Epistemic scores and consistency

Scoring rules measure the distance between a credence and the truth value, where true=1 and false=0. You want this distance to be as low as possible.

Here’s a fun paradox. Consider this sentence:

1. At t₁, my credence for (1) is less than 0.1.

(If you want more rigor, use Goedel’s diagonalization lemma to remove the self-reference.) It’s now a moment before t₁, and I am trying to figure out what credence I should assign to (1) at t₁. If I assign a credence less than 0.1, then (1) will be true, and the epistemic distance between 0.1 and 1 will be large on any reasonable scoring rule. So, I should assign a credence greater than or equal to 0.1. In that case, (1) will be false, and I want to minimize the epistemic distance between the credence and 0. I do that by letting the credence be exactly 0.1.

So, I should set my credence to be exactly 0.1 to optimize epistemic score. Suppose, however, that at t₁ I will remember with near-certainty that I was setting my credence to 0.1. Thus, at t₁ I will be in a position to know with near-certainty that my credence for (1) is not less than 0.1, and hence I will have evidence showing with near-certainty that (1) is false. And yet my credence for (1) will be 0.1. Thus, my credential state at t₁ will be probabilistically inconsistent.

Hence, there are times when optimizing epistemic score leads to inconsistency.

There are, of course, theorems on the books that optimizing epistemic score requires consistency. But the theorems do not apply to cases where the truth of the matter depends on your credence, as in (1).

Syntactic self-reference without diagonal lemma or Gödel numbers

For the proof of Goedel's incompleteness theorem and in work on the Liar Paradox it is usual to use the Diagonal Lemma to secure self-reference. The challenge of self-reference is this. Given a predicate Q, find a syntactically definable predicate P such that

1. (s)(P(s) → R(s))

is provably the one and only sequence of symbols satisfying P. Then (1) says that Q holds of itself. (To get the (Strengthened) Liar Paradox, just make R(s) say that s is not true.) But the proof of the diagonal lemma is hard to understand.

I find the following way of securing self-reference easier to understand. Start with a language that has nestable quotation marks, which I'll represent with ‘...’, and some string manipulation tools. I'll use straight double quotation marks for meta-language quotation. Add to the language a new symbol "@" which is ungrammatical (i.e., no well-formed formula may contain it). For any sequence of symbols s, we define two new sequences of symbols N(s) and Q(s) by the following rules. If s contains no quoted expressions or contains imbalanced opening and closing quotation marks, N(s) and Q(s) are just "@". If s contains a quoted expression, Q(s) is the first quoted expression, without its outermost quotation marks (but with any nested quotations being included), and N(s) is the result of taking s and replacing that first quoted occurrence of Q(s), as well as its surrounding single quotation marks, with "@". Thus:

2. Q("abc‘def‘ghi’’+jkl")="def‘ghi’"
3. N("abc‘def‘ghi’’+jkl")="abc@+jkl".

It is easy to see that Q and N are syntactically defined. Now, let M(s) be equal to N(s) if N(s)=Q(s) and let M(s) be an empty sequence "" otherwise. Again, M(s) is syntactic. Now consider this sentence:

4. (s)(‘(s)(@=M(s) → R(s))’=M(s) → R(s)).

It is easy to prove (given a bit of string manipulation resources) that the only sequence s that satisfies the antecedent of the conditional is (4) itself. So we have constructed the syntactic predicate P(s). It is: ‘(s)(@=M(s) → R(s))’=M(s).

One can also adapt this to work with Goedel numbers and hence presumably for use in proving incompleteness.

[Removed a nasty typo.]

A dilemma for the deflationary theory of truth

Is the claim that truth is completely characterized by the deflationary theory itself a part of the deflationary theory of truth?

If yes, then the deflationary theory of truth has a problematic kind of self-referentiality: it contains the statement:

1. The theory which includes (1) and statements X-Z completely characterizes truth.

But it is not clear that the content of (1) is sufficiently determinate. It seems that (1) suffers from a similar semantic underdetermination to that suffered by the truthteller sentence:

2. Statement (2) is true.

Suppose now that the claim that truth is completely characterized by the deflationary theory of truth is not a part of the deflationary theory of truth. There are now two options. Either the completeness claim is or is not known to be a consequence of the deflationary theory of truth. If it is not a known consequence of the deflationary theory of truth, then we have a problem for deflationary theorists who assert that the deflationary theory of truth is complete. For if they do not know their assertion to be a consequence of the deflationary theory of truth. But if it is not a consequence of it, then the theory is not complete. So it seems unlikely that they know the theory to be complete, and they are asserting something something they do not know.

Suppose that the completeness claim is known to be a consequence of the deflationary theory of truth. But the deflationary theory of truth without the completeness claim consists merely in a claim as to what the truth-bearers are and all the instances of the T-schema. But the completeness claim simply does not follow from these. For there are multiple predicates, with "is true" being only one of them, that satisfy the deflationary theory of truth: any predicate extensionally equivalent to "is true" satisfies the deflationary theory of truth just as well. And such predicates are myriad. For instance: "is true and is not false", "is believed by God", "is true if 2+2=4", etc.

Perhaps if we add to the deflationary theory that "is true" expresses a very natural property, then we can rule out some of the alternate predicates. But "is believed by God" appears very natural as well. So at least the theist can't take this way out.

Deep Thoughts XIV

Every tautology is true.