with this??

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rocqua
When you have an proof that A -> B and indications that A is true, you have
two choices:

\- Accept that B is also true \- Accept that A is false despite the
indications.

In a Bayesian way of thinking you compare which of these has a higher prior
and pick that one. (Glossing over the fact that 'accept X is true' is not
quite a Bayesian thing to say)

~~~
nabla9
That changes nothing because you assume A has higher prior and the other guy
assumes -B has higher prior.

When someone rejects the premise priors, you Bayesian reasoning does not help.

~~~
rocqua
It helps explain the paradox that the same evidence leads to differing
conclusions.

~~~
danharaj
What you described is not essentially different. It's the same phenomenon with
{0, 1} (classical) and [0,1] (bayesian).

What makes bayesian reasoning interesting is its dynamics, how does evidence
change beliefs? But does it even make sense to talk about posteriors
associated with classical deduction?

~~~
gwern
It's hard to say how Bayesian reasoning really works with classical logic
(which is true of all statistical approaches), but you can set up plausible
Bayesian models where you do get 'backfire' or 'polarization' effects in which
people update in different directions based on the same observation because of
the different models/priors they hold:
[https://www.gwern.net/docs/statistics/bayes/2007-bullock.pdf](https://www.gwern.net/docs/statistics/bayes/2007-bullock.pdf)
[http://papers.nips.cc/paper/3725-bayesian-belief-
polarizatio...](http://papers.nips.cc/paper/3725-bayesian-belief-
polarization.pdf)

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nothrabannosir
Great article, thanks for posting this.

In an unlikely clash of worlds, this islamic passage:

 _> Nothing can be soundly understood_

 _> If daylight itself needs proof._

Reminds me of an old SMBC comic by Zach Weinersmith: [https://www.smbc-
comics.com/comic/2010-09-23](https://www.smbc-comics.com/comic/2010-09-23)

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skinner_
Loosely related, one of my favorite aphorisms ever:

“One man’s vicious circle is another man’s successive approximation
procedure.” -- Cosma Shalizi

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JoshMandel
I have a notation question: if I am reading correctly, the article uses "A ⊃
B" to mean "A implies B" \-- but subset/superset notation defined at
[https://en.m.wikipedia.org/wiki/Subset](https://en.m.wikipedia.org/wiki/Subset)
seems to be just the opposite (i.e., "A implies B" should mean B is a superset
of A, as in "B ⊃ A". Are these two conflicting notations, or am I just
confused?

~~~
gwern
It's not subset, but
[https://en.wikipedia.org/wiki/Material_conditional](https://en.wikipedia.org/wiki/Material_conditional)

~~~
repsilat
Sure, though the grandparent is kinda correct in one sense -- if one _were_ to
use a set operator to mean logical implication, you _could_ use "is a subset
of".

Imagine the set universe has just one element, "truth". Every statement is a
set, and that set contains "truth" or it doesn't. Now, if A implies B, then
one of three things is true:

\- A contains "truth" and so does B,

\- B contains "truth" but A does not, or

\- Neither A not B contains "truth".

So implication would mean that B is a superset of A (or A⊆B.)

There are senses in which the reverse operator could make sense, though, like
for logical corrolaries. Any statement provable from B is also provable from
A, but not necessarily vice-versa. If each statement corresponds to the things
it implies, then if B can be proven from A then A⊇B.

I guess this difference is natural. Think of these logical statements as
constraining the universe of possible things. More constraints means fewer
possible universes. (More to the point, a subset of constraints leads to a
superset of universes.) The direction of Set relationships between logical
statements will depend on which you put in your sets.

EDIT: I just re-read this and fixed a couple of errors, leading to maybe the
opposite conclusion!?

