Consistency of LSO with syntactic equality

Abstract

LSO, Logic of Sentential Operators, is defined by extending first- order logic by sentential quantification and sentential operators. Its semantics is defined by a digraph, with kernels reflecting consistent valuations of all sentences and consistency of the language. It is pos- sible, by using self-reference, to construct paradoxes and LSO need not be consistent with arbitrary valuations for its operators. While certain choices for operators allow existence of a kernel, not all do. The syntactic equality operator, only true for syntactically iden- tical sentences, raises the question, whether the language with it is consistent. Analyzing structure and complexity of sentences shows that certain subsets of sentences are consistent and that under some assumption the whole language is consistent. Even without a con- clusive answer to the question of consistency of LSO with syntactic equality, consistency appears to hold and results provide a solid base for a potential proof showing the whole claim.