There can be a pluralism of logics because where is classical negation. This can be justified because  “1 2″ and “1 is not 2” are basically the same statement. However “not 2” has different meanings in different logics, so 1 2 means something different in paraconsistent logic (We’ll use DeCosta’s C1), we mark this difference with 1 2, and claim that the difference between classical negation () and paraconsistent negation () is  marked . Hence .

I mentioned in a previous post that we cannot generalize this statement into because it is a contradiction. Nevertheless, a sense in which is true is the sense .

Now, vagueness is the situation where it is not clear if 1 2. To illustrate, take this curve:

here is vagueness on whether we have two or one “heaps”. This thing could be 2 or it could be 1 and so in a sense 2=1. How is this handled by logical pluralism? Paraconsistent logic would allow that A: “1 is not 2” and ~A: “1 is 2” has a sense in which it is true, while classical logic would explode. The reason for this is entirely based on the difference in negation. C1 creates a new sense each time a true possibility is negated, making the negation of a possibly true row in the truth-table have two senses, one in which the negation is true, another in which the negation is false.

(A  ~A) (A  ~A)

The point is that how you handle negation changes how vagueness is handled (or not handled). A difference in negation also gives rise to an entirely different logic. Vagueness can be described completely as a failed distinction/negation, so that even though we want “1 is not 2” vagueness makes this distinction fail. A different negation yields a different way distinction fails, but no logic “solves” vagueness completely. This is the mathematical relationship between vagueness and logical pluralism.

This may be made clearer with another example. Vagueness is the situation when a distinction fails, which can be described by the failure to distinguish = from , so .

Now for classical logic if vagueness renders 1 = 2 we can prove , likewise in paraconsistent logic there is no problem having as another non-explosive contradiction. this means in particular that and . Substituting, we get that things can get so vague we can’t tell the difference between and , in other words = . and from our previous statement we have and vagueness is now mathematically related to logical pluralism.

At the heart of real analysis and the study of real numbers is a confusion between points and “stuff,” or how points can perfectly describe distance/space/extension. Put the other way around, how can extension be reduced to points? Or how can we know the extension (with points) of a length/width of an object?

This question is caught up in measurement, and the relationship between word and object. I have seen people give (real) number a privileged position between word and object, but I would argue that number suffers from the same flaws that language suffers from, including vagueness. One reason is there is vagueness in what to call “one” (see the classical “heap of sand” in a previous post), further there is vagueness in when something is countable or uncountable. Electron cloud size to water or human height is uncountable, but what about popcorn or puffed rice? These could be counted, but should we be counting the grains? Should we count water molecules? Should we go back to a particle model for electrons so we can count them? This should comes more pointed when we consider the ancient belief that counting/taking measurements about humans directly endangers them (Feyerabend 1999). Why was it believed that being unsure which side of the microscope you are on endangers?

Vagueness runs right through these issues from the finer points of graduate school mathematics to the ethical issue raised above. And of course it does, since we would like number to draw a clear line between word and object, point and “stuff”. Strangely, insight into these distinctions can be garnered by understanding just how troubling (and what) vagueness is.

Enter higher-order vagueness. Now the question has been put to me “Well the vagueness between, say, “high up” and “not high up”, (this can easily be pictured on a cartesian graph with a line gradually going down from left to right) can be dealt with by adding and third “uncertain” value/region. And the trouble here is, as is well known, that adding a region uncertain only adds two new borderline cases between “high up” and “uncertain”, and another borderline case between “uncertain” and “not-high up”, so that new “higher order” uncertainty regions have to be added for these borderline cases. And now we have introduced new sources of vagueness, etc. Ultimately the pursuit of conquering higher-order vagueness by exchanging borderlines(points) with intervals(stuff/extension) is a vagueness between points and intervals.

As that sinks in, realize that a vagueness between points and intervals is a general problem reproducible anytime vagueness rears its ugly head. If we draw the connections from points to words and from intervals to the “stuff” of objects, we find that the line between words and objects is vague, which is also well known. What is new here is that we found this well known vagueness by investigating vagueness itself in a general way.

To summarize I would say that the vagueness between word and object is an essential or “stock” vagueness that crops up anytime we are in vague territory, and is the heart of analysis of “real” numbers. In this sense, vagueness is an important ultimate concept for mathematics, and it ought to be mentioned in analysis text books that vagueness is what the book is about.

—The following is a tribute to my father, Kevin Barnhurst, who passed a year ago this month–

I decided to make this post a tribute to him.

Dad and I were working on this essay (http://journals.sagepub.com/doi/full/10.1177/1464884916689150) when he died.

Dad was a flawed human being, but one comfort is that I almost exclusively remember good things about him, and feel pleasure in remembering him. I know thats good for both of us. We were at odds a lot when I was a kid. I went through a different kind of school system engineered to dumb down the American population, and entered college a logical positivist by default, but underneath all that wash, I was deeply skeptical of my “education”. For dad his family didn’t trust his decision to enter college, and the situation was reversed. For him school was how to become educated, for me what education I have was a result of conversation (with him and many others). I probably would not have gone to college at all if dad hadn’t pushed me hard to apply. That was one of the strange things about dad, he was very forceful, and only made me more stubborn, but he softened later in life and knew how to make his force felt in a strangely soft way.

We kept a long tradition of holding protracted conversations in the evenings and into the night. I owe my intellectual development primarily to him, and it is strange how long it took me, all the way to the last few years of his life, to realize what a gift that was and to reach an understanding that allows respect his for work.