In my previous post I outlined my conjecture that natural languages partially determine a system of logic, based on the observation that there is debate about some aspects of the correct logic, but not others. For example, no-one disputes the correct logic for the AND connective and there are not serious arguments against Modus Ponens.
There is stronger reason to suppose that natural languages do not fully determine a system of logic than simply that no-one seems to be able to agree what it is. If a natural language completely determines its relevant logic, then that natural language determines a formal system that is expressive enough for the arguments that Goedel used in his Incompleteness Theorems to work in that system. This means firstly that the natural language is incomplete (“There is some statement that is true that cannot be proven to be so in that language”) and it cannot be proven in that language that that language is consistent.
These conclusions are difficult to accept.
While it is plausible that natural languages are incomplete, it runs counter to ordinary use that the exact sentences identified in Goedel’s argument are true but not provably so. We can run Goedel’s argument in a natural language for that language, can understand what the sentence means or represents and that the sentence must be true. It is difficult to see how we can accept that the sentence must be true when Goedel’s argument shows that this cannot be proven in the natural language.
To be more precise, Goedel showed that if we can prove the relevant sentence, then the relevant system (in this case language) is inconsistent. So we seemingly have an argument that if natural languages fully determine a system of logic then, following Goedel, they must be inconsistent.
One thing we can never accept is that natural languages are inconsistent and that therefore everything is provable in them. If this is the case, then the whole of human knowledge is a lie as everything is in fact true.