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).