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.
> Two reasonable people with different experiences (and hence different priors) can interpret the same evidence in wholly different ways.
I don't understand how this could possibly be otherwise? Thinking about it more, the problem lies in the juxtaposition of "two reasonable people with different experiences" and "the same evidence", since the evidence you're evaluating is always really the sum total (or distillate, at least) of all evidence you've ever seen. You can't just give someone a new fact in a vacuum and ask them to evaluate it without letting the entire rest of their experience affect that evaluation.
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)
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?
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 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?
The symbol ⊃ is an old notation for material conditional, which is now usually symbolized using ⇒ or →. And yes, use of ⊃ in symbolic logic and set theory is conflicting, as you noted.
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!?
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.
> Two reasonable people with different experiences (and hence different priors) can interpret the same evidence in wholly different ways.
I don't understand how this could possibly be otherwise? Thinking about it more, the problem lies in the juxtaposition of "two reasonable people with different experiences" and "the same evidence", since the evidence you're evaluating is always really the sum total (or distillate, at least) of all evidence you've ever seen. You can't just give someone a new fact in a vacuum and ask them to evaluate it without letting the entire rest of their experience affect that evaluation.
How does Bayesian thinking hep with this??
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)
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.
It helps explain the paradox that the same evidence leads to differing conclusions.
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?
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 http://papers.nips.cc/paper/3725-bayesian-belief-polarizatio...
I was thinking of Bayesian here as a contrast to Frequentist, not as a form of logic with uncertainty.
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
Loosely related, one of my favorite aphorisms ever:
“One man’s vicious circle is another man’s successive approximation procedure.” -- Cosma Shalizi
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 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?
The symbol ⊃ is an old notation for material conditional, which is now usually symbolized using ⇒ or →. And yes, use of ⊃ in symbolic logic and set theory is conflicting, as you noted.
It's not subset, but https://en.wikipedia.org/wiki/Material_conditional
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!?
Could you please include a note explaining that in the text?
I, too, thought it was referring to a subset relationship. Ctrl-F 'Material' also doesn't turn up anything in the page.