I'd not rewrite here classical Gettier cases.
Each of cases hinges on a crucial fact: after obtaining "knowledge" from observable facts via disjunctive introduction or entailment, the context, in which those statements are evaluated, is changed. Invoking the whole story with "true by accident" statements is masking conflation of contexts.
Below I'll try to argue that under mild relativistic assumptions there will be a statement in Gettier case that's both true and false. I'm not touching the subjective part about qualifying questionable truths as knowledge, I'm only taking into account statements made in the case and (somewhat implicit) uses of first-order inference.
Notion of truth is contextual. // (Well, that's pretty clear; "It's raining outside" is no longer true in a context when it's no longer raining. Knowledge about changing world ages poorly.)
Notion of justifiability is contextual. // (If some of presuppositions that were used to justify a statement are no longer true — or, more accurately, not true in current context — still calling a statement justified is absurd: we could just start with a false statement, and using implication derive any statement we want from the get-go and call it justified. Saying "Either we're in Richmond, or all frogs have six eyes" in Richmond and driving to Birmingham does not make every frog grow four extra eyes, neither does it justify that statement in Birmingham.)
If we agree that points 1 and 2 (before parentheses) are true — by that I mean mostly that all assumptions are taken as contextual truths, and all epistemic statements are referring to the current context unless explicitly specified; and Gettier case is sound, then statement "Jones owns a Ford", hereinafter F, will be simultaneously true and false in context of the case.
Explanation of point 3.
Note that we, readers of the case, are in a context where Jones automotive situation is not in the realm of belief, it is taken as a presupposition — and, therefore, can and has to be used for assessment of justifiability; so F must be true to qualify «F entails "F or X"» as just inference. If this inference is not just, case is not sound, so F is true.
F being false is explicitly used to infer X from "F or X" (...and obtain "paradoxical" conclusion that Smith's knowledge cannot qualify as knowledge, but that's not relevant); so F is also false.
Are there any reasonable objections to this argument?
