In the course of the previous 42 chapters we have introduced numerous more or less rational principles which our agent, dwelling in an unknown structure M for L, might choose to adopt in order to address the question

Q: In the situation of zero knowledge, logically, or rationally, what belief should I give to a sentence θ ∈ SL being true in M?

We have argued from the start, via the Dutch Book argument, that it is rational to identify belief with probability, in the sense that it should satisfy conditions (P1–3), At this point the facet of ‘rational’, or at least ‘irrational’, being used is that it is irrational to agree to bets which guarantee one a certain loss. In general however we have offered no definition of ‘logical’ or ‘rational’. Instead we have embraced certain overarching meta-principles, or slogans, which we may feel are ‘rational’, just in the way that we may feel that something is funny without being able to define what we mean by ‘funny’.

We have particularly focused on four such slogans: That it is rational to:

1. (i) Obey symmetries: If, in context, θ and θ′ are linked by a symmetry then they should be assigned equal probability.

2. (ii) Ignore irrelevant information: If θ′ is irrelevant to θ then conditioning θ on θ′ should not change the probability assigned to θ.

3. (iii) Enhance your probabilities on receipt of (positively) relevant information: If θ′ is supportive of θ then conditioning θ on θ′ should increase, or at least not decrease, the probability assigned to θ.

4. (iv) Respect analogies: The more θ′ is like θ the more conditioning on θ′ should enhance the probability assigned to θ.

Of course these are just templates for principles.