We have been motivated to write this monograph by a wish to provide an introduction to an emerging area of Uncertain Reasoning, Pure Inductive Logic. Starting with John Maynard Keynes's ‘Treatise on Probability’ in 1921 there have been many books on, or touching on, Inductive Logic as we now understand it, but to our knowledge this one is the first to treat and develop that area as a discipline within Mathematical Logic, as opposed to within Philosophy. It is timely to do so now because of the subject's recent rapid development and because, by collecting together in one volume what we perceive to be the main results to date in the area, we would hope to encourage the subject's continued growth and good health.

This is primarily a text aimed at mathematical logicians, or philosophical logicians with a good grasp of Mathematics. However the subject itself gains its direction and motivation from considerations of rational reasoning which very much lie within the province of Philosophy, and it should also be relevant to Artificial Intelligence. For this reason we would hope that even at a somewhat more superficial and circumspect reading it will have something worthwhile to say to that wider community, and certainly the link must be maintained if the subject is not to degenerate into simply doing more mathematics just for the sake of it. Having said that however we will not be lingering particularly on the more philosophical aspects and considerations nor will we speculate about possible application within AI. Rather we will mostly be proving theorems and leaving the reader to interpret them in her or his lights.

This monograph is divided into three parts. In the first we have tried to give a rather gentle introduction and this should be reasonably accessible to quite a wide audience. In the second part, which deals with ‘classical’ Unary Pure Inductive Logic, the mathematics required is a little more demanding, with more being left for the reader to fill in, and this trend continues in the final, ‘post classical’, part on Polyadic Pure Inductive Logic.

The idea that it is rational to respect symmetry when assigning beliefs led us in the previous chapters to formulate the Principles of Constant and Predicate Exchangeability, Strong Negation and Atom Exchangeability. Since these have proved rather fruitful it is natural to ask if there are other symmetries we might similarly exploit, and in turn this begs the question as to what we actually mean by a ‘symmetry’. In this chapter we will suggest an answer to this question, and then consider some of its consequences.

First recall the context in which we are proposing our ‘rational principles of belief assignment’: Namely we imagine an agent inhabiting some world or structure M in TL who is required to assign probabilities w(θ) to the θ ∈ SL in an arguably rational way despite knowing nothing about which particular structure M from TL s/he is inhabiting. Given this framework it seems (to us at least) clear that the agent should act the same in this framework as s/he would in any isomorphic copy of it, on the grounds that with zero knowledge the agent should have no way of differentiating between his/her framework and this isomorphic copy.

To make sense of this idea we need an appropriate formulation of an ‘automorphism’ of the framework. Arguing that all the agent knows is L, TL and for each θ ∈ SL the conditions under which θ holds, equivalently the set of structures in TL in which θ is true, suggests that what we mean by an ‘automorphism’ is an automorphism σ of the two sorted structure BL with universe TL together with all the subsets of TL of the form

[θ] = {M ∈ TL|M ⊧ θ}

for θ ∈ SL, and the binary relation ∈ between elements of TL and the sets [θ].