A Logic Without Valid Sentences, Inferences and Metainferences

PAILOS, FEDERICO MATÍAS

Munich

Workshop; Non-Classical Solutions to the Paradoxes; 2017

Munich Center for Mathematical Philosophy (MCMP), Ludwig-Maximilian University (LMU)

I will present a three-valued matrix logic, MMI. MMI, as TS, has no valid sentence or inference. But, unlike TS, has no valid metainference either. This is possible because MMI?s defines only a consequence relation for metainferences. MMI?s consequence relation for metainferences is very similar to TS? consequence relation for inferences. Basically, a metainference is valid in MMI iff for every valuation v, if v satisfies every premise of the metainference according to ST, v satisfies the conclusion according to TS. This determinines a consequence relation for inferences, which is just TS´. As a result, MMI is ?informative? with respect to the meta-metainference: there are valid meta-metainferences in MMI, but there are also invalid ones. Finally, I will present a sound but deeply incomplete proof system for MMI, called PMMI. Though PMMI may be easily expanded with operational rules, it is not obvious how to capture every valid meta-metainference that is not an instance of a structural rule, in a finite set of operational rules.