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Mathematical Relationship between Logical Pluralism and Vagueness

23 Saturday Sep 2017

Posted by nightingale108 in The more technical stuff

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There can be a pluralism of logics because Screen Shot 2017-09-23 at 10.55.59 PM.png whereScreen Shot 2017-09-23 at 10.57.58 PM is classical negation. This can be justified because  “1 Screen Shot 2016-08-19 at 2.33.44 PM 2″ and “1 is not 2” are basically the same statement. However “not 2” has different meanings in different logics, so 1 Screen Shot 2016-08-19 at 2.33.44 PM 2 means something different in paraconsistent logic (We’ll use DeCosta’s C1), we mark this difference with 1 Screen Shot 2017-09-23 at 10.58.22 PM 2, and claim that the difference between classical negation (Screen Shot 2017-09-23 at 10.57.58 PM) and paraconsistent negation (Screen Shot 2017-09-23 at 10.58.22 PM) is  marked Screen Shot 2017-09-23 at 10.58.10 PM. Hence Screen Shot 2017-09-23 at 10.55.59 PM.png.

I mentioned in a previous post that we cannot generalize this statement into Screen Shot 2016-08-19 at 2.33.02 PMbecause it is a contradiction. Nevertheless, a sense in which Screen Shot 2016-08-19 at 2.33.02 PMis true is the sense Screen Shot 2017-09-23 at 10.55.59 PM.png.

Now, vagueness is the situation where it is not clear if 1 Screen Shot 2016-08-19 at 2.33.44 PM 2. To illustrate, take this curve: graph_avg_weight1

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 Screen Shot 2017-09-23 at 10.57.58 PM ~A) Screen Shot 2017-09-23 at 10.58.10 PM (A Screen Shot 2017-09-23 at 10.58.22 PM ~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 Screen Shot 2016-08-19 at 2.33.44 PM, so Screen Shot 2017-09-28 at 1.30.38 PM.

Now for classical logic if vagueness renders 1 = 2 we can prove Screen Shot 2017-09-28 at 1.30.38 PM, likewise in paraconsistent logic there is no problem having Screen Shot 2017-09-28 at 1.30.38 PM as another non-explosive contradiction. this means in particular that Screen Shot 2017-09-23 at 10.57.58 PMScreen Shot 2017-09-28 at 1.34.47 PM and Screen Shot 2017-09-23 at 10.58.22 PMScreen Shot 2017-09-28 at 1.34.47 PM. Substituting, we get that things can get so vague we can’t tell the difference between Screen Shot 2017-09-23 at 10.57.58 PM and Screen Shot 2017-09-23 at 10.58.22 PM, in other words Screen Shot 2017-09-23 at 10.57.58 PM = Screen Shot 2017-09-23 at 10.58.22 PM. and from our previous statement Screen Shot 2017-09-23 at 10.55.59 PM.png we have Screen Shot 2017-09-23 at 10.58.10 PMScreen Shot 2017-09-28 at 1.34.47 PM and vagueness is now mathematically related to logical pluralism.

The symmetry between Completeness and Vagueness

10 Sunday Sep 2017

Posted by nightingale108 in The more technical stuff

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The (Problem of the) Heap of Sand is the classical representation of vagueness, where you create an inference: if you take away a grain from the heap then it remains a heap. This is a good inference for a long time, but eventually breaks down due to the vagueness of what a heap is (eventually you will have taken so many grains away that it wont resemble a heap anymore). And you can’t even tell when the inference will break down, exactly. So, as you can see, vagueness has a very precise and non-vague formulation, something similar to this strange property of vagueness (that it is a clear and distinct concept, even though it is a concept about indistinctness) is common for mathematical concepts. That is, mathematical concepts are chosen and defined so that they contain the terms necessary to get round their own difficulties—their faults or cracks, as all concepts are imperfect. Example: “Completeness”.

The difference between vagueness and completeness, is:

1) that the term completeness is in direct opposition to the claim that every concept has a “fault” or “crack”, in this it is a “perfect” term. The definition of completeness, however, must be vague, because the terms used to define completeness all have cracks.

2) The definition of vagueness, by contrast, is perfect, but the term refers to an essential fault of all terms, and is always perceived. (perception is always vague) Vagueness is defined as how the if, then fails— The “heart of logic” (the paragon of precision) and the most reliable way to preserve truth. It is clear and distinct in its description of how the if, then fails. Since each of the terms in the definition of vagueness have cracks, the definition is precise.

The term vagueness refers perfectly to its referent because it is well defined as an indistinctness, the definition of completeness is vague because the term refers to something that can’t be perceived, a perfect whole.

Stuff, and Higher-Order Vagueness

30 Friday Jun 2017

Posted by nightingale108 in The more technical stuff

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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.

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