I am constantly encountering what I perceive as echoes of Peircean themes in places where acquaintance with or interest in Peirce’s work is slight at best, and that leaves me with a lot of pent up thoughts that I’ve learned through trial and error can’t always be ventilated in the places that stirred them up.
The recent discussion of fuzzy set theory on the Foundations Of Math List is a prime example of that. I see the questions arising there as falling within a larger question about the proper roles of demonstrative reasoning and non-demonstrative reasoning in the logic of inquiry.
That particular theme has recurred so frequently over the years that I’ve decided to give it a name, “Demonstrative And Otherwise” (DAO), and to start collecting the data of its cases in a more deliberate manner.
Here’s another case I view as falling under the rubric of DAO — it arose on a blog devoted to a fundamental problem in computational complexity.
And here’s a blog post where I began to gather a few reflections that bear on this computational aspect of DAO.
Inquiry Into Inquiry 
however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent.
But admit one dyadic predicate, and allow quantified variables to nest up to 3 deep, and you have enough expressive power for ZFC and many other axiomatic set theories (including NGB and NFU), and hence for all mathematics. This is so even without assuming that the dyadic predicate is any of reflexive, symmetric or transitive. The trick is to interpret this dyadic predicate as set membership, and to assume that everything in the domain is a “set”. The amount of axiomatic set theory required to derive Robinson arithmetic (which is Godel incomplete), is no more than extensionality, existence of null set, and adjunction. See Burgess (2005) and Wikipedia • General Set Theory.