imdef @imdef@qoto.org

@ct_bergstrom My impression of ChatGPT-3 (haven't tried ver. 4) is that it is like a bullshitter who has memorized every book in the library. For many tests, recalling stuff you've read and bullshitting to interpolate/extrapolate to novel stuff evidently works very well.

This is an interesting attempt to give the field of #MachineLearning a foundation.
https://royalsocietypublishing.org/doi/full/10.1098/rsta.2017.0237

@gorskon Good write up. As I've argued on my blog, I think the ultimate source of the problem is Nullism - the belief that the scientific method mandates that you have a Null Hypothesis that wins by default, until/unless it is disproven. EBM then ramps up the default thinking to 11. OTOH, if you don't let a hypothesis win by default, but instead carefully weigh all evidence for and against, then there would be no role for an evidence pyramid.

#Cochrane makes statement
https://www.cochrane.org/de/node/81904

@ct_bergstrom A large part of the problem is that evidence-based medicine builds in type of default thinking: interventions are presumed harmful until a RCT proves them beneficial. It's similar to what Sander Greenland calls "Nullism" but ramped up even further.
https://intemittdefault.wordpress.com/2021/07/10/evidence-decisions-and-default-reasoning/

@RanaldClouston At least in simple cases, I think [](P -> Q) says in the object language what P |= Q says in the meta-language. Maybe there's a divergence in complicated cases, I don't know. In the particular case above, I'd take

[](3|x -> 6|x)

to be equivalent to

(for all natural numbers x)(3|x -> 6|x)

@RanaldClouston (With []P meaning "Necessarily, P".)

Vidal also gives the example with a specific value x=4 in the above statement. Here, if I understand the meaning of '2', '4', '6' and the box correctly, the box doesn't really make any difference.

@RanaldClouston I'm sure logicians have written a lot, I'm just not so familiar with this literature. I think identifying "if-then" with strict implication would avoid some, but not all, unnaturalness. For example, we could translate from English to formal language, and manipulate the material implication:

1. "If 2|x and 3|x, then 6|x."
2. [](2|x ^ 3|x -> 6|x)
3. []((2|x -> 6|x) v (3|x -> 6|x))

But then we'd be blocked from translating back into English, because for an if-then form of line 3 we'd need the stronger

3'. [](2|x -> 6|x) v [](3|x -> 6|x)

@RanaldClouston Fair point, simplicity is a point in favor of material implication. What leaves me unsatisfied with the justifications I've read, though, is that they appear to beg the question by not seriously considering what would be different with a non-truth functional "if-then". E.g. a common argument is that it is convenient to allow vacuously true universal statements, but other non-truth functional implications can also allow these.

I should have been more precise. The two formal expressions

(2|x ^ 3|x) -> 6|x
(2|x -> 6|x) v (3|x -> 6|x)

are equivalent. However, it is less clear cut with their ordinary language translations:

"If x is divisible by 2 and x is divisible by 3, then x is divisible by 6."
"If x is divisible by 2, then x is divisible by 6, or if x is divisible by 3, then x is divisible by 6."
#logic #implication #conditional

Should have been more precise. The two formal expressions

(2|x ^ 3|x) -> 6|x
(2|x -> 6|x) v (3|x -> 6|x)

are equivalent. However, it is less clear cut with their ordinary language translations:

"If x is divisible by 2 and x is divisible by 3, then x is divisible by 6."
"If x is divisible by 2, then x is divisible by 6, or if x is divisible by 3, then x is divisible by 6."
#logic #implication

Example 2: write a|x for "x is divisible by a" or "a divides x". Then
(2|x ^ 3|x) -> 6|x
(2|x -> 6|x) v (3|x -> 6|x)

In both cases, the first form is natural and obvious and the second is something you'd normally never write. But, if pressed, maybe you'd bite the bullet and agree it's an equivalent form. I'm still undecided but I enjoyed the paper.
#logic #implication #conditional (3/3)

Because of the thoughts like the above, I found the following paper quite interesting:
https://www.tandfonline.com/doi/abs/10.1080/11663081.2014.911540

Vidal points out that (P ^ Q) -> R is equivalent to (P -> R) v (Q -> R). Both these forms can be seen to be equivalent to ~P v ~Q v R. Specific instances of this equivalence can be awkward/counterintuitive:

Example 1:
("x is a rhombus" ^ "x is a rectangle") -> "x is a square"
("x is a rhombus" -> "x is a square") v ("x is a rectangle" -> "x is a square")

(3/n)

I was never sure what to make of this, because I have yet to read a discussion of why material implication is a better model of mathematicians' "if P, then Q" than other alternatives. For example, why not understand "if P, then Q" in mathematics as "necessarily, if P, then Q" and take it to correspond to [](P -> Q), where [] is an operator of modal logic? I'm sure people already thought of this, I just haven't seen the pros and cons of this alternative (and other alternatives) compared to the pros and cons of the material implication. E.g., what about the implication in relevance logic? (2/n)

Material implication P -> Q is equivalent to ~P v Q. It is generally agreed that the "if P, then Q" construction in ordinary language is not always the same as material implication. However, when you study mathematics, you're trained to think that, in mathematics, "if P, then Q" really is material implication. Here is an in many ways careful explanation: (1/n)
https://gowers.wordpress.com/2011/09/28/basic-logic-connectives-implies/

#logic #implication #conditional