I’m not sure what you would write down (see my comment on the ambiguities of your statement of the problem). But here is one possible version.
Suppose that there’s some set of sentences S (finite, or infinite). We can “believe” some subset of these sentences, B. (For probabilists, we can include statements-of-probability as sentences, EG “it will rain today with p < .4″.)
A justification function J is supposed to map each belief b to a justifying set, J(b). J(b) should be a subset of B (we should believe our justification). The idea is that the sentences in J(b), taken together, should provide a sufficient justification in the belief b.
Think of J(b) as defining a directed graph on B, where x->y if x is in J(y). Now, some assumptions:
There should be no root notes, meaning x where J(x) is the empty set. This codifies the intuition that every belief should have a justification.
Justification should be well-founded: every non-empty subset C of B should have a least element, IE an x such that there is no y∈C such that y->x. This captures the intuition that infinite chains of justification don’t count, including circular reasoning that goes around forever and ever, and also non-repeating infinite chains where you can justify each statement with some other, but “nothing justifies the whole chain”.
Theorem: There is no suitable notion of justification J() which matches the above axioms for non-empty belief sets B.
Proof: Let C=B. Axiom 1 says that there are no root nodes. Since this choice of C is nonempty, Axiom 2 requires that there exist some root node.
As a mathematical theorem, this is not very interesting. What makes it interesting is the independent plausibility of the two axioms with respect to an intuitive notion of justified belief.
For example, formal logic does not try to support a notion of justification following the above axioms. But intuitively, the above reasoning is still behind assertions like “you need to assume something in order to prove anything”.
The fact that Chisholm’s discussion oscillates between these two versions of the Problem of the Criterion and the fact that he seems to be aware of the two versions of the problem help make it clear that perhaps there is no such thing as the Problem of the Criterion. Perhaps the Problem of the Criterion is rather a set of related problems. This is something that many philosophers since Chisholm, and Chisholm himself (see his 1977), have noted.
So I think one could just as well say that Gödel’s incompleteness theorems formalize the problem of the criterion, or perhaps even cite the Löbstacle as a formalization of the problem.
I’m not sure what you would write down (see my comment on the ambiguities of your statement of the problem). But here is one possible version.
Suppose that there’s some set of sentences S (finite, or infinite). We can “believe” some subset of these sentences, B. (For probabilists, we can include statements-of-probability as sentences, EG “it will rain today with p < .4″.)
A justification function J is supposed to map each belief b to a justifying set, J(b). J(b) should be a subset of B (we should believe our justification). The idea is that the sentences in J(b), taken together, should provide a sufficient justification in the belief b.
Think of J(b) as defining a directed graph on B, where x->y if x is in J(y). Now, some assumptions:
There should be no root notes, meaning x where J(x) is the empty set. This codifies the intuition that every belief should have a justification.
Justification should be well-founded: every non-empty subset C of B should have a least element, IE an x such that there is no y∈C such that y->x. This captures the intuition that infinite chains of justification don’t count, including circular reasoning that goes around forever and ever, and also non-repeating infinite chains where you can justify each statement with some other, but “nothing justifies the whole chain”.
Theorem: There is no suitable notion of justification J() which matches the above axioms for non-empty belief sets B.
Proof: Let C=B. Axiom 1 says that there are no root nodes. Since this choice of C is nonempty, Axiom 2 requires that there exist some root node.
As a mathematical theorem, this is not very interesting. What makes it interesting is the independent plausibility of the two axioms with respect to an intuitive notion of justified belief.
For example, formal logic does not try to support a notion of justification following the above axioms. But intuitively, the above reasoning is still behind assertions like “you need to assume something in order to prove anything”.
But, really, I think the problem of the criterion is supposed to be a vague cluster and the above comes nowhere near capturing the extent of it. Internet Encyclopedia of Philosophy on the subject:
So I think one could just as well say that Gödel’s incompleteness theorems formalize the problem of the criterion, or perhaps even cite the Löbstacle as a formalization of the problem.