The Gettier Problem

Abduction problems allow us to problematise the notions of knowledge and belief. Consider Bob who believes himself to be in possession of twenty euros. Observe that this is true, and that he is justified in his belief because he recalls placing it in his wallet earlier that day. However, unbeknownst to Bob the dastardly Moriarty has replaced his wallet with a convincing fake of com- parable content including twenty euros. Gettier argues that despite the fact that Bob’s belief was justified and true, he cannot claim to know he possesses twenty euro.

We note two things about this scenario: (1) If we accept Gettier’s argument then when faced with the choice between the My-wallet and Moriarty’s wallet we are insisting that we may only achieve knowledge just when the Moriarity’s wallet is impossible i.e. that we cannot later learn that Moriarty tricked us. In other words for anything to be known our belief must follow by neces- sity from the justification we develop on the basis of our premises. Seen alongside the grue-problem we can argue that the grue-case is only paradoxical because we conflate the notion of confirmation with the notion of knowledge. Both the grue-hypothesis and the emerald-hypothesis are equally confirmed by our evidence. This is not in itself problematic, but just seems so if we expect our methods of confirmation to converge (without exception) on one particular conclusion. (2) From the minimal premise that knowledge is the desired goal of reasoning we may draw the conclusion that justifications should provide a sufficient conditions for the acceptance of the justified claim. The need to exclude the Moriarty case would suggest that a justification need also be a necessary condition. Hence the Gettier argument seems to motivate the requirement that justification ought to provide necessary and sufficient reason for the acceptance our conclusion. This is not a solution, since the move is illegitimate by the fact that it makes JTB-definitive of knowledge and this begs the question on the table. Namely, is knowledge justified true belief?

We can also represent a Gettier problem which undermines even the notion that safe-belief is prompted by justified true belief. We take the example from the paper of Baltag et al5 in which they develop a model to include operators for knowledge, belief and justification. We treat this according to their standard model: M = <W, ∼,, E, V> where W is the set possible worlds, V is a valuation map, ∼ is an equivalence relation for knowledge and≥ is a plausibility preorder as usual. We adopt the notation that [[φ]] is the set of φ-worlds determined by V in W. Belief is defined in terms of the plausibility order ≤, while E is their evidence function. In addition to the usual modal operators we have justification terms and an admissibility predicate relating

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In [16] and [84]

4_{e.g. its causal, mereological or supervenience structure} 5

justification terms with propositional formulas.

Imagine an agent Alice hears of our plight because Moriarty boasts of his evil plot telling her he stole our wallet and replaced it. Unbeknownst to him he has mistakenly replaced our wallet with his own which contains 20euros, before he threw our wallet in the rubbish bin. Let p be defined as “I have 20 euro” and q similarly as the claim“Moriarity has 20 euro”. Moriarty boasts of his heist and his comparable wealth of 20euro. Being a reasonable person Alice accepts his testimonial evidence, which implies (a) she finds his evidence admissible and (b) accepts it. Let the actual world be denoted @, then we represent the facts as follows. We omit the arrows ensured by the reflexivity and transitivity.

@

w

w’

w”

The plausibility of the worlds goes from right to left with w’ and w” being the most plausible worlds in our model. Hence immediately we can see that Alice at the actual world believes either myself or Moriarty have 20e. So M, @ |= Ba(p ∨q)∧(p ∨q). Which is to say that Alice believes

a true fact since I do in fact possess 20euro as a consequence of Moriarity’s foolish plan. Further- more allow that the Evidence function E is such that testimonial evidence is always considered an apt justification for the claim testified, i.e. we have [j : q], then since all the axioms of classical reasoning are also justified Alice holds that [x : q → (p ∨ q)] where x denotes the justification variable for the classical right weakening axiom. But then of course by the application axiom there is justification for our conclusion namely that [x·j :(p∨q)]. In short the initial justification stems from Moriarty’s testimony, and the latter two follow as a rule of logic, or by an operation of evidence combination. As such she can derive a justification for her true belief B(p∨q), but in no sense does Alice know a true fact because M, @ |=¬K(p∨q). Worse still she does not even have safe belief in the claim that Moriarty has any money, since an update with the true information that Moriarty has no money, would ensure that all the q-worlds are removed.

@

w

Resulting in a picture where the most plausible worlds serve to invalidate Alice’s testimonial justification for her belief. Worse she neither believes q or p and M, @ 2 a(p ∨ q) since despite

the fact that @ validates the disjunction, w is equally plausible and invalidates the disjunction contrary to the hypothesis that we have safe belief. In any case, this is a counterexample to the definition of knowledge in terms of justified true belief.6

The insight that this suggests is that knowledge as defined by the K-operator is a far more idealised concept than we had previously thought. In practice we may never attain absolute knowl- edge of the kind required to defeat every Gettier case, but that doesn’t do anything to take away the confidence accrued to reliable reasoning. The Gettier problems suggest that we should not try to account for the reliability of reasoning by simply taking our best estimates as infallible, or true because “known”. But we do need some kind of measure by which to prioritise certain lines of reasoning. Otherwise any and all intellectual activity is inexplicable, given the judicious paralysis we would face in any abduction problem.

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If this example seems too artificial, consider the real life Gettier case involving the communication between Galileo and Kepler which has passed into the folklore of science. Upon discovering Saturn’s rings Galileo sends Kepler notice of this discovery in terms of an anagram, thereby patenting the discovery. Kepler (incorrectly) deciphers this anagram to mean that Mars has two moons. The later claim is also true, but would only be discovered two hundred years later. An odd coincidence. Even stranger when Galileo sends news of his next discovery that Venus has phases i.e. that it must be going around the sun. Kepler again deciphers this claim incorrectly to mean that Jupiter has a red spot. This is again true, but not yet discovered. Note the presence of true, justified belief and ask yourself whether Kepler really knew either of his conclusions?

Underdetermination problems are horrendously toxic. Every proposition you care to mention has a gerrymandered competitor by Quinean reasoning. Every single underdetermination problem can be transformed into a Gettier problem! Observe that we are justified in thinking that the emerald we buy tomorrow will be green - we are justified in this belief by the law that all emeralds are green, and better yet, it is true that the emerald will be green. However Gettier’s case stems from the fact that we can’t be said to know that that the emerald we see tomorrow will be green. Since we are unaware that tomorrow’s emerald is rendered green by the time-indexed property of grue-gems. Without providing a reason to reject underdetermination arguments, we risk destabilising the notions of knowledge and belief altogether.