Is there a difference between trivial and nontrivial negation? It occurred to me that we could think of the following series of negation operations/relations:
- Empty negation = primordial double-negation elimination (DNE). Modulo (2), this can be construed as empty alterity.
- Alterity = a kind of trivial negation = the "other than" relation, "A is other than B."
- Inequivalence of absence and opposition/contrariety: ¬(¬A = A¬) (the occurrence of the negation sign on the right of A, there, mirrors the distinction between a modal operation Mx and another xM (as when we prefix and/or suffix some "M" by "¬"). Might not be as trivial as (2).
- The greater than/less than relation: more nontrivial, since incommensurable terms are logically possible. (See also e.g. the SEP article on continuity and infinitesimals for depictions of infinitesimals as neither larger nor smaller than zero, but not equal to zero, either.)
- Whichever demi-negations we admit into our system.
- Based on the conditions of (5) or not, all the inverse arithmetical operators of nth-order (or more) arithmetic, e.g. subtraction is traced back to complementation, and then both stand under division, which is below logarithms and roots, which are below the counterparts for tetration, and so on and on.
One of the SEP articles on contradictions or negation or dialethism (I don't remember which) mentions Graham Priest's objection that analyzing ordinary-language "not" as an incompatibility/incompossibility marker is to prejudge the linguistic situation in favor of the LNC, which in this dialectical context is in dispute. Now that might be a fair complaint on its own but it also seems as if we simply do have an incompatibility/incompossibility operation anyway, i.e. there are times when, "A and not A," must be false.
With that in mind, this is my question:
- If we iterate the alterity operation, as in, "A is other than other-than-B," does DNE then hold trivially? It seems perhaps not, since:
- If we differentiate existential from universal alternation, we seem to allow for a potentially false localization of DNE and a necessarily true one:
- Letting "❧" mean "other than," take A = ❧❧∃A. This would be read as, "There exists an A that is other than other-than-A, which equals A." But this seems unnecessary (see below).
- By contrast, A = ❧❧∀A seems (trivially) true, since if the given other-than-A's are all the others-than-A, we are just left with A again.
Regarding (3), imagine starting with some x for the initial element in the set of finite ordinals and writing x' to signify something different from x, and substitute this signification process for the typical construction of Zermelo or von Neumann ordinals by then writing {x, x'}' to mark out the next number, and then {{x, x'}'}', and so on. But so again, if we are simply referring to different x's, it doesn't then seem as though "other-than-other-than x" automatically refers back to just x, but could refer to something other than x and other than our initial example of something else besides x.
Is DNE trivially true for universal alternation and nontrivially possibly false for existential alternation?
Revision: vertex-free hypergraphs as logical operations considered by themselves
Assume that nodes in a logical graph represent propositions, with edges representing inference relations. But it is reportedly possible to imagine two free-floating edges that connect to each other without constituting a first-order node (although, on a second-order level, any system of edges, vertex-free or not, can be collapsed into a node). Let such a hypergraph then be a representation of a logical operation "by itself."
Interpretation: then this is the intuition that intuitionists have about DNE and proof-by-contradiction: if we use the cancellative type of "not," then if we write ¬¬A we should end up with A, yet why are we compelled to write down A? So then there is an even emptier alterity, if you will, than reported above, but there is the vertex-free hypergraph of two ¬-operations, the one acting on the other, which leaves behind no nodes A. So there is a dimension of pure negation which does not support DNE as useful for establishing affirmative propositions "after the fact."
