Truth without satisfaction

In his famous paper [7], Tarski gave a definition of truth for a formalized language. Unable to perform a direct recursion on the concept itself, he gave a definition in terms of satisfaction. This makes it natural to ask if such an indirect procedure is necessary or whether a definition of truth can be given without using or somehow invoking the concept of satisfaction. The question, as it stands, is vague; and later we shall be concerned to make it more precise. But even as it stands, it has an obvious technical interest. The situation that Tarski found himself in is common in mathematics. We wish to define a certain concept, but unable to perform a direct recursion on the concept itself we perform a recursion on a related concept of which the given concept is a special case. It would therefore be desirable to know when the related concept is necessary, both in the case of truth and in general. The question may also have some philosophical interest. There is a fundamental difference between the concepts of truth and of satisfaction. The former merely applies to certain linguistic units; the latter connects language to an ontology of objects, typically extra-linguistic. A negative result on defining truth without satisfaction may perhaps constrain formal attempts to implement non-referential conceptions of truth. In the present paper, however, we will not be concerned in detail with the philosophical aspects of our question, although we will from time to time mention some points of contact between our discussion and the philosopltical literature. Interest in our question dates back to Wallace [9]; and the topic was subsequently taken up by Tharp [8] and Kripke [3] (especially Section 10). We have made our presentation self-contained, though the reader may consult the earlier work for general background and for elucidation of particular points. The plan of our paper is as follows. Section 1 sets out the general framework in which our question and its cognates are posed. Section 2 solves the questions in case the meta-theory is not required to be finitely axiomatized;