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.

In Part 1 we placed no conditions on the arity of the relation symbols in L, the restriction to unary only happened in Part 2. In this third part we shall again allow into our language binary, ternary etc. relation symbols. As we have seen, despite the logical simplicity of unary languages, for example every formula becomes equivalent to a boolean combination of Π1 and Σ1 formulae, Unary PIL has still a rather rich theory. For this reason it is hardly surprising that with very few exceptions (for example Gaifman [30], Gaifman & Snir [32], Scott & Krauss [132], Krauss [69], Hilpinen [46] and Hoover [52]) ‘Inductive Logic’ meant ‘Unary Inductive Logic’ up to the end of the 20th century.

Of course there was an awareness of this further challenge, Carnap [12, p123 -4] and Kemeny [61], [64] both made this point. There were at least two other reasons why the move to the polyadic was so delayed. The first is that simple, everyday examples of induction with non-unary relations are rather scarce. However they do exist and we do seem to have some intuitions about them. For example suppose that you are planting an orchard and you read that apples of variety A are good pollinators and apples of variety B are readily pollinated. Then you might expect that if you plant an A apple next to a B apple you will be rewarded with an abundant harvest, at least from the latter tree. In this case one might conclude that you had applied some sort of polyadic induction to reach this conclusion, and that may be it has a logical structure worthy of further investigation.

Having said that it is still far from clear what probability functions should be proposed here (and possibly this is a third reason for the delay).

Having derived some of the basic properties of probability functions we will now take a short diversion to give what we consider to be the most compelling argument in this context, namely the Dutch Book argument originating with Ramsey [122] and de Finetti [25], in favour of an agent's ‘degrees of belief’ satisfying (P1–3), and hence being identified with a probability function, albeit subjective probability since it is ostensibly the property of the agent in question. Of course this could really be said to be an aside to the purely mathematical study of PIL and hence dispensable. The advantage of considering this argument however is that by linking belief and subjective probability it better enables us to appreciate and translate into mathematical formalism the many rational principles we shall later encounter.

The idea of the Dutch Book argument is that it identifies ‘belief’ with willingness to bet. So suppose, as in the context of PIL explained above, we have an agent inhabiting some unknown structure M ∈ T L (which one imagines will eventually be revealed to decide the wager) and that θ ∈ SL, 0 ≤ p ≤ 1 and for a stake s > 0 the agent is offered a choice of one of two wagers:

(Bet1p) Win s(1 − p) if M ⊧ θ, lose sp if M ⊭ θ.

(Bet2p) Win sp if M ⊭ θ, lose s (1 − p) if M ⊧ θ.

If the agent would not be happy to accept Bet1p we assume that it is because the agent thinks that the bet is to his/her disadvantage and hence to the advantage of the bookmaker. But in that case Bet2p allows the agent to swap roles with the bookmaker so s/he should now see that bet as being to his/her advantage, and hence acceptable. In summary then, we may suppose that for any 0 ≤ p ≤ 1 at least one of Bet1p and Bet2p is acceptable to the agent. In particular we may assume that Bet10 and Bet21 are acceptable since in both cases the agent has nothing to lose and everything to gain.