Stalnaker proposes that a possible worlds analysis of propositions allows us to escape the dilemma facing reconstructions of arguments used to explain a process of reasoning. The idea is, supposing someone sees a footprint in the sand, and they immediately infer that a person has been walking in the sand within the last few hours, one may explain this inference by constructing a deductively valid argument of this process of reasoning. The restructured argument adds some suppressed premises, e.g. such impressions are made only by human feet, to the explicit premise (the perceptual belief). Now, the dilemma faced by the reconstruction concerns its correctness as an explanation. The suppressed premises must be accounted for, but how exactly must they have entered into the initial inference? The reconstruction either imposes on the agent implausible unconscious processes, or it fails to be an explanation, but is rather a model of how the inference might have happened.
Stalnaker’s account constructs belief states as sets of possible worlds, and individual beliefs as negative properties of belief states. The explanation above is correct, since the belief state will be one relative to which the premise entails the conclusion, and the suppressed premises can be though of as properties of the initial belief, i.e. properties which show that that belief state is one relative to which the explicit premise entails the conclusion. An argument may be extracted for further analysis as follows:
1. if for all possible worlds compatible with the initial belief state of the agent in which the premise is true, the conclusion is also true, then initial belief state of the agent is one relative to which the premise entails the conclusion.
2. The suppressed premises are properties of the initial belief state which show that the belief state is one relative to which the explicit premise entails the conclusion.
3. (1) and (2).
4. if (3) and the initial belief state of the agent contains no possible worlds in which the premises listed are false, then the reconstructed argument is a literal description of the situation.
5. If the reconstructed argument is a literal description of the situation, then it is a correct explanation.
6. The reconstructed argument is a correct explanation.
The objection I wish to raise here concerns the bloated premise (4). Essentially I’ve packed the key properties which an argument must have in order to be considered a correct explanation. A great advantage of this account allows a complicated reconstruction to have many suppressed premises, while still being a literal description – as Stalnaker notes. The problem, however, lies in the multitude of reconstructed arguments that could be assessed as literal descriptions of the situation. The problem, then, is not that it can’t explain the situation, but rather that it has too many explanations.
A weakened Benacerraf dilemma seems to face Stalnaker’s explanations. The suppressed premises may be few in number or many; depending on how complicated we wish to make our explanation. But which explanation is the correct one based on the explicit premise, inferred conclusion, and stipulated suppressed premises? They are all correct, it seems, according to Stalnaker’s account. But that doesn’t seem right. We weren’t hoping to just create an imaginative model of how the inference might have been made, but rather to correctly explain it. But an imaginative model is what we appear to be left with, for the suppressed premises, which show that the belief state is one relative to which the explicit premise entails the conclusion, may be near infinite in number. And since these suppressed premises are what differ from one argument reconstruction to another, then we are left with near infinite possible explanations. A weakened Benacerraf dilemma would conclude that for any explanation, we should not accept it.
The conception of beliefs as negative properties of a belief state was supposed to elucidate this problem of reconstructing arguments. Instead, it has left us with no explanations which we can accept. Perhaps, then, there is something wrong with considering beliefs as negative properties of a belief state.