An Abduction Problem/Solution

Let T be our theory7 , and `, be our consequence relation. We have an abductive problem when there is a contention χ before us which is not entailed by our theory. Two potential versions suggest themselves. In one case we also know that T ` ¬χ, and in the other T0¬χ. In the latter case we can imagine the grue-paradox as a good example. Intuitively, we wish to able to conclude that the emerald-hypothesis is true, but we have no way of proving our contention either way. In the former case we might imagine the case of an informal paradox, wherein we have arrived at a unintuitive conclusion, i.e. T ` χ, but we seek to motivate T ` ¬χ as this is the more plausible result. These kind of cases crop up in philosophy all the time. Assume for the sake of argument a premiseφsuch that T∪{φ} `χ. This is not areductio argument against φsince no contradiction need emerge, nevertheless you feel obliged to adapt your theory in such a manner so that¬χcomes out as the more plausible result. That is to say, you disbelieve χ despite evidence, and seek to amend the theory so as to promote a belief in¬χ. A number of amendments can achieve this result. Either you add further information to your theory such that ¬χ becomes a consequence of your information - in which case`is definitively non-monotonic. More modestly you might attempt to block the derivation ofχby removing some premises from our theory. Or perhaps reason indirectly to the conclusion ¬χ by the sheer absurdity of the conclusion χ and its consequences. For this latter strategy to work we would have to show that χ` ⊥or more likely T∪χ` ⊥. In any case an abduction problem is a choice problem.

A bit more precisely. Let an abductive problem be aχ-problem where we wish to concludeχ, but it is not immediately derivable from our theory T. So for instance we have a choice between the a set of explanatory or justificatory claims (γ, γ1, ...γn). This set may or may not contain

mutually exclusive elements. An abductive solution to theχ-problem is a choice function over {γ,

γ1, ...γn} with one free variable i.e. c(? {γ, γ1, ...γn}) = γi such that γi : χ for one particular

choice which justifies χ. The choice function may be specified in such a way so as to be indepen- dent or dependent upon broader details of the theory T. In most cases, this choice function is best understood as an operation which takes inputs from our active theoretical commitments.

In any case we assume that there are operations we can perform on our theory so as to converge on the result T ` χ. An abductive problem is as we described above, but an abductive solution is a sequence of operations which ensures that χ ∈ T. Obviously we can amend our theory with further propositional information or particular justification-operations as in justification logic. Let

γ be either an operation performed on, or a premise included in, our theory. We make three classifications.8 An abductive solution is

Consistent If T ∪{γ}₀⊥

Fully Explanatory Ifγ `χ∧ ¬W

(γ1...γn).

7_{Thought of as abstraction from our beliefs and knowledge however these are cashed out.} 8

Minimal If for all abductive solutions γ’ ifγ `γ’ thenγ’`γ.

These criteria are apt if not exhaustive constraints on abductive solutions. Recall the Gettier case involvingMoriarty’s wallet. It is clear that the argument works by undermining the explana- tory power of any justification you might endorse. No typical justification γ can be such that T ∪{γ} 0 χ, where χ is the Moriarty-hypothesis. For any typical scenario, there is an extreme counter-example to be found in logical space. As such Gettier concludes that justified true belief, no matter the particular nature of the justification, is not definitive or even explanatory, hence not sufficient for our understanding of knowledge. In short no addition of vindicating premises can ex- clude an underdetermination problem if we allow that W is the entirety of logically possible space. Even restricting W often allows for the generation of a reasonable underdetermination problem. This is, despite appearances, a positive result. Such an observation is crucial for underwriting the non-monotonic species of explanation.

The idea that a justification or an explanation must be such that it rules out all alternatives is too strong. This confusion stems from the idea that knowledge is that which holds in possible worlds, and explanations and justifications are apt to induce knowledge. This constraint insists that an explanation would ensure that no ¬χ-worlds are possible, after an explanation has been deployed. But this only really holds if our justification is permissible in every possible world. Or put another way, the function c( ?, {γ, γ1, ...γn}) = γi such that γi : χ where γi is consistent,

fully explanatory and minimal is very hard to find. Sceptical arguments go towards separating the notions of explanatory and justificatory claims from knowledge. We might prefer to say that an explanation (or justification) is apt to induce knowledge or belief just when it suffices to rule all other similarly plausible hypotheses. Construed in terms of plausibility orders we might say that an abductive solution to the χ-problem is an explanation i.e. an operation which re-orders the plausibility ranking so that only the χ-worlds appear among the most plausible worlds/theories. Or rather we might simply adopt the following standard: An abductive solutionγ is...

Explanatory ifT∪γ`χ

Fully T-Explanatory ifT ∪γ `χ∧ ¬W

(γ1...γn).

Preferred ifγ is Explanatory and for allγ’, we have a ranking such thatγ is preferred to γ’

On this setting we have the agent consider an abductive solution solely with respect to his total information, but we might further want to say that only true information is considered. What reason do we have to think that grue advocate cannot derive his desired conclusion from his total information. To avoid such possibilities we should have to specify further constraints on the calibre of an abductive solution. Minimally, we would hope to converge collectively on a single solution.

Minimal constraints on a Solution

You might think that abductive solutions to a χ-problem are primarily subjective. They are the operations required to enhance a theory in a way which ensures the derivability ofχfrom the newly available premises. But this “solution” is no solution at all. Add a contradiction to your theory and we can derive any number of trivial abductive solutions to aχ-problem. Easier still, addχto your theory and we’re done. To avoid trivial problems we must impose a restraint against such solutions.

We say that a theory T commands assent just when true information regarding the theory is such that only one conclusion will be inferred by any rational agent in the χ-abduction prob- lem. Allow that the notion of rationality is here somewhat context dependent, and we insist that rational reflection on theory T is performed with acknowledgement of the appropriate context. Contexts are incorporated in our reasoning by the addition of information and rules of procedural

inference to our base beliefs. If Tcommands assent, then disagreement over theχ-problem is only possible if two agents reason on the basis of distinct assumptions about the context i.e. that they are working with relevantly different theories T1 and T2. We call reasoning from inappropriate

contextual assumptions to be defective reasoning.9

An abductive solution is intersubjectively rational just when it is agreed that our theory is (a) contextually appropriate and (b) commands assent. If we let γ be the conjunction of all the contextually appropriate information and rules of justification, γ can be added to our theory. So we say that γ is an inter-subjectively rational solution to theχ-problem if T∪{γ} `χ. This is far preferable to the merely subjective solutions to theχ-problem, since the intersubjective constraint forces us to defend our solutions. In this respect, there is an effective check on the viability of our abductive solution.

Any hope of putting an entirely objective check on candidate solutions flounders if we link objectivity with knowledge, due to the pervasive nature of Gettier-style counter examples. But in analogy with the notion of safe belief, we might think of an abductive solution as being objectively confirmed if under any true change of the context our solution χremains derivable. When we dis- cuss an abductive solution we would hope that it at least be intersubjectively rational. Ultimately, we wish to be able to say how the “?” input of our choice function, c(? {γ, γ1, ...γn}) = γi such

that γi : χ, is determined.