conceptual

We must not forget that variables are a very recent development within modern mathematics (from late XVI century). We are so used to them that it is easy to forget how artificial they are. Natural language does not have variables. How does it manage? Most people respond with “pronouns”. But there are also reference shifting markers. Traditionally, reference shifting markers track whether and how the referents of two related clauses are coordinated. The most basic sort of coordination is plain coreference, but there are reference shifting markers for other sort of relations, like disjointness, simultaneity, etc. In any case, markers are linguistic resources we have not exploited in formal languages, but we could, i.e., we could use them instead of variables.  So, for example, instead of the formula  (P → Q ) → (R → P),  we could just have the sequence  →2 → →   where that 2 is a reference shifting marking that tells us how the arguments of the basic operato...

One of the main philosophical peeves of mine are philosophers confounding symbols and what they represent, and in particular mistaking features of logical notation with actual logical facts. For example, is d ouble negation  an actual logical rule or is it actually a notational convention of certain logical notations? It seems to be the later, since we can have expressively equivalent logical notations where double negation is not even expressible, for example, if we consider diagrammatic systems where negation is represented by a reversible transformation of diagrammatic elements (Monroy forthcoming). For similar reasons, one might consider the   c ommutativity  (of disjunction or conjunction) not as an actual logical rule but a notational convention of certain logical notations, since we can have expressively equivalent logical notations where commutativity is not even expressible , for example, if we consider expressions that are not sequences but sets of symbol...