Triviality Theorems

Triviality Theorems are results concerning indicative conditionals. They show that the only way that the probability of a proposition can consistently track the conditional probability of an indicative conditional is with trivial probability distributions, distributions that are too simplistic to be an adequate model of human belief.

The reason for interest in the triviality theorems is that there is a plausible principle called Adam's thesis, named after Ernest Adams, that the acceptability of an indicative conditional, A→C, is the subjective conditional probability P(C/A). The evidence for Adams' thesis comes from the fact that it seems to explain a wide variety of conditionals that people find acceptable plus, A factual proposition is a proposition that does not contain any conditional. For any factual proposition Q, the acceptability of the proposition is just the subjective probability of Q, i.e. the degree to which one believes Q to be true. If one thinks that conditionals have truth values, then it seems reasonable to believe that their acceptibility would also be the subjective probability of the conditional being true. So, the simple explanation, called Stalnaker's hypothesis, for why Adam's thesis is true is that there is something about the truth conditions of → that guarantees that for any rational credence function P, P(A→C)=P(C/A). The triviality theorems show that this explanation cannot hold except for extremely restrictive (trivial) credence functions. Thus the simple explanation cannot be right. This result calls into question whether truth is the right semantic concept for understanding conditionals.

The first triviality theorem was by David Lewis, "Probabilities of Conditionals and Conditional Probabilities," Philosophical Review 85 (1976): 297-315. Many papers since then have generalized the theorem and given alternative proofs of triviality.

Contents 1 Failures of Tracking 1.1 Dynamic Triviality Results 1.2 Static Triviality Results 2 Failures of Finitude

Failures of Tracking

These proofs show that differences in P(C/A) cannot systematically match differences in P(A→C). One can see this informally by looking at the figure where the yellow shaded area corresponds with the space of possibilities where C is true, the green where A is true, and the yellow green area where A and C are both true. The fraction of area enclosed by a curve corresponds to its subjective probability for some person. P(A→C) is just the area enclosed by the dotted line (as a fraction of the total area of the box). P(C/A) is the ratio of the yellow-green area to the green (including yellow green) area. Notice how the area for A→C represents the fact that A→C can be true or false when A is false. It has been drawn so that A→C is false when A is true and C is false, which is uncontroversial, and so that A→C is true when A&C is true, which is slightly controversial but doesn't affect the basic argument here.

Simple inspection shows that you can wiggle certain segments of the curve to increase or decrease P(A→C) without affecting P(C/A) and vice versa. The only way they could track one another is if there were some additional constraint that is not being represented in the diagram. The problem is that it is hard to see how such a constraint could exist. This failure of the two probabilities to track one another just illustrates the basic idea underlying the following rigorous proofs. (Alan Hájek showed me this useful illustration and I believe he was the inventor.)

The rigorous proofs about failure of tracking fall into two classes: those that depend on claims about belief dynamics--rational response to evidence--and those that depend on belief statics--constraints about what people can believe at one time.

Dynamic Triviality Results

Lewis's original article assumed that learning occurred by conditionalization and that one's rational beliefs should be closed under conditionalization by any proposition. But one can think conditionalization is too strong because it implies that one becomes absolutely certain of the evidence, and that one's change in belief needs only to be responsive to propositions that characterize evidence, not all propositions. In "Probabilities of Conditionals and Conditional Probabilities II," Philosophical Review 95 (1986): 581-589, Lewis proves the triviality results without using these questioned assumptions. Lewis uses Jeffrey Conditionalization, which represents rational response to not-totally-certain evidence, and restricts the conditionalization to propositions that are members of a finite partition of evidence propositions.

Static Triviality Results

The first static triviality result appears in Carlstrom and Hill's Review of Adams' The Logic of Conditionals. They argue that two people can have the same degrees of belief P(A∧X) and P(A) for any X, which means they have to agree on P(X/A), and specifically P(C/A) for any C. But they can disagree about the probabilities of ¬A∧X. Because, for any proposition C and any proposition X that is not entailed by ¬A, they disagree about X, but agree about P(C/A), there can be no X such that P(X)=P(C/A). The proof's assumptions about the rationality of disagreement have been questioned by Simon Blackburn, "How Can We Tell Whether a Commitment has a Truth Condition?" and Jonathan Bennett.

But the static argument was extended by Alan Hájek in "Triviality on the Cheap?" where he only assumes that → is a binary connective linking the propositions A and C.

Failures of Finitude

Another more problematic discovery by Alan Hájek in "Probabilities of Conditionals--Revisted," Journal of Philosophical Logic, 18 (1989), 423-8, shows that Stalnaker's hypothesis fails by assuming only that a probability function can exist over a finite number of possible worlds and the theorem . That is, we assume that we can model human thinking with a finite number of possibilities. This is arguably not too severe a restriction because our belief states are not infinitely fine-grained, nor do we have an infinite number of different beliefs that cannot be clumped together as a single propositional attitude.

While you need to look at Alan's paper for the full proof, the basic idea is pretty simple. It is illustrated with an example involving three possible worlds, each of which is 1/3 likely. As shown, C is true in the first two worlds, and A is true in the last two. Because these are distinct worlds, the only possibilities for P(A→C) is 0, 1/3, 2/3, and 1. But P(C/A) is 1/2. So they are not equal. (Thanks to Frank Jackson for showing me this illustration.)

The theorem uses the principle, P(Q)=P(Q∧R)+P(Q∧¬R), which entails the law of excluded middle, and you may have independent reasons to deny excluded middle.

One extremely important point about this proof is that it doesn't involve the probabilities being the probabilities of the truth of the conditional; they could be just acceptability (assentability, assertibility, etc.) values. Thus, the triviality result is not merely that the concept of truth is inappropriate for conditionals.