Foundation system may be considered as having two aspects, their logical content, and their ontological content. In general these are not cleanly separated, this being the case only for `free' logics (REFERENCES). In the case of first order this is almost a clean separation, the only ontological content in the logic being the presumption that at least one thing exists.
Concerning the former aspect, the purely logical, there is (almost by definition no dispute that these aspect are properly logical, and that theorems derived by these means alone (where this is possible) are logical truths. Concerning the ontological aspects there is dispute, and this dispute provides the grounds on which logicism is often held to be refuted. In the case of Principia Mathematica, the flagship of logicism, objections are levelled at the axiom of infinity and the axiom of choice. The axiom of infinity is sometimes supposed to have synthetic content and hence to be essentially empirical in nature (even Russell appears to have taken this view). The axiom of choice is looked upon as suspicious in more mysterious ways, and is rejected as a logical principle by some opponents of logicism.
In the following material I wish to let it be known that no statements are intended to speak in any way about the real world, our intended subject matter is entirely abstract, and the axiom of infinity, if it is in need of justification must be supported by other than empirical means. In fact our domain of discourse is the natural numbers, whose existence I am content is no less controversial than that of the terms of the formal language which we present. Ultimately, the existence of natural numbers is something which I would prefer to regard as true a priori, but on a matter so basic as this I do not expect that I will be able to offer convincing arguments to those who find this proposition doubtful. I hope it will be the case that the material presented would be equally useful if accepted on the basis of postulating the existence of natural numbers. We also consider it to be reasonable in the semantic framework proposed to accept that via a suitable encoding of rules for selecting natural numbers we may use natural numbers to represent collections of natural numbers which may be infinite (the rule "take a natural number and accept it without caveat or condition" determines the set of all natural numbers).
This paper promotes several distinct causes. At its most specific it presents a class of logical systems which may be suitable as foundations for mathematics, particularly for us in computer assisted mathematical activities. These systems cannot be adequately presented without motivation, justification and some background. An important element of such explanations is my position in general terms on the philosophical aspects of the relationship between mathematics and logic. The best description of my position is "Logicism", and this paper therefore contains some arguments in support of a logicist position. Of course there is really only one logicist position, that mathematics is reducible to logic, but a defence normally involves putting forward a formal system and arguing that this system is at once a logic and an adequate foundation for the development of Mathematics without need of further non logical postulates.
It is appropriate to undertake some discussion of logicism partly because one of the main technical objectives of the proposed formal systems is to re-establish that freedom of abstraction present in Frege's foundational system in a context in which this is not spoiled by inconsistency.