Bayesian Confirmation / Justification

Question

Bradley in his book titled " A critical introduction to Formal Epistemology" mentions the grasshopper example (page 122) where:

Hypothesis: Unobserved grasshoppers live south of Canada

Instance: An observed grasshopper 3 inches south of Canadian border.

Additional information: You previously believe grasshoppers could only live in relatively mild climates. However if there are grasshoppers living 3 inches from the Canadian border, there are almost certainly grasshoppers in Canada. So your hypothesis is disjustified

I am puzzled by this example. I would really appreciate it if someone could tell me how this is supposed to work. (Formalized solution of the example is also highly appreciated)

Example can be read here

Answer 1

The example, and the way it is phrased, is very confusing, I had to read it several times to parse it correctly. (It also originally had a small but decisive typo in your version of it in the question, I have edited to fix that.)

The hypothesis is that (all) unobserved grasshoppers live south of Canada --which is to say, there are no unobserved grasshoppers north of the Canadian border. But if you observe a grasshopper close to the border, it "disjustifies" (which is a confusing way to say "lessens the justification for") the hypothesis. Seeing a grasshopper that close to Canada makes it much less likely there are no grasshoppers in Canada.

Answer 2

The idea is that there is an epistemological margin of error for whatever might be considered a habitable region. Therefore, wherever a grasshopper is observed to be living, we know that there is at least some minimal sphere within which the grasshopper may live. We don't know how big it is, but it must have a radius which is greater than 2 inches.

I cut some corners in the following proof, but it at least gives you an idea of how it might be formalized. The premises are:

1. If a given species lives within a minimal habitable sphere, that sphere is a subset of the region habitable by that species.
2. If two regions intersect, there is a region that is a subset of both of them.
3. The transitivity of subsets.
4. A grasshopper lives in a minimal sphere, designated m.
5. Canada intersects the minimal sphere within which the grasshopper lives.

The conclusion might be read as: There is a region habitable by grasshoppers which has a subset within Canada.

• Hxy = x is a habitable region for the species of y
• Mxy = x lives in the minimal habitable sphere y
• c = Canada
• g = grasshopper

{1}            1.  ∀xy[Mxy → Ǝz[y⊆z & Hzx]]    Prem.
{2}            2.  ∀xy[x∩y → Ǝz[z⊆x & z⊆y]]    Prem.(Set Theory)
{3}            3.  ∀xyz[(x⊆y & y⊆z) → x⊆z]     Prem.(Set Theory)
{4}            4.  Mgm                          Prem.
{5}            5.  m∩c                          Prem.
{1}            6.  Mgm → Ǝz[m⊆z & Hzg]         1 UE
{1,4}          7.  Ǝz[m⊆z & Hzg]               4,6 MP
{8}            8.  m⊆h & Hhg                   Assum. TD(h)
{8}            9.  m⊆h                         8 &E
{8}            10. Hhg                         8 &E
{2}            11. m∩c → Ǝz[z⊆m & z⊆c]         2 UE
{2,5}          12. Ǝz[z⊆m & z⊆c]               5,11 MP
{13}           13. r⊆m & r⊆c                   Assum. TD(r)
{13}           14. r⊆m                         13 &E
{13}           15. r⊆c                         13 &E
{8,13}         16. r⊆m & m⊆h                   9,15 &I
{3}            17. (r⊆m & m⊆h) → r⊆h           3 UE
{3,8,13}       18. r⊆h                         16,17 MP
{3,8,13}       19. r⊆h & r⊆c & Hhg             10,15,18 &I
{3,8,13}       20. Ǝxy[x⊆y & x⊆c & Hyg]        19 EI
{2,3,5,8}      21. Ǝxy[x⊆y & x⊆c & Hyg]        12,13,20 EE
{1,2,3,4,5}    22. Ǝxy[x⊆y & x⊆c & Hyg]        7,8,21 EE