<K I 26/73a> All the axioms of arithmetic have the form: ‘‘If something corresponds to the concepts b1, b2, …, then it is valid that f(b1, b2, …) = 0.’’
The collection of the axioms of arithmetic can be symbolized in the following way:
In which to each must be added the condition that: If an object corresponds to bλ, if an object corresponds to bκ, etc. We chose the various indices because not every equation necessarily contains all and the same b.
Well now: is it possible to derive contradictions from two or more such hypothetical relations, each of which is quite clear from the nature of the respective concepts? It is manifestly impossible, if the concepts are not absurd. This is because, if two such equations are in contradiction, this simply means that the following truth obtains, directly or indirectly: if there is a bμ, then it is valid that F = 0 and at the same time it is valid that not-F = 0; from which directly follows: a B can not exist. Contrariwise, we see that if a B is impossible, then there can be evidently valid contrasting relations or ones that implicitly contain a contradiction, based on the concept B (sc. in hypothetical form, under the assumption that there is a B). Of course, it does not follow from all this that, for an impossible B, there can only be valid relations that would contradict each other directly or indirectly. For instance, the judgments ‘‘a round square is round’’ and ‘‘a round square is square’’ do not contradict each other at all. And furthermore it is clear that certain truths, which are evident and compatible for the sphere of objectivity (and eo ipso must be compatible, because they concern objective concepts and are evident), can become incompatible for the sphere of objectlessness. This can occur if concepts become objectless under certain conditions, more precisely, if concepts arise from certain connections of other concepts and become ‘‘impossible’’ for certain relations, whose possibility derives from the nature of the component concepts.
<K I 26/73b> Let us now take precisely this case as the starting point for our further observations and pose the question: May we use the domain of objectless- ness as bridge to obtain knowledge for the domain of objectivity? And when is this the case?
If, without regard for objectivity, I were to connect the propositions that were immediately evident for bλ, or if I would apply complications that involve concepts bλ, then I should expect contradictions as soon as I consider the objects as belonging to the domain of objectlessness, i.e., when thinking the concepts under those specific conditions that guarantee their objectlessness. Due to contradicting relations for the impossible, we would then also obtain contradicting relations for the possible as soon as I switch over from one domain to the other. Such transitions are thinkable in the following way: in the respective propositions are to be found, besides the possibly objectless concepts, concepts that have unconditional objectivity. By connecting various of these propositions we could formally derive a new proposition that contains only objective concepts. If I choose as a premise now one, now the other among two contradictory or formally contrasting propositions, then as consequences I obtain two different and mutually contrasting propositions for objective concepts.
When is such a case excluded? When does every inference that takes the basic principles as unrestricted (in the sense that it allows even contrasting concepts) and that leads to a proposition containing only objective concepts lead to a conclusion that cannot contradict any other formally valid proposition of the domain? We can answer that it is the case, either if the basic principles do not contrast at all, despite the objectlessness of the concepts, or if we take precautions to the effect that those objectless concepts that make the basic principles contradictory are (explicitly or implicitly) excluded. Indeed, a priori it is thinkable that not all objectless concepts that occur lead to a contradiction among the basic principles. In other words: we allow all objectless concepts for which the basic principles are demonstrably free from formal contradiction, and exclude all those for which this is not the case. Having done so, the expanded domain can serve as foundation for any and all inferences. All derivations are correct, no longer under the restrictive presupposition that they involve only objective concepts, but under the far less restrictive one that no objectless concept is allowed that would render the basic principles contradictory.