# Are "A ∧ A" and "A ∨ A" degenerate expressions?

Although some time ago I had become somewhat familiarized with the notion of degeneracy in mathematics and physics, in my musings on the trivial/nontrivial distinction I found that both Wikipedia and Wolfram thematically correlate triviality and degeneracy. Though they don't seem to be absolutely identical states, they are (oddly?) similar enough in some sense to merit further musings in this vein.

So far, "Degenerate Epistemology" is the only philosophy paper I've seen in which the concept of mathematical degeneracy has been adapted to a philosophical subject. Granting that logical degeneracy is not very far from the mathematical kind, my question for now is just this: whether "A ∧ A" and/or "A ∨ A" are logically degenerate? There is something "squished triangle" about them, so to speak; or so it seems to me. (Or, perhaps better, are these expressions akin to digons?)

• Based on Wolfram's notion of degenerate Cases- I think it makes sense to think of a degenerate well formed formula of the LST to be a formula that is logically equivalent to one if its subformula. Under that lens, if φ = ψ ∧ ψ, then ψ is a subformula of φ and it is logically equivalent to φ, and so φ is "degenerate" in that it degenerates ( is equivalent) to one of its own subformula. Similarly, for the disjunctive case. Nov 15, 2023 at 0:42
• Those expressions may not be equivalent to A in some substructural logics. So they are not entirely trivial. Nov 15, 2023 at 6:32
• @Bumble you implicitly bring up a consideration that I ought to have appreciated more beforehand: triviality may well be relative (if also sometimes absolute! at least, the phrase "absolutely trivial" does not seem entirely pointless?). Or, then, this is yet another distinction to track, in triviality theory... For the sake of this question, though, I would be willing to accept Michael Carey's comment as an answer... Nov 15, 2023 at 15:11
• They discussed trivial vs degenerate on Math SE. Apparently, "trivial" is more along the lines of superlative like "the simplest", and "degenerate" along the lines of comparative like "simpler than the rest" to a point of 'collapsing' some features. Your formulas are degenerate in the sense of 'collapsing' a binary operation into an unary one by substituting the same item into both positions, for example. Nov 15, 2023 at 20:10
• Trivial seems to be more background relative and subjective, while degenerate is more intrinsic, degenerating some structure. So all four combinations are possible: trivial and degenerate (0 matrix), non-degenerate but trivial (identity matrix), degenerate but non-trivial (nilpotent Jordan cell), non-degenerate and non-trivial (rotation matrix). But you have to name the background and the structure to specify them. For example, A → A is trivially true (given the definitions, say) and a degeneration of a binary propositional schema. Nov 16, 2023 at 1:49