What are non-classical metatheories like? How do you use a metatheory that isn't classical?

It is straightforward to use a classical system to analyze simple non-classical ones. However, this is the only way I know to work with non-classical systems, by mapping them back to a system that's more expressive, but classical above a certain level of metaness.

Apologies for the long-winded example, it's meant to illustrate what I mean by the metatheory being "essentially classical" even in cases when the object of study is nonclassical.

A word on notation: I will use Łukasiewicz notation for connectives in the object language and English words or, and, implies for meta-level stuff.

When talking about first order logic I will use ordinary symbols (⇒, ∧, ¬) for the object language and English words or, and, implies = if ... then for meta-level stuff.

I'm interested in nonclassical logic, particularly propositional logics because you only have one type of non-logical symbol. Also, what they're trying to achieve is interesting: how do atomic, non-analyzable things-capable-of-bearing-truth-values interact?

I know of a handful of ways of talking about nonclassical logics, but they're all embedded in a classical substrate.

The most powerful for propositional logics, in my opinion, is just translating formulas inductively into a classical first order logic with a single sort.

Let K be conjunction, A be disjunction, C be implication, N be negation, and F be bottom.

We can turn well-formed formulas in our propositional calculus into closed terms in classical first order logic with a single one-place predicate t for designating the chosen truth values. Note that the quantifiers are all universal and all constrained to appear on the left edge of the wff. When we induct, we construct our matrix and a quantifier train to close our formula separately. I'm doing this so I don't need to talk about the truth values of open formulas in FOL.

x  (where a is a variable)    --->   ∀a.    t(x)
Kxy                           --->   ∀xy.   t(k(x, y))
Axy                           --->   ∀xy.   t(a(x, y))
Cxy                           --->   ∀xy.   t(c(x, y))
Nx                            --->   ∀x.    t(n(x)))
F                             --->   t(f)

In parcticular, KaNb maps to ∀ab.k(a,n(b)). KaNb does not map to ∀a.k(a,∀b.n(b)).

I can come up with logics with more interesting relationships between truth-values than some are chosen and some are not, such as defining an order on them <=. E.g. F <= U <= T would be true in most three-valued logics. I can even state interesting facts like K and A are covariant in both their arguments, C is contravariant in its first argument and covariant in its second argument. In this translation procedure, the kinds of things I want to say about truth values on the left hand side affect my choice of predicates on the right hand side.

I can represent things like inference rules by using classical connectives in my FOL on the right hand side.

  a  b
  ----       --->        ∀ab.(t(a) ∧ t(b)) ⇒ t(k(a,b))

I can also talk about semantics on the left hand side by talking about models on the right hand side.

How do you work with a foundation that's nonclassical though? If my meta-level reasoning includes things like uncertainty, vagueness, modality, or rejects classically valid inferences like double negation elimination, how do you actually use it in practice rather than just studying it as a mathematical object?

  • 2
    How do you use a metatheory that is classical? Classical logic, with its material conditional and the law of explosion, isn't natural logic. Nor is it more expressive than alternatives, just more familiar. We learn it by first "mapping" to the vernacular, and then developing new habits after enough practice. New languages are learned the same way, first everything is mentally translated into the native tongue, but eventually new patterns take hold directly. People evolved to train and master, if we can master one artifice why not another. – Conifold Dec 15 '20 at 5:31
  • The "usual" meta-theory is not formalized: it is standard mathematics. For example, there are meta-theoretical results that are constructive, like e.g. Godel Incompleteness Theorem, and others that are not. – Mauro ALLEGRANZA Dec 15 '20 at 9:20
  • If you instead formalize meta-theory, you have many choices: second order arithmetic, set theory, etc. In this case, the choice of the set theory is an option: a "classical" one, or an intuitionistic, constructive, etc. one. – Mauro ALLEGRANZA Dec 15 '20 at 9:21

"What are non-classical metatheories like? How do you use a metatheory that isn't classical?"

I think it rather depends on what kind of logic you are considering. For example, a dialethic metatheory would be quite odd, since without constraints it would allow that two explicitly incompatible logics are both correct. It seems to conflict with the idea that at some level there is one and only one reality.

"I know of a handful of ways of talking about nonclassical logics, but they're all embedded in a classical substrate."

This is indeed typical of accounts of non-classical logics. Beall and Restall, in their book Logical Pluralism, advocate for a particular kind of logical pluralism, however they appear to stick to a classical metatheory. Stephen Read, in his paper Monism: The One True Logic, criticises this approach and claims that Beall and Restall are in practice classical logicians for whom non-classical logics are just an exercise in applying logic to particular activities. Such an approach is consistent with logical monism together with Carnap's principle of tolerance. What Beall and Restall's response to that claim is, I'm not qualified to say. Read himself is of the view that it only makes sense to use the same logic in the metatheory as one uses in the object language. His own preference is for relevance logic.

"How do you work with a foundation that's nonclassical though?"

Graham Priest's advice in his "An Introduction to Non-Classical Logic" (2nd edition, 2008) Postscript: A Methodologial Coda (pp 584-586) is that if you think classical logic is incorrect, don't use it at the meta level. Instead, you might shape your meta-level procedures and semantics in a way that is consistent with the logic of the object language. Or possibly you could consider classical logic to be a special case of your chosen logic that applies in certain contexts, and that the metatheory is one of those applicable contexts. Either way, there is no simple solution. Priest considers it to be "one of the most important technico-philosophical issues to which non-classical logics give rise".

This probably doesn't answer your question as well as you would have liked, but hopefully it was of some use.

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