Propositions and Rational Belief

Hannes Leitgeb and Stephan Hartmann

Personal notes from lectures of Week 3: Rational Belief

(These are personal summaries and paraphrasings of some of the major points of the lectures that I felt to be important. They are not meant to be comprehensive records nor intended to be reproductions of copyright materials. I encourage you to participate in the course for better understanding.)

Propositions and Possible Worlds

Propositions are true or false at possible worlds, with one of these possible worlds being the actual world. It is possible to show that worlds are identical if and only if the same propositions are true at them; and propositions are identical if and only if they are true at the same worlds.

Propositions can be defined as sets of possible worlds, such that every proposition is identical to the set of worlds at which it is true. Using set theory, we can visualize negation ('not'), conjunction ('and') and disjunction ('or') of propositions in the form of Venn diagrams. Also, the relationship of logical implication between propositions (proposition X implies proposition Y etc.) can be understood as a subset relation (X is a subset of Y).

Postulates of Rational Belief:

For an inferentially perfectly rational person 
1. If capital W is her set of entertainable possible worlds, then she believes capital W. (This is expressed by the logical form 'A or not A'.)
2. She does not believe the empty set. (This is expressed by a logical contradiction of the form 'A and not A')
3. If she believes X, and if X is a subset of Y, then she also believes Y.
4. If she believes X, and if she believes Y, then she also believes X-and-Y, the intersection of X with Y.

From these four postulates, a theorem can be derived stating:
There is always a least or smallest believed proposition ('B_W'), and the person in question believes a proposition if and only if B_W is a subset of that proposition, i.e. B_W logically implies that proposition.

Philosophical implications: 

* Rationality has an information-compressing effect (if we just know the B_W, we can determine whether a particular proposition is believed by the person, is disbelieved, or the judgment has been suspended.)

* Rational belief has a kind of mathematical structure.

All-or-None Belief vs Strength of Belief

While we can categorize beliefs in terms of believing, disbelieving, and suspending judgment on a proposition ('all-or-none'), we can also think of belief in terms of a scale in which we assign a numerical degree to the strength of belief in a position. This means that we can analyse a perfectly rational person’s degrees of belief over propositions in terms of rules of mathematical probability. The strength of belief ranges from probability value of 0 to 1.

However, there are problems in relating the all-or-none belief categorization with the degree of belief categorization. This is manifested in the Lottery Paradox. 

Lottery Paradox:

Considering a fair 1000 ticket lottery that has exactly one winning ticket. It is therefore rational to believe that one ticket will win. 

For a joint theory of rational belief and rational degrees of belief, we’d need to set a probability limit for strength of belief beyond which we can say that the proposition is ‘believed’. Let’s suppose the limit is 0.9. If the strength of belief is more than 0.9, the proposition is believed.

Based on this, we do not believe that ticket 1 of the lottery will not win (Probability of winning is 0.001). We also do not believe that ticket 2 will win. We also do not believe that ticket 3 will win. And so on for all the 1000 tickets individually. This entails that it is rational to believe that no ticket will win! This is of course contradictory with what we saw earlier, that it is rational to believe that one ticket will win.

How to resolve this paradox is still a matter of some dispute.