TheGrammarBolshevik wrote:By "inference," do you mean "inductive inference"?

I don’t have any strong opinions on the definition of logic, or any of its subsets. As long as whatever we want to call “deduction” plus whatever we call “inference” adds up to our definition of “logic” that should be sufficient.

Zamfir wrote:@Tristan, is your "logic" part still doing much useful work there? For pretty much any case involving the real world, you've just shifted the problematic parts into the "observation", with the logic as a possibly trivial step at the end.

Exactly, at this step we’re just rewording “justified” and part of that term is the ability to use logic, so it has to be included for completeness, but that’s all it has to do.

Zamfir wrote:But the Gettier problem kicks in at knowing there is one dog, which in your account becomes entirely an issue of observation.

I definitely agree, I think that the whole point of Gettier problems is to point out that there was a problem with the way we were using “justified” in the JTB definition.* Rewording “justified” this way allows us to stick all the problems in one spot (mostly in “observations” but possibly in “inferences” as well.) But it should still leave the definition to be functionally equivalent to JTB in all cases, we haven’t added or taken anything away, we’ve just shuffled around the terms in “justified.”

The next step is to add “true” to the third requirement, so we end up with:

1. S believes p

2. p is true

3. It's possible to logically conclude p based on S's true observations and inferences

I don’t think anyone would call a belief based on false observations or false inferences knowledge, so adding “true” here shouldn’t cause a conflict with our everyday usage, LTB and JTB should still be equivalent in all day to day usages. But it should remove a lot of Gettier style problems, and it may remove all of them. One big benefit either way is that it forces us to be explicit in our describing our hypothetical situations to be able to apply this definition. It’s not possible to just create a somewhat vague situation, point to a problem somewhere in “justified” and be done with it. If we want to use LTB we have to be explicit about what observations and inferences are being used, which would hopefully force us to ask questions about what’s actually going on in our brains. Some of those questions can only be answered empirically i.e., it may be possible to come up with a hypothetical situation which wouldn’t seem to work with LTB, but in those situations we would have to ask “is this a realistic situation, would this actually be possible?” Which seems like an interesting and worthwhile question to ask.

Zamfir wrote:And at the same time, your account doesn't do much about the Fermat's Theorem issue either. It's clearly possible to know some mathematical facts without knowing every deducable conclusion of them. Your account still makes guessing correctly about a mathematical theorem the same as knowing.

I think I might’ve addressed this a few times already, but here goes:

1. I believe that I personally use “knowledge” to include cases where a belief could be consciously logically concluded, but hasn’t definitively.

2. I know that not everyone agrees with this, but I suspect given enough examples that some/many people actually would.

3. If you don’t agree with it under any situation then the definition you’re using should include #3 as “S

has been logically concluded based on S's true observations and inferences." I'm assuming that concluding p from an observation p is acceptable here. Or someone could even include both possibilities with an "either" option if that's how they were using "knowledge."

4. I honestly don’t understand the FLT well enough to say if I think a mathematician would “know” FLT in this hypothetical situation. It seems that the same point could be made with arithmetic instead, it’s not a question on difficulty, it’s a question of simple logic. If in one instance I claim that 15 x 20 = 300, but I make a mental error such that I appear to have consciously used incorrect logic to get there, do I know that “15 x 20 = 300” in this instance? I would say yes. If other people would say no, that’s probably an example of us using different internal definitions of “knowledge.”

------------------------------------------------------------------------------------------------------

*I could also imagine other styles of examples that would point out problems with the way we use “belief” and “true” even if those don’t seem as popular. In fact, some of them may have been brought up and just lumped in with Gettier problems as “Gettier style problems.” I’m ignoring the possibility for these other kinds of problems since the whole point has been Gettier problems and the issues we run in to with hypothetical situations. I think JTB works great in day to day usage since usually we can’t tell if an example of knowledge is an example of a Gettier problem outside of hypothetical situations