Logical Syntax

Abstract

The syntax of formal languages in logic can be studied using the tools of mathematical linguistics, just like that of natural languages. Notably, the standard hierarchy of grammatical complexity makes sense for logical formalisms too. In particular, it is proved here that the language of predicate logic, though itself contextfree, has some natural dialects which are non-context-free. On the other hand, some reasonable variants are considered whose syntax is regular. From a logical point of view, it is natural to go on from here, and extend the analysis so as to include the complexity of semantic interpretation of syntactic structures. For this purpose, the usual automata hierarchy is used, with machines now being not mere recognizers, but transducers outputting semantic values. Amongst others, it is shown that any putative finite state evaluator for predicate logic must make two kinds of error: evaluating ill-formed input, but also mis-evaluating well-formed input. Finally, there is a number of results on related issues, such as the complexity of logical laws, or at a higher textual level of aggregation, of logical systems of proofs. For instance, it is shown that, on a fixed finite vocabulary, the purely propositional tautologies form a contextfree set, whereas the set of universal validities of modal propositional logic is more complex than that.

Software package requirements and procurement

“…The topic of ranking packages according to the degree to which they satisfy requirements is strongly related to the topic of verisimilitude in logic (Brink, 1989 andBenthem, 1987), where theories are ordered according to how close they are to a notional "truth". But there are several ways in which the package requirements and procurement problem differs from the standard problem of verisimilitude.…”