I've been posting this around a lot lately (including on "Gettier in Zombie World"), still looking for a solid response.

I think Bayesian probability actually resolves Gettier problems, completely and (ironically, because Bayesian probability doesn't concern itself with this in the slightest) satisfyingly. Understanding that we only know likelihoods, not facts, is enough.

Situation: I know John had 10 coins in his pocket. I think he got the job. I don't know that Smith had 10 coins in his pocket. Do I "know" that the person who got the job had 10 coins in their pocket?

Classic Gettier Interpretation:

• Belief-Jones had 10 coins in his pocket
• Belief-Jones got the job
• • Conclusion-The person who got the job had 10 coins in their pocket

Bayesian Gettier Interpretation(Example numbers used for ease of intuition; minimal significant digits used for ease of calculation):

• Belief-It's likely (90%) John got the job
• Belief-It's likely (90%) John had 10 coins in his pocket
• • Conclusion-It's likely (81%) John got the job with 10 coins in his pocket

...and...

• Belief-It's unlikely (10%) someone other than John got the job
• Belief-It's possible (50%) a given person had 10 coins in their pocket
• • Conclusion-It's unlikely (5%) that someone other than John got the job with 10 coins in their pocket

...thus...

• Belief-It's likely (81%) John got the job with 10 coins in his pocket
• Belief-It's unlikely (5%) that someone other than John got the job with 10 coins in their pocket
• • Conclusion-It's likely (86%) that someone got the job with 10 coins in their pocket