Logic was traditionally conceived as:
Many systems that deny 3 have been studied. Putnam argued that 1 is incorrect - our discovery of logic is at least partly empirical (a posteriori) and might need to be recast according to Quantum Mechanics (and the logic thereof).
I will attend to 2 - logics that self-modify, that change.
This class of logics is not be confused with well-known formal systems that pertain to belief revision (like AGM) or logic systems that model temporal relationships (like modal-based temporal logics). Instead, these logics change.
Classical Logic was and is most often still held to be:
Non-Eternalist Logic is not explicitly non-Classical (in that every change could still be Boolean - e.g. as a superset of Classical Logic) but it is a repudiation of these often implicit, dogmatic, assumptions about Classical Logic that permeate the Philosophy of Logic (to echo Graham Priest's criticism elsewhere).
Non-Eternalist Logics are logics that change. There are not to be confused with logics that model change (but which are themselves static and fixed) nor logics that are time- or context-sensitive (also fixed and static in terms of grammar, axioms, truth-conditions, etc.).
In what ways can a logic change? Some ideas:
Such that a specific wff may change in its truth-assignments (e.g. - T or F in Classical Logic).
This is not so radical:
When two logics comprise a dual pair they can be modeled as cross-logics and the specific operations between them in terms of computing their propositional truth-values can be modeled as transactions (e.g. - discrete operations - though these could be understood as financial operations - perhaps a natural evolution to information and credibility markets).
I believe that this is less a philosophical proposal to radically replace some system of thinking with another. Instead, it's an attempt to more accurately model what’s already being done, in human life, in our ordinary day-to-day lives.
We can replace at least some epistemic notions (particularly those that use logical operators or predicates) with the idea of an interface (and add some other notions) while reducing some of the grammatical baggage (trading an enlarged semantics for a smaller overall language footprint).
Here, the temptation is to treat two so-interfaced logics as one (continuous but semantically or at least proof-theoretically richer) logic - each self-consistent and contained, but sprawling and overlapping.
The potential scope of application includes epistemic modeling of belief change (like AGM) and logic revision (axiom revision like Euclidean to Hyperbolic geometry).
We can take an alternative approach here too - shedding the epistemic wrapper to review changes to the raw underlying logic itself.
See: logical module
Applied to two logics as an interface and allowing for:
Classical Logic is well-known to be limited in at least the following two ways:
i. Note for instance the distinction the imperative logic condition IF p DO x (used in programming) and the material conditional IF p THEN q (in Classical Logic).
ii. Control-flow programming is also trivially a mixed-logic scenario - in that it combines Boolean as well as imperative logics to perform operations.
One other potential advantage of this approach is to more-effectively model changes in the logic of one's thoughts rather than just the thoughts or beliefs themselves (which AGM, for example, captures).