# Is it there a "completely expressive" formal system / logic language?

I wonder whether it exists a formal system such that all (or a considerable number of) the others can be considered as a subsets or fragments of it.

I would say that, for instance, First-Order logic is a subset of Second-order logic (and so on if we keep with higher-order logics); on the other side, I would argue that, for instance, temporal logics also extend the logics in other directions, adding temporality operators.

So my question is: which is the most expressive logic that you have seen (even if it has no Computation uses, but just philosophical or even recreational) and the most expressive logic that you have seen that actually is used in any application in Computation.

This is obviously not a single answer question (as I use "expressivity" to say just, "what can be said"), just would like you to express what you think.

• There is no "most expressive logic" for the simple reason that already some first order logics are incompatible with each other, e.g. classical, intuitionistic and paraconsistent ones. Already second order logic is not effectively axiomatizable, so of limited computational value, and it gets worse in higher orders, but they get ever more "expressive", and there is no upper bound to that. Mar 17, 2021 at 19:40
• Hello, and welcome. I have to point out that asking for opinions is generally considered bad practice here. This one is a bit of a border case, though, since it basically asks for expert experience, which could be seen as on the "good subjective" side. Will leave it to the community to decide upon that :) Mar 17, 2021 at 19:47
• Hey, sorry then, I thought this question's answers can provide interesting knowledge: it can be subjective whether it is the "most expressive or not", but the point is reasoning about arithmetics and so on :) Mar 17, 2021 at 20:26
• Note that "complete" has a different technical meaning than the informal way you use it to mean "expressive". It's easy to find complete logics that aren't all that expressive. In fact completeness in this technical sense is in tension with how expressive a logic is. E.g. 2nd order order logic is incomplete in this technical sense. Mar 18, 2021 at 2:11
• @polcott Just stop confusing logical boolean values with pragmatic epistemological truth, would you? All you end up with is an idiosyncratic language. You cannot have it both ways: Either you allow for it to be about something outside of it (the world), then it is not a closed system because the fundamental truths are not defined to be true within, but from without the system. Or you do not, then it may be complete, but idiosyncratic. Mar 18, 2021 at 13:45