Layers vs Levels

Written by  on September 6, 2017 

In a discussion of this paper, I argued that the virtus dormitiva is not “viciously” circular because it restates the proposition in different language. Of course the different language might be trivial or it might be significant. So, one could argue that if the different language were only trivially different, then it really is vicious. But whatever. My point was that that different language is a layer that has to be reduced or eliminated in order to demonstrate the circularity. Apparently, this word “layer” presents a problem for some people. Many people seem to think in terms of hierarchy when talking about real or logical systems. E.g. an axiomatic system allows propositions to be composed of “lower level” elements (axioms and, even lower, the alphabet). E.g. an organization like a corporation is a higher level collective, composed of lower level departments or people. Etc.

The concept of levels assumes that directional hierarchy. It’s effectively a partial order where any level is ≥ or ≤ any other level. But I posit that some systems may not submit to a partial order, where there is no cumulative relationship between any 2 components. Or, more likely, the relationship between any two components is not as simple as ≥. My favorite middle ground example is the onion. There’s no up or down with respect to the center. You can vary your direction and stay at the same layer. Of course, all you need do is switch to polar coordinates and you get your partial order. But that’s not the point.

I don’t want to impute the properties I’m trying to discover. So, why bias the conversation by using levels when we could use the more generic term layers just as effectively?

Anyway, the point that caused me to write this log entry is: What does it mean to have a logic that requires the more generic concept of layers and does not succumb to the concept of levels? What does a non-hierarchically layered logic look like?

Well, my (largely ignorant) guess at an answer is paraconsistent logic. But it’s useful to first consider non-monotonic logic, which I think (in my ignorance) would be a partial order. Here, when you add a new proposition to an extant argument, it’s truth value could change. When that happens, you have to have a handler that resolves the situation. E.g. is the truth value of the new proposition weighted more heavily than the older ones? Can you find a single old proposition that contradicts the new one? Etc. But the objective is to accumulate propositions in a singular argument.

Paraconsistent logics allow persistent contradiction. You still have to try to resolve any conflicts, because your purpose is not to allow any old nonsense to pollute your argument(s). But you have alternatives for how you handle the inconsistency. One example might be to unify as many of the propositions as possible into a “lower level” argument, then allow 2 lines of inference at a “higher level”, where the mutually incompatible propositions are appended to the unified part of the argument. This would still be a partial order. But another example might be to maintain 2 entirely independent lines of inference.

In the latter case, where we maintain 2 independent arguments, transformations and consequences of the arguments flow out in one dimension (forward in time, if you like). So we have a 2 dimensional construct. Argument # vs. consequences. And if we also consider that new propositions can be added, then we have a 3rd dimension. Movement in any of those 3 dimensions might change how you view the arguments. And there may not be a simple relation (like ≥) that characterizes the differences as you move in those dimensions. Hence, levels is no longer a useful concept, in general.

As usual, I hope I don’t sound like too much of an idiot in what I’m saying. But I thought I’d document it before I forget and have to rethink the thought.

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