In the last chapter we presented several systems which are essentially equivalent to first order quantification theory with identity. In this chapter we will discuss a natural generalization of those theories which is slightly stronger than standard quantification theory. The viewpoint developed in the last chapter is that logic is the study of operations on sets of sequences and the ways in which those semantic operations can be represented in languages. These operations take the interpretations assigned by a model to the predicate letters and assign sets of sequences to the complex formulas. If one begins from the type of language found in quantification theory where atomic formulas are written in the form F n x 1...x n then it is natural to interpret predicate letters by assigning sets of n-tuples. However, if we take a fresh look at the language which was developed at the end of the last chapter, it is clear that there is no reason to make this restriction. The only remaining trace of the fact that each quantificational predicate letter has a specified number of arguments is in the superscript on predicate letters. Thus in the system to be presented now we will drop the superscripts and also the assumption that predicate letters are assigned sets of n-tuples for some fixed n. Instead, predicate letters will be assigned sets of finite sequences.
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- The system of anadic logic first appeared in Grandy, ‘Anadic Logic and English’, Synthese (1976).Google Scholar