TITLE: Using the multiverse approach to decide the law of the exluded middle?
QUESTION [9 upvotes]: Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ is true (and some in which $CH$ is not a relevant hypothesis?).
In formulating logical languages, there has been an outstanding problem: that of deciding the Law of the Excluded Middle $p\vee\neg p$. In classical logic $p\vee\neg p$ is true, but in intuitionistic logic this is not the case. With $CH$, the pragmatic mathematician tries to avoid invoking $CH$; if he assumes $CH$ or $\neg CH$, he will state it clearly. In everyday mathematics practice, the mathematician does use $p\vee\neg p$, but we do see an effort to give  effective constructions, and hard estimates in analysis.
It seems that people are using classical logic basically because it's a core logic, in analogy to the constructible universe $L$ in which $CH$ is true. It is the logic first discovered (and often wrongly attributed as the Platonic choice) and the consistency and strength of other languages proven in terms of this core logic. Afterwards people come up with other logics, like Brouwer coming up with intuitionistic logic, in which a fundamental principle, the Law of the Excluded Middle, does not hold. It seems to me that this debate regarding the Law of Excluded Middle can be formalized using the multiverse approach.
So has anyone tried to use the multiverse approach towards considering these plurality of languages? Perhaps by considering a multiverse of topoi?
PS. The difficulty for arithemetizable syntax to describe continuous properties of geometric/measure-theoretic space is central (may I say?) to the difficulty of deciding $CH$. In a like fashion, the difference between classical and intuitionistic logic plays up in comparing the Dedekind-reals and the Cauchy-reals. In other words, continuous properties have different descriptions in these two logics. I'm hoping that work on the Law of the Excluded Middle will shed some light on trying to use discrete languages to model continuous properties. There's more where this comes from, but it is enough for now...

REPLY [2 votes]: I offer a different perspective on the question, which I hope will be found
useful.
One view of logic is strongly syntactic:  develop a language, and develop
rules for manipulating this language and classifying parts of this language
(terms, well-formed formulae, sentences, derivation, proofs, theories, and so on).
One of course would like to arrange things so that from a syntactic perspective
one can easily define such subsets so as to say "for each x in this collection
of objects, I can interpret x in this environment so that the interpretation is
at the same time true, meaningful, and tells me something about the environment." .
Indeed, such a viewpoint helped in various areas of scientific and technological
development.  Compiler theory, software development, fuzzy logic are just three
areas of "real-world" applications.  In this "real-world", the focus is often on 
solving a relatively narrow class of problems with a small set of tools, 
as opposed to considering many alternative problems with a wide variety of tools.
I call this an "engineering" perspective.
In my view, what you propose is a comparison of several logical systems in the
context of some unifiable framework, loose enough to encompass all of the
possibilities that you want to consider, but tight enough that you can resolve
some of the potential questions you see.  I call this a "scientific" perspective.
Speaking from ignorance, the closest I see in attempting such things are in fields
like modal logic, algebraic logic, and certain forms of general proof theory. I don't know enough of modal logic to comment intelligently, but I see various 
logics being compared and contrasted in some papers on modal logic.  I haven't 
worked in algebraic logic, but relational algebras, cylindric algebras, 
Heyting algebras, and more occurred in my studies in universal algebra: these 
structures were meant to put a strong algebraic component in studying and modeling 
logical systems.  From my naive perspective, one of the intentions was to provide 
an algebraic way to "define, true, meaningful, and useful subsets of a language". 
I know (almost) less than nothing of proof theory, but I recall Kleene's 
"Introduction to Metamathematics" where he develops formalisms for both classical 
and intuitionistic logic in parallel, using notation similar to those found
in proof theory, and I dimly recall Lindstroem's result on first-order logic being the 
strongest logic system in a sense (something about having both compactness and 
downward L-S theorems). So attempts have been made which are (as Gerry Myerson 
might say) "not a million miles away" from your proposal.
Of course, people who have spent time studying such things should weigh in and
clarify the situation.  Andrej Bauer has given a nice philosophical perspective,
but I think he missed the aspects above.  In particular, I think the "engineering"
perspective can be brought more in line with the "scientific" perspective, in that
while a "full" multiverse of logics could be studied, emphasis should be placed on
the "favored" multiverse of logics that are currently or soon will be applied to give
us some of the technological benefits mathematical logic can give society.
Gerhard "Ask Me About System Design" Paseman, 2015.10.08