Adam I. Gerard

Addressing Metalogical Skepticism

I wanted to address something I worked out a while back and think should be commented on.


Generally, skepticism (of the philosophical variety - e.g. Pyrrhonian Skepticism) comes in narrow or global versions. Global versions (like Descartes' Evil Demon Argument) apply against knowledge generally (all of it). Local versions attack specific domains of knowledge (like a specific field in science, subject, or a subset of human inquiry).

Skeptical arguments typically take aim at some underlying, but necessary, presupposition and purport to demonstrate that such presumptions are false or that conclusions derived from them are invalid knocking out some piece or region of knowledge.

For example, if you can’t figure out whether you live in the Matrix, or aren’t being deceived by any Evil Demon, or what have you with absolute certainty, contends the philosophical skeptic, then you don’t really know something. That delicious apple you just ate might just be a fancy computer illusion and so you can’t, with absolute certainty, conclude that you did, in fact, eat an apple, enjoy it, etc. (since it, the apple, might not even exist).

Metalogical Skepticism

So, skepticism about logic (itself) is somewhat rare. The idea being that if you refuse to accept logic (at all), it’s difficult to then convincingly argue that one shouldn’t use logic (since, in doing so, one often uses logical argument). Either that or they repudiate logical inference entirely (and so are disposed to wanton irrationality and don’t then care about logical argument whatsoever and aren’t in the business of trying to convince you about logic or justification at all).

More narrowly construed attacks allow for some semblance of reasoning or attempt to draw out weaknesses in the scaffolding or architecture of logic (it’s systems, assumptions, dependencies, and so on - e.g. the nature of proof and judgment, theories of rationality and evidence, the nature of symbols and meaning, Truth, and so on).

I think the best-articulated one, and perhaps the most concerning one, in our present times involves challenges to logical justification by way of the metalanguage and object language distinction (near-universally employed through logic, mathematical logic, and mathematics):

  1. We justify a logic L (prove and characterize its properties: deduction theorem, soundness, completeness, consistent) within a metalogic L+.
  2. But, what then justifies L+? Supposedly, another metalogic L++ that has L+ as its object language.
  3. But, what then justifies L++? Supposedly, another metalogic L+++ that has L++ as its object language.
  4. And so on, ad infinitum.
  5. But, justification cannot transmit across infinite sequences.
  6. Therefore, all logics lack justification (are unjustified)

How does one address this intuitively persuasive argument?

Justifying Logics (A Sketch)

Suppose every logic was, in fact, isomorphic to each other. And, for the sake of simplicity, that each logic was classical. If we were able to prove that any logic were sound, complete, and consistent then we’d have proven that for all of them. And, we’d have proven that as such for all of them regardless of their relationship to each other.

Now, a metalogic within which we can prove properties about some primitive (zero order) logic like classical logic must have additional linguistic expressiveness - must be equipped with additional linguistic machinery (to be able to talk about soundness, truth, completeness, and consistency) since these don’t exist in zero order logic (by itself). So, the simple example I give above simply can’t defuse the metalogical skeptical argument given previously (since it depicts a scenario where every logic is strictly isomorphic to the others). We also know that adding extra “stuff” to a formal system can transform a previously sound and complete system into one that’s not (take the Liar Paradox, Hume’s Law, and other notorious Semantic Paradoxes).

So, we quickly arrive at two takeaways:

  1. If we’re going to address metalogical skepticism, we may need to break our argument into multiple stages: address primitive, base logics (classical sentential logic / propositional calculus / Boolean algebra) showing how each can be justified. We could use a superset of L, call it L+, to justify L. If L+ is semantically closed, L+ can justify itself (given enough linguistic machinery - reflection principles, idempotence, predicativity all come into play).
  2. The semantic paradoxes (including all of the alethic paradoxes: Curry’s paradox, Liar cycles, Liar sentence, Yablo sequences, etc.) block metalogical justification since their presence in any formal system L+ implodes it.

Can we arrive at a formal system capable of expressing proof-theoretic concepts, consistency (and hence, consistent Truth), and so on all within a language that’s semantically closed (or at least one that’s not necessarily not semantically closed to use Tarski’s vernacular)?

Yes, my previous proposal regarding Truth, Truth Grounding, and the Liar Paradox. It’s consistent and has sufficient machinery to capture consistent conceptions of Truth.

Proof theoretic notions might be made more consistent (using the same procedure applied to Gödel's Provability predicate - e.g. restricting either the predicate or unrestricted diagonalization both of which play an essential role in the famed Second Incompleteness Theorem) and so might be added to it without much fanfare. If so, then classical zero order logic could then be justified in FOLT++ (First Order Logic with Truth Grounding - FOLT+ - and the rest of the proof-theoretic machinery, +) and FOLT++ can demonstrate its own justifiability.