Raven Paradox Revisited

The Raven Para­dox Set­tled to My Satis­fac­tion is a pretty good post, but there’s a few things that we can note to make this prob­lem even clearer:

  • We can sim­plify this prob­lem so that there are only two col­ors (black and white) and two kinds of ob­jects (ravens and lap­tops). The post sort of did this, but not very con­sis­tently. I just thought this was worth men­tion­ing as non-black and non-raven are slightly more ab­stract and so slightly harder to rea­son with.

  • Ac­tu­ally, we can make this even more leg­ible. Make the ob­jects medicine and food; and the prop­er­ties fresh and ex­pired. It’s then im­me­di­ately clear that if we want to check that [All the medicine is fresh], we can ei­ther check each item of medicine to see if it is fresh or we can check all the ex­pired ob­jects and see that none of them is medicine. We’re used to this kind of prac­ti­cal rea­son­ing, so it’s much eas­ier for us than deal­ing with ob­jects and colours.

  • If we have sim­plified the prob­lem so there is only one shade of black, then A: [All ravens are black] is equiv­a­lent to B: [The first raven is black] plus C: [All ravens are the same color] apart from the de­gen­er­ate case. This makes it rather clear why A is more likely to be true when there are less ravens. In par­tic­u­lar, if there’s only one raven, then we only have to worry about B since C is triv­ially true.

  • We tend to con­fuse the fol­low­ing [An ob­ser­va­tion of a white lap­top is in­de­pen­dent of the color dis­tri­bu­tion of ravens] with [An ob­ser­va­tion of a white lap­top is in­de­pen­dent of how many ravens of a par­tic­u­lar colour we will ob­serve]. The first is true apart from the re­stric­tions im­posed by the num­ber of black ravens hav­ing to be in­te­gral, but the sec­ond is only true if we knew it was a lap­top be­fore this ob­ser­va­tion. If it could have been a raven then it can in­fluence the num­ber of ravens of a par­tic­u­lar color by in­fluenc­ing the num­ber of ravens in to­tal.

  • We tend to ex­pect some cor­re­la­tion be­tween the color of an­i­mals. For ex­am­ple, we are quite ready to guess that all ravens are black af­ter only see­ing quite a small ran­dom sam­ple. On the other hand, sup­pose that all the non-black things keep be­ing non-liv­ing crea­tures, whilst many of the black things are non-raven liv­ing crea­tures. We might guess that be­ing black pro­vides an evolu­tion­ary ad­van­tage in this world and so guess that all ravens will be black with­out ever hav­ing seen a sin­gle one. The point is that, given par­tic­u­lar pri­ors, you may have ad­di­tional ev­i­dence be­yond merely the nu­mer­i­cal re­duc­tion.

  • Hem­pel’s re­s­olu­tion from Wikipe­dia is worth high­light­ing as it sub­tly re­frames the prob­lem to make the as­sump­tions more ob­vi­ous. Con­sider the state­ment [All sodium salts burn yel­low], with the con­tra­pos­i­tive [What­ever does not burn yel­low is not a sodium salt]. Burn­ing some ice and find­ing it does not tell yel­low would be ev­i­dence to­wards the con­tra­pos­i­tive and hence also the origi­nal state­ment. This seems para­dox­i­cal, but con­sider if the chem­i­cal makeup was un­known at the start. If some­thing doesn’t burn yel­low and then we analyse it and dis­cover it is a sodium salt then we would have dis­proved the hy­poth­e­sis. By con­ser­va­tion of ev­i­dence, if we dis­cover it is ice we would gain ev­i­dence for the hy­poth­e­sis. It’s easy enough to calcu­late the amount of ev­i­dence us­ing Bayesian tech­niques. From here, it’s easy enough to see that this ob­ser­va­tion only pro­vides no in­for­ma­tion if we im­plic­itly as­sume that we know the chem­i­cal com­po­si­tion (or the type of ob­ject in the raven prob­lem).