By Paul Lorenzen

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What is going on here is that people are proposing axiomatic theories that use signs like x e M as formula symbols, where e is uninterpreted. Because we can propose as many axiomatic theories as we like, the only objection to this is that the significance and purpose of this activity remain completely obscure. On the other hand, if in the future someone were able to construct levels of sets for which the axioms are true, then an axiomatic foundation for set theory would be superfluous. What are we to make of the fact that axiomatic set theories are recognized in mathematics as the only acceptable theories, despite the obvious unintelligibility of such theories?

In introducing such rules, more and more predicates are encompassed by the rules. These further predicates are themselves more precisely specified by rules, and in this way systems of predicates are created. 2 for the predicates "play" and "work," then we cannot at the Page 11 same time call an action both play and work, even if the exemplary introduction of the predicates suggested such an equation. It is, however, always possible to change the systems of rules by introducing new examples. , polarity systems).

Logic, that is, the theory of logical operators, represents a further step that has not yet been taken. How can logical operators be introduced methodically into the lan- Page 13 guage that we have constructed so far? The starting point for this is found by returning again to the practical situations in which we speak. We must imagine two people, both of whom use the same conceptual system, perhaps the system above, and who engage in dialogue with each other. What does it mean when one of the two asserts, for example, that "all ravens are living beings"?

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