The ConLog project investigates logical systems in which contradictions are provable. These are systems in which for some statements not only the statement but also its negation is provable. Such logics are called "negation inconsistent" or "contradictory", and they are required to be non-trivial, i.e. they must not be absolutely inconsistent in the sense of validating arbitrary statements.
The project radically breaks with a time-honored Aristotelian tradition and assumes that it is theoretically rational to work with logics that, while being non-trivial, admit provable contradictions. According to classical logic, contradictions have a trivializing effect on logical inference. With classical logic, any statement whatsoever logically follows from contradictory assumptions: ex contradictione quodlibet. The 20th century already has seen a change of perspective on contradictions, first among a minority of philosophical logicians and later also among computer scientists dealing with the processing of inconsistent information. As of the 1940s, inconsistency-tolerant, “paraconsistent” logics have been developed in which contradictory premises do not trivialize reasoning. Still, these logics do not allow for provable contradictions. More recently it has been observed that there indeed exist non-trivial contradictory logics that are not at all artifacts and have motivations completely independent of exploring the very possibility of non-trivial negation inconsistent logics.
The overall objectives of the project are (i) to investigate the formal properties of various non-trivial contradictory logics and to thereby obtain insights into their negation inconsistency, (ii) to see whether or not the inferential patterns and axiomatic principles that lead to provable contradictions are endorsed by speakers of natural languages, and (iii) to inquire into the consequences working with non-trivial negation inconsistent logics has or may have within the philosophy of logic. To achieve its goals, the ConLog researchers make use of advanced proof-theoretic and model-theoretic methods and techniques, methods from experimental philosophy of language and linguistics, and detailed conceptual analyses.
If the project achieves its aims, our attitude towards contradictions in scientific theories and theory formation will have to be revised or at least heavily nuanced. We have to distinguish the provable contradictions we have to accept from those that indeed hamper our theorizing.
