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Michael Dorfman stressed the following unavoidability in many answers:

Note that due to Agrippa's Trilemma, there are only three things logic could possibly be founded upon: unsupported axioms we take on faith, circular reasoning, or an infinite regress. Or, of course, a combination of the three.

I was trying to make use of the trilemma in the context of sound deduction systems for monadic second-order logic with standard semantics. Then I noticed that the presence of an (incomplete) axiom system (which I considered as given and hence didn't want to question in that context) might add further possibilities not covered by the trilemma.

Let's be clear from the start that transfinite induction needs an axiom system to work reliably, so any proof using transfinite induction might already be covered by the "unsupported axioms" part of the trilemma. My confusion/question is whether the "combination of the three" part is also correct. The "unsupported axioms" proof-method can be used both "top-down" and "bottom-up", but both "circular reasoning" and "infinite regress" look like they can only be used "top-down". Because "transfinite induction" looks like it can only be used "bottom-up", the truth of the "combination of the three" part seems to imply that any deduction system trying to use transfinite induction to overcome "incompleteness" will be forced to ultimately boil down to only "unsupported axioms", and hence will ultimately be covered by the normal "first-order" incompleteness theorems for second-order logic.


Appendix: Normal (mathematical) induction allows to capture certain cases where it is known how to systematically prove each instance in a series of successively more complex statements in a compact way. But (the simplest) transfinite induction allows to assume that all infinitely many statements that can in principle be proven true by a given (normal) induction scheme are already proven true, and hence can be used to prove a statement based on the simultaneous truth of all these "prior" statements.

  • What does monadic mean in the context of 2-logic? I'm aware only of monads in Category Theory, and at a fairly superficial level. Unless I've missed something - you haven't said what other possibilities there may be (apart from the ones in the trilemma) given your incomplete axiom system. – Mozibur Ullah May 30 '13 at 21:05
  • I think also one should distinguish the means and strategy of proof (your 'top-down' & 'bottom-up') from the directionality of logic used in all three cases of the tri-lemma. That is they are all one way. In foundations one can actually accomplish bottom-up, in the other two cases one starts in media res, and in circular moves 'up-around' & in infinite regress we move 'up'. In all three situations one can have the strategy of proof construction (I think) that allows movement in both directions by having contingent points of proof. – Mozibur Ullah May 30 '13 at 21:16
  • @MoziburUllah "Monadic second-order logic (MSOL) is a restriction of second-order logic in which only quantification over unary relations (i.e.: sets) are allowed." This is not really a serious restriction, because both set theory and natural numbers can encode tuples without problems, but it makes comparison between Henkin's and standand semantics more convenient. – Thomas Klimpel May 30 '13 at 21:44
  • Transfinite induction allows conclusions based on infinitely many assumptions, while the trilemma only considers conclusions based on a single assumption. Finitely many assumptions can be reduced to a single one by combining them with "and", but this is not possible for infinitely many assumptions. Now you could of course posit arbitrary means for proof by "unsupported axioms", but I don't see transfinite induction this way. I just want to use it to defer justification to the meta-set theory. But I want to do it in a way to avoid the "first-order"/"set theory in sheep's clothes" issue. – Thomas Klimpel May 30 '13 at 21:56
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Its a no & yes answer, as in -

No: Agrippas Trilemma seems to me a result in deductive formal thought. Formally there are three cases. And formally its possible that a combination may be used.

Normal induction operates over countable chains and transinfinite over chains of higher ordinality. Since you are using an axiomatic system then (as you state) the first part of Agrippas Trilemma applies: the use of an axiomatic system by faith.

Mathematics done formally always resorts to this particular move.

Yes: However in the practise of mathematics as it-is-done opposed to mathematics done formally I don't think Agrippas Lemma has much purchase. The reasoning (and inspiration) is of a different order. One interesting possibility is coherentism where various parts of a theory are shown to 'hang together'. (One could say this is inspired by circular reasoning - there is a metaphorical similarity - if that is the right word). I'd also throw in aesthetics - which explains why some mathematicians talk about beauty, elegance, profundity as well as simplicity & prettiness.

Perhaps your 'top-down' & 'bottom-up' is a strategy to perform this coherence?

But this probably true of all systems where reasoning applies. For example architecture & physics. Compare for example how one comes up with the design of a new architecture and how one is built.

  • Just checking: Did you indeed mean to write "ordinality", or perhaps "cardinality"? – user3164 May 30 '13 at 19:31
  • ordinality. if you don't have order how can you perform the induction? How do you propose to choose the next element? – Mozibur Ullah May 30 '13 at 19:34
  • "top-down" means first proving the final result from some other assertions, before trying to prove these assertions themselfes. "bottom-up" means to start from "foundations" providing proofs of additional true statements until the final result can be proved from these true statements. So this is not related to coherentism, but my goal concerning monadic second-order logic is. I want a sound deduction system that depends on the underlying meta-set theory in a coherent way relative to how the semantics depends the underlying meta-set theory. – Thomas Klimpel May 30 '13 at 20:17
  • @Klimpel: Sure. But presumably before you attempt a proof you have reasons why this is an interesting result and why it is plausible? If that is so, then that is what I was trying to suggest with 'coherentism'. The 'top-down' & 'bottom-up' appears to be a strategy to accomplish a proof in the context of a foundational point. But of course one may be aware of the instability of foundations - in the sense that foundations can perhaps be dug further down. In which case (depending on your beliefs about foundations) one may be in the situation of infinite regress. – Mozibur Ullah May 30 '13 at 21:00
  • @Klimpel: I suppose that it is a trite observation that the Trilemma is roughly analagous to the possibities of 1-dimensional manifolds with boundaries: circle, line and line with one boundary point. Its interesting that the line with two points is excluded. Presumably this reflects the (perhaps unjustified) that knowledge can always increase. – Mozibur Ullah May 30 '13 at 21:03

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