In the initial investigations of Johnson and Carnap the language was taken to be unary. That is, in the notation we are adopting here the relation symbols R1, R2, …, Rq of the language L all had arity 1, and so would more commonly be referred to as ‘predicate’ symbols. Carnap, [12, p123–4], and later Kemeny [64, p733–4], did comment that generalizing to polyadic relations was a future goal but with a few isolated exceptions there was almost no movement in that direction for the next 60–70 years.

There seem to be at least three reasons for this. Firstly the subject, and particularly the notation, becomes much more involved when we introduce non-unary relation symbols, a point which Kemeny acknowledged in [64] (appearing in 1963 though actually written in 1954). Secondly one's intuitions about what comprises rationality appear less well developed with regard to binary, ternary, etc. relations, possibly because in the real world they are not so often encountered, so the need to capture those fainter whispers which do exist seems less immediately pressing. Thirdly, for those working in the area there were more than enough issues to be resolved within the purely unary Pure Inductive Logic.

For all of these reasons it seems quite natural to concentrate first on the unary case. In addition many of the notions that this will lead us to study will reappear later in the polyadic case where this prior familiarity, in a relatively simple context, will provide an easy introduction and stepping stone to what will follow.

Our method in this, and the next chapter, will be to introduce various principles that one might argue our agent should observe on the grounds of their perceived rationality, though we will not set very high demands on what this amounts to. It will be enough that one is willing to entertain the idea that these principles are somehow rational. As already indicated these principles will largely arise from considerations of symmetry, relevance and irrelevance.

Before a cricket match can begin the tradition is that the umpire tosses a coin and one of the captains calls, heads or tails, whilst the coin is in the air. If the captain gets it right s/he chooses which side opens the batting. There never seems to be an issue as to which captain actually makes this call (otherwise we would have to toss a coin and make a call to decide who makes the call, and in turn toss a coin and make a call to decide who makes that call and so on) since it seems clear that this procedure is fair. In other words both captains are giving equal probability to the coin landing heads as to it landing tails no matter which of them calls it. The obvious explanation for this is that both captains are, subconsciously perhaps, appealing to the symmetry of the situation.

At the same time they are, it seems, also tacitly making the assumption that all the other information they possess about the situation, for example the weather, the gender of the referee, even past successes at coin calling, is irrelevant, at least if it doesn't involve some specific knowledge about this particular coin or the umpires's ability to influence the outcome. Of course if we knew that on the last 8 occasions on which this particular umpire had tossed up this same coin the result had been heads we might well consider that that was relevant.

Forming beliefs, or subjective probabilities, in this way by considering symmetry, irrelevance, relevance, can be thought of as logical or rational inference. This is something different from statistical inference. The perceived fairness of the coin toss is clearly not based on the captains' knowledge of a long run of past tosses by the umpire which have favoured heads close to half the time. Indeed it is conceivable that this long run frequency might not give an average of close to half heads, maybe this coin is, contrary to appearances, biased.

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.