Neither constants nor variables

This Friday, after a long chat with Christian Romero (and some participation of Andrés Villamil), I think I got convinced that there constant-variable distinction leaves room for a third semantic option. The context is the discussion on the nature of propositional terms, but the morals apply to any referential domain. In classic bivalent propositional logic, it is customary to distinguish between propositional constants that have a fixed truth value and propositional variables that can take either. This is just the application of a general distinction between constants that have a fixed value and variables whose reference can range over the whole domain of possible values. It just happens that because the domain in classical logic has only two possible values, there is no room for a third option. However, any even slighter larger domain immediately would suggest at least a third option: referential terms that have neither a fixed semantic value, nor can take any semantic value in the domain of interpretation, i.e., referential expressions that can take more than one, but not any semantic value. Thus, for example, in a four-valued logic like L4 aka LE, we can have four propositional constants corresponding to the four possible semantic value assignments. Also, we usually have an infinite set of propositional variables that can take any of the four value assignments; but now it is not that difficult to see that we can have also other sorts of restricted variables that can take some but not all of the four value assignments. So we may have a different set of variables that range only on the classical values, another different set of variables that range only on the values that contain truth, another for those values that contain falsity, another for the non-classical values, etc. A little bit of combinatory can tell us that we can easily enrich our language with eight new sorts of restricted variables: four that can take two different values and other four that can take three different values. Logics with even more possible semantic values will result in even more possible types of restricted variables.

San Marcos Chapell, this fall, just a couple of days after the end of DIAGRAM2022

So far, so good; now: so what? According to Romero, enriching thus the language gives it more expressive power, so that there are truth-functions in LE that are expressible in the new extended-language that could not be expressed in the traditional one (i.e, the one with only one propositional constant T and only one kind of (unrestricted) variables). But of course, expressibility claims must be evaluated always in relation to the availability of other expressive resources.  Yes, adding a new type of restricted variables to this language increases its expressibility, but the same expressive power could be achieved also by adding other propositional constants!

Now, as is well known, there is a deep connection between modality and semantic constancy which can be easily expressed in a relation between moral operators and propositional constants. Thus, for example, in any system that contains a proposition constant T that only allows for a true interpretation and a modal constant ◻︎ that is true when applied to propositional terms that are true in every interpretation, it is a truth of normal modal logic that

⊧ ◻︎T

In general, for every strong modal operator S there is a corresponding constant C such that S(C) is logically true (in any system that contains both S and C, obviously). This must be obvious enough:  in any system that contains a proposition constant C that only allows for a single semantic value X as interpretation and a modal operator S that is true when applied to terms that have that same semantic value X in every interpretation, S(C) is going to be true in any interpretation. Now, thanks to restricted variables, this can be generalized to other sorts of universal modal operators like the determinacy operator △. For any universal (but not strong) modal operator M, there is a syntactic type of restricted variables A={A, A2, A3, …}, as characterize above, such that for all A in A, 

⊧ M(A) 

For example, given that the determinacy operator △ is true only and whenever it is applied to a propositional expression whose interpretation is always one of the two classical truth values, then any system that includes in its language a set of restricted propositional variables A={A, A2, A3, …} that cannot take non-classic values as interpretation, for any A in A, it will be logically true that

⊧ △A 

In general, in any system that contains a proposition variable type A={A, A2, A3, …} of variables that range only over semantic values that satisfy some given condition R and a modal operator M that is true only and whenever it is applied to terms that have some semantic value that satisfies the same condition R in every interpretation, M(A) is going to be true in any interpretation and thus logically true.

This allows for a language with modal operators to be able to mimic, so to speak, the syntactic distinction between variables. In general, given a language LA that contains a proposition variable type A={A, A2, A3, …} of variables that range only over semantic values that satisfy some given condition R and a language LM that contains a modal operator M that is true only and whenever it is applied to terms that have some semantic value that satisfies the same condition R in every interpretation,

Γ ⊧ Σ in LA only if Γ[A/P]  ∪ MAΓ  ⊧ Σ[A/P] in LM

where, given a set of formulas Ε in LA, Ε[A/P] is the set of formulas in LM resulting from uniformly substituting every variable in A in Ε for a new non-restricted propositional variable not occurring in Ε, and MAΕ is the set of formulas of the form M(Ai) for every restricted variable Ai of type A occurring in E.