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We say that a probabilistic credence function $c : \mathcal{F} \rightarrow [0, 1]$ is

*regular*if $c(A) > 0$ for all propositions $A$ in $\mathcal{F}$ such that there is some world at which $A$ is true.

**The Principle of Regularity (standard version)**If $c : \mathcal{F} \rightarrow [0, 1]$ is your credence function, rationality requires that $c$ is regular.

I won't specify which worlds are in the scope of the quantifier over worlds that occurs in the antecedent of this norm. It might be all the logically possible worlds, or the metaphysically possible worlds, or the conceptually possible worlds; it might be the epistemically possible worlds. Different answers will give different norms. But we needn't decide the issue here. We'll just specify that it's the same set of worlds that we quantify over in the Dutch Book argument for Probabilism when we say that, if your credences aren't probabilistic, then there's a series of bets they'll lead you to enter into that will lose you money

*at all possible worlds*.

In this post, I want to consider the almost-Dutch Book Argument for the norm of Regularity. Here's how it goes: Suppose you have a credence $c(A) = 0$ in a proposition $A$, and suppose that $A$ is true at world $w$. Then, recall, the first premise of the standard Dutch Book argument for Probabilism:

**Ramsey's Thesis**If your credence in a proposition $X$ is $c(X) = p$, then you're permitted to pay £$pS$ for a bet that returns £$S$ if $X$ is true and £$0$ if $X$ is false, for any $S$, positive or negative or zero.

So, since $c(A) = 0$, your credences will permit you to sell the following bet for £0: if $A$, you must pay out £1; if $\overline{A}$, you will pay out £0. But selling this bet for this price is weakly dominated by refusing the bet. Selling the bet at that price loses you money in all $A$-worlds, and gains you nothing in $\overline{A}$-worlds. Whereas refusing the bet neither loses nor gains you anything in any world. Thus, your credences permit you to choose a weakly dominated act. So they are irrational. Or so the argument goes. I call this the almost-Dutch Book argument for Regularity since it doesn't punish you with a sure loss, but rather with a possible loss with no compensating possible gain.

If this argument works, it establishes the standard version of Regularity stated above. But consider the following case. $A$ and $B$ are two logically independent propositions --

*It will be rainy tomorrow*and

*It will be hot tomorrow*, for instance. You have only credences in $A$ and in the conjunction $AB$. You don't have credences in $\overline{A}$, $A \vee B$, $A\overline{B}$, and so on. What's more, your credences in $A$ and $AB$ are equal, i.e., $c(A) = c(AB)$. That is, you are exactly as confident in $A$ as you are in its conjunction with $B$. Then, in some sense, you violate Regularity, though you don't violate the standard version we stated above. After all, since your credence in $A$ is the same as your credence in $AB$, you must give no credence whatsoever to the worlds in which $A$ is true and $B$ is false. If you did, then you would set $c(AB) < c(A)$. But you don't have a credence in $A\overline{B}$. So there is no proposition true at some worlds to which you assign a credence of 0. Thus, the almost-Dutch Book argument sketched above will not work. We need a different Dutch Book argument for the following version of Regularity:

**The Principle of Regularity (full version)**If $c : \mathcal{F} \rightarrow [0, 1]$ is your credence function, then rationality requires that there is an extension $c^*$ of $c$ to a full algebra $\mathcal{F}^*$ that contains $\mathcal{F}$ such that $c^* : \mathcal{F}^* \rightarrow [0, 1]$ is regular.

It is this principle that you violate if $c(A) = c(AB)$ when $A$ and $B$ are logically independent. For any probabilistic extension $c^*$ of $c$ that assigns a credence to $A\overline{B}$ must assign it credence 0 even though there is a world at which it is true.

How are we to give an almost-Dutch Book argument for this version of Regularity? There are two possible approaches.

On the first, we strengthen the first premise of the standard Dutch Book argument. Ramsey's Thesis says: if you have credence $c(X) = p$ in $X$, then you are permitted to pay £$pS$ for a bet that pays £$S$ if $X$ and £$0$ if $\overline{X}$. The stronger version says:

**Strong Ramsey's Thesis**If every extension $c^*$ of $c$ to a full algebra $\mathcal{F}^*$ that contains $\mathcal{F}$ is such that $c^*(X) = p$, then you are permitted to pay £$pS$ for a bet that pays £$S$ if $X$ and £$0$ if $\overline{X}$.

The idea is that, if every extension assigns the same credence $p$ to $X$, then you are in some sense committed to assigning credence $p$ to $X$. And thus, you are permitted to enter into which ever bets you'd be permitted to enter into if you actually had credence $p$.

On the second approach to giving an almost-Dutch Book argument for the full version of the Regularity principle, we actually provide an almost-Dutch Book using just the credences that you do in fact assign. Suppose, for instance, you have credence $c(A) = c(AB) = 0.5$. Then you will sell for £5 a bet that pays out £10 if $A$ and £0 if $\overline{A}$, while you will buy for £5 a bet that pays £10 if $AB$ and £0 if $\overline{AB}$. Then, if $A$ is true and $B$ is true, you will have a net gain of £0, and similarly if $A$ is false. But if $A$ is true and $B$ is false, you will lose £10. Thus, you face the possibility of loss with no possibility of gain. Now, the question is: can we always construct such almost-Dutch Books? And the answer is that we can, as the following theorem shows:

**Theorem 1 (Almost-Dutch Book Theorem for Full Regularity)**Suppose $\mathcal{F} = \{X_1, \ldots, X_n\}$ is a set of propositions. Suppose $c : \mathcal{F} \rightarrow [0, 1]$ is a credence function that cannot be extended to a regular probability function on a full algebra $\mathcal{F}^*$ that contains $\mathcal{F}$. Then there is a sequence of stakes $S = (S_1, \ldots, S_n)$, such that if, for each $1 \leq i \leq n$, you pay £$(c(X_i) \times S_i)$ for a bet that pays out £$S_i$ if $X_i$ and £0 if $\overline{X_i}$, then the total price you'll pay is at least the pay off of these bets at all worlds, and more than the payoff at some.

That is,

(i) for all worlds $w$,

$$S\cdot (w - c) = S \cdot w - S \cdot c = \sum^n_{i=1} S_iw(X_i) + \sum^n_{i=1} S_ic(X_i) \leq 0$$

(ii) for some worlds $w$,

$$S\cdot (w - c) = S \cdot w - S \cdot c = \sum^n_{i=1} S_iw(X_i) + \sum^n_{i=1} S_ic(X_i) \leq 0$$

where $w(X_i) = 1$ if $X_i$ is true at $w$ and $w(X_i) = 0$ if $X_i$ is false at $w$. We call $w(-)$ the indicator function of $w$.

*Proof sketch.*First, recall de Finetti's observation that your credence function $c : \mathcal{F} \rightarrow [0, 1]$ is a probability function iff it is in the convex hull of the indicator functions of the possible worlds -- that is, iff $c$ is in $\{w(-) : w \mbox{ is a possible world}\}^+$. Second, note that, if your credence function can't be extended to a regular credence function, it sits on the boundary of this convex hull. In particular, if $W_c = \{w' : c = \sum_w \lambda_w w \Rightarrow \lambda_{w'} > 0\}$, then $c$ lies on the boundary surface created by the convex hull of $W_c$. Third, by the Supporting Hyperplane Theorem, there is a vector $S$ such that $S$ is orthogonal to this boundary surface and thus:

(i) $S \cdot (w-c) = S \cdot w - S \cdot c = 0$ for all $w$ in $W_c$; and

(ii) $S \cdot (w-c) = S \cdot w - S \cdot c < 0$ for all $w$ not in $W_c$.

Fourth, recall that $S \cdot w$ is the total payout of the bets at world $w$ and $S \cdot c$ is the price you'll pay for it. $\Box$

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