That is, are there certain conceptual primitives, such as object, action, structure, property, logic, event, quantity, partial, paradox, system, concept, etc, or connectives/judgements, such as for all, in which, or, false, there exists, such that, true, and, etc that rather than being merely tools to discuss concrete matters of logic and mathematics are themselves invariant and as such matters of study in their own right, and admissible to a concrete semantic model (or set thereof)?
If such a universal logical language exists, it would be subject to some very peculiar limitations which were developed by Alfred Tarski. His undefinability theorem puts some very interesting limitations on such a language. In particular he considered a language which:
- Is a formal language (its particularly hard to provide a concrete semantic model for non-formal languages)
- Was self referential (necessarily for arguing that a language is truly universal.)
- Contains a negation operator (we like to think negation exits in logic, so its a reasonable requirement)
- Is powerful enough to prove all the truths in arithmetic (otherwise discussing matters of mathematics could be tricky)
He demonstrated that any such language cannot define its own semantics. He argued that, to define its own semantics, such a language would need to define a
True(n) predicate which returns true if and only if
n was some form of a sentence in that language. His particular proof involved encoding a sentence using Godel numbers, and making
True(n) be a predicate that accepts a number as an argument. He then used the diagonalization lemma to demonstrate that there must exist a sentence which is true but
True(n) is false.
There are a few subtle limits. Dan Willard, for example, explored mathematical systems where multiplication was not a total function, which was just enough of a tweak to prevent the diagonalization lemma from being proven in the language. However, those systems are not as mainstream as the ones you and I learned in school.
Now, amusingly enough, this suggests that the semantics of English may not be fully sufficient for us to agree upon what "logic" and "mathematics" are, semantically, so I cannot claim this proves that such a universal language cannot exist. However, hopefully those limitations provided by Tarski will help you explore your own answer to the question.