International Journal of Advances in Philosophy

2018;  2(2): 38-43



Higher Order Vagueness

Marcia R. Pinheiro

IICSE University, DE, USA

Correspondence to: Marcia R. Pinheiro, IICSE University, DE, USA.


Copyright © 2018 The Author(s). Published by Scientific & Academic Publishing.

This work is licensed under the Creative Commons Attribution International License (CC BY).


A discussion on this so debatable and debated concept, Higher Order Vagueness. After going through the issues that make Higher Order Vagueness an object of scientific attention, we conclude that vagueness is undesirable in logical systems but desirable in language, and the Semiotics Universe splits into three other universes inside of the universe of the Computer Science Systems. We create a new way of addressing problems with sigmatoids, and define vagueness in an almost mathematical fashion.

Keywords: Vagueness, Higher order, Logic, Language, Sorites

Cite this paper: Marcia R. Pinheiro, Higher Order Vagueness, International Journal of Advances in Philosophy, Vol. 2 No. 2, 2018, pp. 38-43. doi: 10.5923/j.ap.20180202.02.

1. Introduction

What is a concept?
A concept is a set of two elements: sigmatoid (Pinheiro 2015, p. 19) and world reference.
A sigmatoid is a string of unspaced symbols.
Xxxooo is a sigmatoid.
Xx is a sigmatoid. Xooo is a sigmatoid as well. Therefore xx xooo is one sigmatoid followed by another.
We use sigmatoids to point at world references, and those can be material or immaterial.
Material world references are things such as what we identify in the world as being a bottle when we say bottle of water.
Immaterial world references are things such as what we identify in our imagination as being sound when we say the sound of silence.
Higher Order Vagueness, three sigmatoids that are found together, like this, in discussions involving Paradoxes of Language, such as The Sorites (Pinheiro 2016a), is what we call concept when used in that context, of the paradoxes of language, because both the expression, which is the three sigmatoids together, and the immaterial world reference, which is the idea of blurred line and talk about the own talk, must be in our Inner Reality (Pinheiro 2016b, p. 68) as a whole for us to understand the content of discourse.
Whilst vagueness (Pinheiro 2017b, pp. 1-4) came about as an ideal token to explain the shock we get when confronted with paradoxes of the language, such as The Sorites, higher order just seems wrong, out of place.
In this paper, we will prove that it is indeed wrong and out of place: it should be lower order instead.
It is probably better saying indiscernibility of boundaries due to indeterminacy inherited from the meta-language axioms.
We could also say logical systems with identified gap (Pinheiro 2013, para. 1) and/or glut (Pinheiro 2012, para. 1) area in the meta-language.
The point that made us follow a different path from that of Priest and Hyde in 2000 was exactly this: we thought that progress was a machine becoming a human being, but they thought that progress was a human being becoming a machine (Pinheiro 2016c, para. 12).
We try to have the lowest number of axioms as possible this far in Logic, so that we are after the smallest generator or generation set.
Higher Order Vagueness (HOV) is something that happens when we attempt to apply the logical system at hand.
The metalanguage could tell us that an atom (Yamamoto 1991, p. 73), in the logical system we have, is considered to be true if and only if its attribution is 3, and it is considered to be false otherwise.
Now we are given an atom that has attribution 2.999.
Since the metalanguage does not mention what to do in this case, we are left in the gap.
That is when we say that we are facing The Problem of Higher Order Vagueness.
If we think about the concept of god, Catholic god, and therefore the world reference creator of all, we can think that god is the generator of our logical system, so that, if there is a mistake with the thinking of god, there is a mistake of the higher order or the order above us.
In this paper, we will discuss this idea further.

2. Development

The reason behind the creation of a logical system has to be making the machines interact with us in a different way, so that the logical system is their persona, thinking here that we could define a Mechanical Persona (in analogy with the Human Persona (Pinheiro 2017, pp. 21-24), which could be regarded as part of the set of the Inhuman Personas.
A mechanism created by us has to be inferior to us because it is created according to our thinking, and because of our actions, similar to the way things are between humans and god.
In this sense, writing the metalanguage is explaining how things work to the machine, but the closer we go to the human language, the higher we go in orders in Computer Science (Computer Science GCSE GURU 2018, para. 1-11).
Since Logic is either philosophical or computational, and every logical system in the sense of Graham Priest (NCL) bears no completeness or soundness issues (Priest 2000), and therefore is fully computational, we are going closer to the machine when translating our thinking into what they can understand or metalanguage, and therefore we are going lower in Computer Science.
Because it cannot be the case that we refer to the purely philosophical logic when putting logical systems together, the issues we find when analysing The Sorites Paradox have to do with lower order vagueness, not higher.
Maybe our thinking processes go higher in levels of complexity (Pinheiro 2015b, p. 136) when trying to explain our logical system to others or to a machine, but we ourselves then suffer from higher order problems, not the logical system under analysis.
In this case, when we say Higher Order Vagueness, we could be right only if referring to Philosophy of Mind or Psychology or something of the type.
In order to correctly decide on how to refer to the issues we here mention, it is worth assessing some extracts of texts found in scientific journals:
1) In (Hu 2017, p. 1)
“all vague predicates are higher-order vague.”
>>If we talk about higher-order vagueness of predicates, then we talk about language, and that means that we are inside of Philosophy of Language or Logic, but not Philosophy of Mind or Psychology or the alike.
2) In (Burgess 1990, p. 427):
>>Here we see order as level of operation in a string of operations or level of the function in a string of functions: it is the same sense we have when we talk about the fourth derivative of a function. That is the derivative of the third order derivative, which is the derivative of the second, which is the derivative of the first (Houghton Mfflin Harcourt 2018, para. 1-2).
We could say that we have the first order definition of truth, then the second order, which would be the definition of the truth of that definition of truth, and so on.
The problem here is that the metalanguage stops all on the second order: all else seems to be misunderstanding in the mind of the thinker.
In each logical system, we only have three levels of reasoning or existence (Cambiucci 2008, para. 1-25): the metalevel (where we tell the machine what to do to cope with our logical system), the system level (where we describe the system itself, our particular view of things), and the application level (where we try to use the logical system to handle the issues we have at hand, so say solving a problem in Mathematics, MAPLE systems).
If we are not applying the system, we only have two universes where we could be stepping: metalevel or system itself.
We could then think, as Bobzien (Bobzien & Keefe 2015) points out, that we talk about vagueness of vagueness of bald, so second hierarchical order of vagueness, or unmentioned vagueness of the elements that form the basis of the metalanguage.
The problem is that, in this case, the metalanguage would be either one level above or one level below the own language of the logical system, and that is a mistake in the definition, which makes the system unacceptable, and therefore that is a mistake that makes it be a non-system.
We could never have third order vagueness in this case, since we should have realised the mistake and simply deleted the system, or fixed it/its metalanguage axioms, on the second level of vagueness.
We do not mathematicize language: whatever is vague should remain outside of the scope of Computer Science.
Vagueness is not regression in linguistic knowledge; vagueness is a progression instead, since it allows us to do more with the sigmatoids than we have done before (mouse from the past -> an animal versus mouse from nowadays -> an animal or a computer item).
We always prefer, if we have only 10 dollars, buying something that can do lots of things (Harper College 2018, para. 1-40) instead of something that does just one thing: say we have a machine that is fax, photocopier, printer, and scanner (Nashuatec Global 2018, para. 1-2) for 10 dollars and another that is only fax (Officeworks 2018, para. 1-8).
Both do the same jobs in the same way when it comes to the fax option.
Which one are we going to buy?
We are lazy: we prefer learning one sigmatoid that does the job in ten situations to learning ten sigmatoids that do different jobs in different situations.
Optimisation is a universal tool: we all prefer things that have been through optimisation processes, so that everyone prefers the 2i ( 2018) to 2 in cars or the four wheel traction to two or one.
Vagueness is evolution because we seek comfort, pleasure, and efficiency (Pinheiro 2016c, p. 2).
Whatever is machine-like, we decided a long time ago, is lower-level of reasoning or utility, and whatever is vague is closer to what we, as purely human creatures, wish for.
In this case, heap is good for us, for our daily lives, but is bad for computers, and therefore should be avoided in the same sense that driving is good for responsible adults but can be a hazard for kids.
Let’s keep vague vague and therefore agree that whatever is vague has got multiple world references, and we could then even talk about Worlds of Discourse.
Sigmatoids that bear non-null vagueness score are definitely not suitable for use with logical systems because there will always be confusion that the computer or its users cannot deal with originating from that inclusion.
Consider The World of Mathematics.
Select the sigmatoid function.
We have at least real and complex functions.
If we are not aware of the vagueness involved in the match (function; its world reference), we may find only the real intersects of our function, and deal with those as if they were all the zeroes available.
That may lead to decrease in our marks at school.
That may be something we cannot deal with.
The user might think that their machine can deal with complex numbers. After entering the function in the computer, the computer crashes.
With logical systems, only what is perfectly defined, so only what has got vagueness zero, so say 0, could be part of the system (Pinheiro 2015, pp. 19-22).
3) In (Hyde 1994, p. 37):
To make it easier, say that
P(yellow) = 1 if hue = 0.15 in the scale xxy.
P(orange) = 1 if hue = 0.16 in the scale xxy.
The yellow set contains a borderline case: what to do with 0.155?
The orange set contains a borderline case: what to do with 0.155?
In this example, we observe that we talk about lower-order vagueness, not higher, because detailing the figures we have further would be going down on the 0-1 scale, not up (way less than 0.15, and therefore in the lower range of the interval [0,1]).
Using the language of Hyde, we can then say that vagueness is the quality that equates having borderline cases between the positive and the negative cases of application of a sigmatoid to world objects.

3. Conclusions

The phenomenon of Higher Order Vagueness could as well be called Lower Order Vagueness instead: it happens in the lowest level of the conversation between computers and human beings, and also in the lowest level of the possible differences between positive and negative cases of a predicate.
Vagueness is not an unwanted occurrence when it comes to purely human language, since it provides us with more freedom and fulfillment, but it is an unwanted occurrence when it comes to logical systems, since we really do not want computers or human beings who use those to face the consequences of the confusion caused by those terms: impossible computations, misleading results, and so on.
The Linguistic Universe is divided into Worlds of Discourse, so that we have the World of Mathematics, the World of Religion, and therefore what could be called Big Worlds, and the World of Arithmetic, the World of Algebra, the World of Catholicism, and therefore what could be called Small Worlds.
Some sigmatoids might have a single reference in a particular world, say the World of Arithmetic, and more than one reference in the Linguistic Universe.
Vagueness does come by degrees, and we’d better have the meter on zero when we talk about sigmatoids that are part of logical systems.
We should now establish a better way to describe the relationship between signs and our world: (sigmatoid; world reference; world of pertinence) seems to be a first step in that direction.
Vagueness is not a regression in human knowledge: it is a progression instead, since we are increasing utility/application, which is something seen as progress in all other areas (multifunction machines are preferred everywhere on earth to single-function ones).
In each logical system, we have only three levels of reasoning or existence: the metalevel (where we tell the machine what to do to cope with our logical system), the system level (where we describe the system itself, our particular view of things), and the application level (where we try to use the logical system to handle the issues we have at hand, so say solving a problem in Mathematics, MAPLE systems).
We could never have third order vagueness in this case, since we must realise the mistake, and simply delete the system, or fix it/its metalanguage axioms, on the second level of vagueness.
We do not mathematicize language: whatever is vague should remain outside of the scope of Computer Science.
The metalanguage is closer than the logical system to the machine language. In this case, it has to be more objective than the language of the logical system. If we use MAPLE to operate in the World of Mathematics, then we cannot have definitely true in the metalanguage of MAPLE, since that is vaguer than belongs to.
A question of the type how true is your definition of truth is a question about the metalanguage, not the logical system, so that it must be a question of lower order, not higher.


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