Dilemmas of Ambiguity and Vagueness

Note: this essay remains unfinished and will be modified as time permits.

"All the limitative Theorems of metamathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. Godel's Incompleteness Theorem, Church's Undecidability Theorem, Turing's Halting Problem, Turski's Truth Theorem -- all have the flavour of some ancient fairy tale which warns you that `To seek self-knowledge is to embark on a journey which...will always be incomplete, cannot be charted on a map, will never halt, cannot be described." - Douglas R. Hofstadter

Introduction

1. How to 'Explain' what you mean

Often we cannot. Many terms are inherently vague and cannot be precisely explained. Many definitions are circular. Examples abound in the dictionary and the law books. We develop an understanding of what we think the terms mean through culture and context. I will try to convey to you what I know about the subject. The explanation, of course, will be vague :-)

In the natural world, almost everything is analog not digital, continuous not discrete. Humans in a natural environment have little difficulty with this (but it gives computers and bureaucrats fits!). If it gets too cold, we put a coat on. But there is no precise definition of "too cold" -- so how do we know when to don a coat? While environmental temperature is a continuum but a coat on or off is a discreet event, we have no serious difficulty in managing the problem. There is no precise temperature for putting on a coat and that is just fine with us.

In the figure below, I will illustrate three possibilities for how phenomena may go from "Cold" to "Hot", "Tiny" to "Large", "Bad" to "Good", etc. (For more on this type of representation of vagueness, see "A Fuzzy Fairly Happy Face" by Patrick Grim.)



The "C" curve is the way many of our leaders -- particularly those pushing "absolute morality" -- would like for you to think the world runs: things are either black or white with no shades of gray. For example, the government sets a maximum cholesterol level beyond which you will be unhealthy. Same with alcohol percentage in the blood: you're drunk if it is more that 0.1% (now being changed to 0.08%). More than that level is bad, less is good or at least "OK".

The "A" curve is the way most things are: a smooth transition from cold to hot, black to white, bad to good. No particular point is obvious as a point to set a break. An example would be the qualifying term, "bald headed". The number of hairs on head varies from none to many. At what minimum quantity of hairs do you become bald? We call people that still have hairs "bald headed", so "none" must not be the answer.

Curve "B" represents a situation where the parameter of interest varies smoothly but there is a small area in which it changes rapidly. When we have such a situation, it would be reasonable to set the "Good/Bad" break point somewhere in that region of rapid change. An example of this type of phenomenon is the temperature my body perceives while in my hot tub. There is a very narrow range between hot and cold. An even better example is death. While we would like to think that organisms, particularly humans, are either dead or alive, there actually is a small region is which there is vagueness. In fact, legal issues result from this narrow range of vagueness.

Problems develop when we have a situation that is best described by Curve "A" but we arbitrarily force it to look like Curve "C" by assigning an arbitrary breakpoint. An example is a speed limit on a road. Above a certain speed, the government says it is "BAD" and below that speed, it is "GOOD". Now, really, the damage to property resulting from a crash varies smoothly over quite a wide range of speeds. Therefore the speed limit as assigned by the government, is purely arbitrary.

"So what", you say. Unfortunately, the forcing of continuums (typically, Curve A) into a step function (Curve C) does cause problems in large structured societies such as those we live in. Left to free individuals, putting on a coat is not a problem but what about controlled groups such as the military? Soldiers, as a group, are ordered to put on their coats. To avoid problems, this order will most likely be based on some arbitrary but fixed temperature (or an arbitrary date). A much more serious example is the arbitrary assignment of the legal age for voting. The right to vote probably ought to be determined by a person's knowledge of the issues -- not some arbitrary age. Note that the government, in its infinite wisdom, has determined that it is ok to get killed in some meaningless, politically inspired, war at a younger age than the legal age for voting!

Most of us have heard the old saw about a lecherous man propositioning a young lady, that goes something like this:

A man offers a virtuous young lady a million dollars to go to bed with him. She says, "Well, it would be a disgusting act, but for a million dollars, I will do it just this once, as I do need the money". But then he says, "Well then, would you do it for twenty dollars?". She screams back at him, "What do you think I am? -- a whore!". He responds, "What you are has already been established -- what we are doing now is negotiating the price."

In general, even though the world is naturally continuous, we find we must quantize (or discretize, if you prefer) our view of much of nature in order to avoid difficulties in rule-based societies. We quantize the age for buying alcoholic drinks, getting married, getting into school and the military, drawing pensions, watching movies, driving cars, and leaving home. We quantize test scores to determine school grades. We quantize poverty, illness, pornography, speed limits, and allowed intoxication levels.

All this quantization is done to make it easier for humans to get along with each other, bureaucrats to do their job, and/or to avoid the strain of having to make a judgment. But there is a price to pay: you would like to put on your coat when you feel cold not because some temperature had been met. And so it is with all the rest of the quantized decisions. Persons receive welfare not because they need it but because some arbitrary monetary income has been met. Students receive grades not based on their true understanding but because of some arbitrary test score.

Further, all quantizations result in the Vagueness Dilemma which is:


Whenever a continuous event or characteristic is converted to a discreet function, the value chosen for the breakpoint is arbitrary and cannot be justified.

(This aspect of vagueness is examined in more detail in the section, "Understanding Vagueness").

Let us look at some easy examples:

• Exceeding the speed limit
  
  Let us say that the speed limit is 65 mph and it is a clear day and a straight road (note that the speed limit will -- most likely -- also be 65 on a cloudy day!). Let's say you're doing 65.1 mph and a cop pulls you over. Should you get a ticket? Certainly. What about a "tolerance", you say? OK, lets have a 5 mph tolerance. All that means is that you must get a ticket at 70.1 mph.

What if the cop says, "Alright, I suppose I will allow you to go one tenth of a mph faster than 65". "Well then", you say, "if 65.1 has, in effect, now become the speed limit, how would you feel if I exceeded that by .1 mph?". The cop, like most bureaucrats, quickly realizes the errors of her ways. While the limit is purely arbitrary, it must be absolutely complied with! To allow the slightest inflation is to invite an infinite series of incremental increases. So, like the "Voter's Paradox " a dilemma results from the fact that the problem can be viewed microscopically or macroscopically. That is, for all practical purposes, one small incremental change is insignificant but the accumulation of many of these small increments can cause trouble.

All the other quantified aspects of our lives are subject to the same difficulties. Depending on what it is being quantified, the results can sometimes be very serious.


• The "Communications Decency Act"

In this act, recently passed by the US government, there is a prohibition against "indecent" materials being transferred over the internet. The word "indecent" is used as if it were as precise as "dead". In this case, it appears that they haven't tried to define the term. That is the way such laws start. As lawsuits and bad justice proliferates, they will find it necessary to try to define the term. More laws will be created. Of course, they will be unsuccessful but will only make matters worse since the term -- as currently used -- is undefinable.

• Click here for a few additional vague terms we use daily:

2. Why is Vagueness a problem?

For several reasons -- a big one being that it is notoriously abused by bureaucrats, particularly the government bureaucrats and politicians. I will provide a couple of examples:

• Vagueness fosters growth of things that are good for them but bad for you
  
  Like the national debt. How much is too much? Surely a bit more will not hurt. And how much should we spend on welfare and public health care? The answer is purely arbitrary and will never be enough to satisfy those who promote it and too much for those who despise it! How much control should governments have on the environment? Surely declaring a piece of your property as a WETLAND will not hurt. Surely making you replace all the asbestos in your building will not hurt. Etc., etc.
• Vagueness allows arbitrary condemnation and harassment of individuals
  
  When does a work of art become "pornographic"? How much physical punishment can you apply to your own children? At what level of activity does an action become "Sexual Harassment"? At what level of activity does a child become "Sexually abused"? A photograph of the child in the nude? Maybe. It is all arbitrary and vague. Which means that the government can set the levels at any place they want (and then incrementally change it!).
  
  A good example on this issue is the proposed change to the "intoxication level" as defined by the state laws. At present the value used by most states is .1 percent alcohol in the blood. The new law proposes to reduce it to .08, claiming that it would save thousands of lives. Well, of course it would. But so would reducing it from .08 to .06. There is no logical point to set this value. It is purely arbitrary.

• Laws created to eliminate vagueness tend to suppress Self-Reliance

In the natural state, humans have no serious problems with making judgments appropriate to the many vague situations they encounter. This builds self-reliance as it should. On the other hand, in a government controlled society, judgement must be reduced as much as possible. While we might say, "eating too much fat is bad for you", the government must say, "eating an excess of x amount (x varies from time to time) of fat is bad for you. Being a little lazy, we tend to let the government make such decisions for us. After all decisions can be work!

3. There is no General Solution to the Vagueness problem

Governmental "break points" are established to prevent abuse -- often at the urging of the public. Bureaucrats have spending levels invoked on them to keep them from cheating. But, as we see, the quantization of a continuous phenomenon has some serious drawbacks. Many people suffer because they are on the wrong side of some arbitrary and heartless break point. It is a no-win situation.

If you think this is an easy problem or you don't see the seriousness of it, then I challenge you to tell me how much should a public hospital spend on a patient before they let him/her die. Let us say the patient is a "homeless person", nee "bum", or a criminal from death row who has had heart failure. Is a heart transplant appropriate? Don't forget that the public hospital has a fixed budget. These are the kinds of arbitrary break points that must be defined in any bureaucratic situation.

4. Philosophical Underpinnings

I must admit that I haven't had the time or the resources to do extensive research on this subject. In the Reference section I have identified a couple of references that I found to be somewhat helpful. Russell's short essay is identified primarily because it is on the internet. It doesn't really get into very much depth.

Russell attributes vagueness to being mostly a problem of language. Of course language is part of the problem, but I do not agree that it is the main problem. There would still be vagueness even if we had a very precise, logically structured, language. As I discuss elsewhere in this essay, the principle source of difficulty seems to be in making discreet statements about continuous phenomenon.

This aspect of vagueness is discussed at some length in Williamson's book, Vagueness;. He uses the standard example, the 'sorites syllogism' of the "Paradox of the Heap" to illustrate the difficulties. When does a conglomeration of sand particles become a "heap"? How many hairs on a man's head would allow him to escape being called 'bald'? These problems are just symbolic representations of many real-world problems and it is unfortunate the philosophers use them instead of any of a multitude of real-world problems since many people will just dismiss their arguments as just so much academic horse pucky.

In my view, the problem with vagueness is common to the problem of Social Dilemmas and that is, the paradox comes about as a result of having two different views of the same phenomenon: the microscopic view and the macroscopic view. When you switch from discussing a heap to discussing a grain of sand, you are switching views. The two views require different expressions to properly discuss and should not be mixed. I believe it to be comparable to the physicist's discussion of fluids. When the physicist is discussing fluid flow he imagines the fluid to be infinitely divisible. To talk about a particle of fluid is meaningless. We know that in reality, a fluid is composed of molecules and we could discuss the physics of the interaction of these molecules. Either way of looking at fluids can be useful.

Real world examples such as 'indecent materials', recently made illegal on the internet, are just as interesting and useful as heaps of sand and that is what I will concentrate on in this essay.

To learn more about the theoretical aspects of vagueness, go to my essay, "Understanding Vagueness"

Vague or Ambiguous Ideas and Concepts

"Everything you've learned in school as "obvious" becomes less and less obvious as you begin to study the universe. For example, there are no solids in the universe. There's not even a suggestion of a solid. There are no absolute continuums. There are no surfaces. There are no straight lines." - Richard Buckminster Fuller (1894-1983)

Society somehow plods on in a sea of vagueness and ambiguities! Humans routinely live, love, work, go to prison, and die based on vague or ambiguous definitions! The most amazing aspect of it is that it works so well -- in spite of the vagueness and ambiguities. It is not a trivial matter -- in fact it is a very serious matter. People rot in prison and die in wars based on vague definitions and criteria. Let us examine just a few of the vague terms we live by:

1. Rights

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. -- Declaration of Independence, July 4, 1776

The term, "rights", best illustrates my point here -- namely, that people live and die for concepts in which there is only a vague understanding of meaning. "We hold these truths to be self-evident" makes a claim which is totally unprovable. What the term,"rights", means and what it consists of is a long way from being "self-evident"! (As far as what does equal mean, see my essay, "The Equality Myth".)

If we were trying to define "dogs" and were having trouble coming up with a workable definition, one solution would be to make a list of all animals that we would call "dogs". This list would include big ones, little ones, red ones, black ones, gentle ones, mean ones, and so on but most probably would not include any cats or ducks. There would be general agreement on the contents of the list. Now try this with the term "rights". No two lists would be alike and further, we would never agree to a common list! The fact is, while we might die defending our "rights", we have little idea as to what it means.
I will not go into further detail now since there is an essay, "The 'Right' Approach" that discusses this in some detail.

2. "Family Values"

Now, what the hell does that mean? There are all kinds of families! There are extended families, wealthy families, crime families, sexually open families, religious families, atheist families, etc. Values? What kind of values? My dictionary defines it as: "the social principles, goals, or standards held or accepted by an individual, class, society, etc.". Then values would include the principles followed by primitive tribes that were very sexually promiscuous. Or tribes that were war-like, or practiced cannibalism, etc.

I wonder if "Family Values" doesn't have a variety of meanings, depending who the listener is!

3. Patriotism

My country, right or wrong? My president, right or wrong? Should we be enthusiastic about going off to a war, with a chance of great suffering or death, to embellish someone's political career? What are the limits to patriotism? What is the purpose of patriotism? Actually it is just another meme and, like all memes, has the potential to be either good or bad. Therefore, it is suspect. Before you commit your life to patriotism, you might want to give it a little rational thought.

4. Legitimate Government

Just like pornography, we can't define it but we know a legitimate government when we see it! Oh really? What is the definition? The criteria for the U.S. government seems to be this and this only: "it has been there a while". That is, we may not recognize a ruthless dictator who takes over a country by wiping out a democracy -- right away. But if he stays in power long enough and he is powerful enough, we will give him recognition. Consider Communist China and the former Soviet Union. On the other hand, look at Fidel's Cuba. He pulled the lions tail and he -- and Cuba -- are being punished.

Let us do a little mental exercise on this problem to see if it can be solved. Consider a small island, previously uninhabited. Let us say that a couple of hundred families move to the island to get away from an "oppressive government". Is there a legitimate government? No, not yet -- we just got here. Now, things go ok for awhile but then disputes start to develop. Some of them are serious, a few fights ending in death and that sort of thing. The folks meet and decide we need a police force. They hire a bunch of fellows and give them guns and token pay.

Before too long, it occurs to these "policemen" that they could do a lot better. One of them decides to be the Grand Leader and the others go along since they have been assured of a cut of the rewards. They establish a dictatorship. There is nothing anyone can do about it since it turns out these fellows have a hefty store of armament and the other islanders have practically none.

Is this a legitimate government?

Does a legitimate government require establishment by a majority vote of the people? If so, most country's governments would not qualify -- including the U.S.A..

So, maybe a government can be called legitimate if it is somewhere between a dictatorship and a freely established democracy? Where? I think we have to conclude that the concept of legitimate government is vague. Therefore, in principle, there is no difference between a legitimate government and a group of thugs running a country for their own amusement. It is simply a matter of degree.

5. Legalities

5.1 Crime

[Much of the following material was inspired by the book, Bad Acts and Guilty Minds.]

Is the law always clear cut? Certainly not! Can you have your life destroyed for an action that has been deemed a crime but for which you are innocent? Absolutely -- happens all the time. Can laws be made definitive? Well, yes and no. The God of the Jewish/Christian bible had the right idea -- don't get bogged down in details. Don't get into the bottomless pit of "why's" and "circumstances". Being God, he was too smart for that. He just said, "Thou shall not kill". To go any further gets you in a quagmire of never ending vague refinements.

But we humans figured that only a god could get away with such ruthlessness. We care and we are full of compassion and understanding. What if someone killed someone else in self defense? What if it was an accident? What if the perpetrator was mentally deranged at the time? Is it fair to punish persons who happen to fall in those situations? Surely we can do better than a simple, "Thou shall not kill".

Maybe, maybe not. Maybe there is much suffering by innocent people as a result of our good-hearted efforts. Maybe criminals go free due to obscure contradictions in the law. Maybe, with lots of confusing and vague laws, a good lawyer can always get you off no matter what you have done!

Consider the following examples (some from the aforementioned book):

• John attempts to kill his wife Annie but only wounds her in the arm. She goes to the hospital. There she contracts scarlet fever and dies. Is John guilty of murder?
• If a drunken driver intentionally driving head on towards oncoming vehicles "just for fun and excitement". He comes at you but you manage to dodge him. Should he be tried for murder? OK, let's say he hits you and kills your wife. Now should he be tried for murder? He didn't do anything different. The only difference between the two cases was your skill in getting out of his way.

Do we punish for the result or the intention? If it is the result, then why do we have all these degrees of murder, manslaughter, etc.? If it is intention we punish, why don't we hang people who fire guns at others but miss or only wound them?

5.2 Legal Complexities and Contradictions

"Sweer`s Impossibility Theorem:
Nothing can be both completely general and internally consistent at the same time."

As laws are put in place, contradictions with previous laws will arise. Further, terms will be used that are vague. These two facts make it necessary to create more laws! These laws, in turn, will also be contradictory and vague. Therefore, the legal code grows exponentially! But the contradictions and vagueness are never removed.

One of the best examples of this is the United States income tax code. I will quote from a recent article, "Why you Can't Trust the IRS" published in "Real Money" magazine.: "When the income tax law first took effect in 1918, it had just 170 pages of law and regulation. Since then the tax code has grown to encompass over 17,000 pages. On top of that, hundreds of thousands of pages of court decisions interpret that law."

The article goes on to describe Money Magazine's annual survey of private-sector tax preparers in which an imaginary but realistic financial profile is provided to various professional income tax preparers and they prepare the returns. Fifty preparers come up with 50 different income tax returns! No two are alike. In 1992, for example, the amount owed to the government -- as determined by these preparers -- varied from $16,219 to $46,564! The truth is, no one knows what the correct amount should be! Not even the IRS.

Another example of a troublesome contradiction in law is the rule emphasized by the Nuremberg trials at the end of World War II in which soldiers are held individually responsible for any crimes committed against humanity. Yet, soldiers can also be tried for treason if they disobey an order. So here we have contradictory laws based on an ambiguous definition! Violating either interpretation can get you shot!

It seems that life in general, legal systems in particular, is plagued by some sort of universal law like Gödel's Incompleteness Theorem (as suggested by Sweer's quote above). That is, it seems impossible to design a set of laws that are internally consistent, that lack contradictions. Maybe someday, some brilliant philosopher will prove this but the best I can do is only suggest that it sure seems to be that way!

5.3 Precise Definitions by the Government (Ha!)

An example of the government's efforts at making precise definitions is given by the food labeling rules and laws. You probably think, for example, that "zero fat" means ZERO FAT. Wrong. "Zero fat" or "no fat" or "fat free" means "less than 0.5 g per serving". "Fresh" can be used only on a food that is raw, has never been frozen or heated, and contains no preservatives. (Irradiation at low levels is allowed.) "Fresh frozen," "frozen fresh," and "freshly frozen" can be used for foods that are quickly frozen while still fresh. Blanching (brief scalding before freezing to prevent nutrient breakdown) is allowed." A fish left lying on the dock a couple of days in the hot sun is, therefore, "fresh". ;-(

I am sure they are working on defining "indecent" as we speak.


6. Consensus

The term consensus is often used in political discussion and in legalities in a way that suggests "a general agreement". But what percent of a group has to agree before you have consensus? 100 per cent? 51 per cent? I checked the dictionary. It says, "1. majority of opinion. 2. general agreement or concord; harmony". Not much help there. If it only means "majority" then the term is not only not needed but is misleading.

Taking Advantage of Vagueness

We all use vagueness to our advantage sometimes in dealing with other people. Our intention is not necessarily to do harm or to take advantage but maybe to avoid hurting someone or to improve communication or a situation. A humorous example is the problem most of us have to face from time to time of being forced to give a "Letter of Recommendation" on someone you may not think too highly of. Check out the link, and you will see how using a tad of vagueness can be helpful.

While in our personal and natural lives, vagueness may not be a problem and, in fact, is quite useful, when it is used in the field of governance, there are serious consequences. Following this essay is an expose of that particular subject called, "Vagueness in Governance". It is strongly recommended that you read it for it could save your life. I'm serious.

1. Use of Vagueness by Businesses

Using vagueness intentionally becomes more harmful when used by businesses to promote products and downright sinister when used by the government to further its own ends. Businesses have had to become experts in the use of vagueness since they are repeatedly harassed for making false claims. I just bought a weed puller the other day (other day??) that has a "lifetime guarantee". Whose or what lifetime? My lifetime? The weeder's lifetime? The manufacturer's lifetime? The long distance telephone providers routinely advertise that if you will sign up with them, they will give you 20% off. 20% off of what? Apparently the public falls for this incredibly simple minded deception for if they did not AT&T and the rest would not spend millions of dollars on such ads.

2. Use of Vagueness by the Government [Note 1]

"The warning message we sent the Russians was a calculated ambiguity that would be clearly understood." -- Alexander Haig

VAGUENESS is the basis for more government growth and abuse than any other factor. Essentially all programs that the government is involved in rely on definitions that are vague. In general, vagueness cannot be avoided as this is the nature of life -- it's just that the government takes advantage of vagueness more than other entities do and the consequences are so great.

All government operations grow on the basis of vague requirements. In fact, the problem is so serious that the amount of annual funding of government agencies is primarily determined by the amount they had the previous year (See my essay, "Government Spending and Financing)". Every level of government bases its size and needs on purely arbitrary values -- not any kind of absolute values. For the most part, they actually assemble their list of requirements based on their budget, rather than their needs. Did you think it was the other way around?

Now, surely, you say, there must be justification for some agencies. Well, let us look at one of them.

• The US Defense Department

The Defense Department consists of much more than troops and their equipment. If you look at the US Defense budget you see that the total is around 280 thousand million dollars with a component for military personnel of about 80 thousand million dollars. The majority of military spending actually goes for "maintenance and operations", "research", etc. The majority of military spending actually goes to civilians; civil service, contract personnel and to government contractors. So, we have many opportunities for the use of ambiguity and vagueness in the military!

The basic question is, "How much defense is enough?" Of course, nobody knows. In fact the interpretation is wide open. But "what we need" just doesn't matter. That argument is given just to help justify their expenditures which are actually based on other reasons. These reasons include personal job security, profit making, and, in general, selfishness of the individuals involved.

Another question is, "How good does the equipment have to be?"

Better than any potential enemy's? Of course, but the answer is, again, wide open. You and I don't know too much about what the enemy has -- only our government has any real idea (from their spy organizations). So, how do they really determine how good our equipment must be? It is determined by the current technology and the marketing capabilities of our government contractors! We can spend billions for a few missiles and find ourselves buying a new batch if the contractor claims that major technological improvements have been made. That is the way it works.

Yet, Somehow We Cope in this Vague and Ambiguous World

In a natural state, humans convey meaning even though the terms are vague. We understand "dangerous", "sexy", "wealthy", "skinny", "friendly", and "happy" without knowing the precise meaning. It is the structured society that is managed by a legal structure that has a problem with vagueness. Bureaucrats and computer programmers have a difficult time with vagueness. The rest of us have a general feeling about what the terms mean which is adequate for communications.

The directions that society is going, which is to make every term and definition precise, is a mistake. It appears that Gödel's Theorem of Incompleteness may have more general application. Complex systems seem to get more complex and more inconsistent the more we try to remove the vagueness and inconsistencies. But, sadly, that will not keep us from trying.

Notes:

[Note 1] The discussion on government in this essay is on the US government in particular. Other governments, however, are probably similar.

References:

BLACK, Max: Perplexities: Rational Choice, the Prisoner's Dilemma, Metaphor, Poetic Ambiguity, and Other Puzzles. Cornell University Press. 1990

On the net: FUZZY SYSTEMS - A TUTORIAL, James F. Brule', 1985

On the net: Otávio Bueno and Mark Colyvan, 'Just What is Vagueness?'

Connolly, William E.: Politics and Ambiguity. University of Wisconsin. 1987

George C. Christie, "Vagueness and Legal Language", 48 Minn. L. Rev. 912 (1963-1964), online at http://eprints.law.duke.edu/247/

Katz, Leo: Bad Acts and Guilty Minds. University of Chicago Press, Chicago. 1987

Keefe, Rosanna and Peter Smith, eds.: Vagueness: A Reader. MIT Press. 1996

On the net: Justin Needle has a wonderful page on Vagueness with lots of references. (Already it looks great, but I think there is much more to come!)

"Fuzzy Logic Introduction", Jeyakody Parthiban, 1996

Pilla, Daniel J.: "Why You Can't Trust the IRS", Real Money magazine, Spring, 1996

On the net: Vagueness, Bertrand Russell, 1923.

On the net: Stewart Shapiro, Primer on Vagueness, May 20, 2005

Sorensen, Roy, Vagueness and Contradiction, Oxford: Oxford University Press, 2001.

Sorensen, Roy, "Vagueness has no function in law", Legal Theory, 7/4: 385-415, 2001.

Sorensen, Roy, A Vague Demonstration, Linguistics and Philosophy(2000) 23: 507-522

Williamson, Timothy: Vagueness. Rutledge, London. 1994.

On the net: Williamson, Timothy: "Vagueness in Reality",

Back to "the Paradox page".
Forward to "Understanding Vagueness"
Forward to "Vagueness in Governance"
Back to my philosophy page.