Understanding Vagueness

By Leon Felkins

Email: leonf@perspicuity.net

Written 9/8/1996

Revised 11/16/2012

Hard at Work!Note: this section is in work with much to be added if and when time permits.

"Man is ready to die for an idea, provided that idea is not quite clear to him." - Paul Eldridge

  1. Introduction
  2. In the real world, almost all physical objects have characteristics that vary continuously, not discreetly. That is, the real world is analog -- not digital. No one is 5 foot, 11 inches tall -- but there are many who are "roughly" of that height. While it is useful and reasonable to "round-off" measurement to a meaningful accuracy, we get into difficulties when we "round-off" to "1 or 0", "true or false" for other common characteristics such a person's age, weight, intelligence, etc. I'm talking about such assessments as "old", "fat", and "smart". While saying a person is 5 foot, 11 inches tall is just roughly correct, saying that he is "tall", while convenient, does not convey much information.

    Yet such vague terms as "tall" can still have meaning (the meaning varies with the locale, of course). We agree that some people are tall and some people are short. But there's a huge number of people that we would have difficulty in deciding whether they were tall or short. These are the borderline cases and they are the source of difficulty with vagueness. Philosophical text books talk about the "Sorites Paradox" and usually give the classic example of the sand heap. When is a pile of sand a heap? We all agree that one grain of sand is not a heap but that a million grains is a heap. Somewhere in between the pile must change from "non-heap" to "heap". The precise location of that point is impossible to logically define. How many hairs must a man lose before he is determined to be "bald"? How many seconds make up a "moment"? How much money do you have to earn to be "rich"?

    While these problems may seem trivial, vagueness, in general, has some very serious consequences (some of these are discussed here). In my view, vagueness has not received nearly the attention that would be appropriate for its importance.

  3. The Basics
    1. What is Vagueness?
    2. "No man means all he says, and yet very few say all they mean, for words are slippery and thought is viscous." - Henry Brooks Adams (1838-1918) "The Education of Henry Adams", 1907
      Philosophers disagree on whether vagueness exists in the real world or is just a problem of semantics. What I will be discussing here falls mainly in the latter category. If you are interested in the concept that some objects in the world may be vague, consult the essay "Quantum Objects are Vague Objects" by Steven French & Décio Krause in the August, 1996 issue of Sorites.

      The vagueness of interest here, however, result from a language problem of trying to refer to an object as having an absolute value when in fact the object has a continuous range of values. An example would be the sentence, "Press reports state that the senator is a known liar". This sentence is vague in several ways. What press reports? All? Some? What is meant by "known"? Who knows? Everyone? Some? What is meant by "liar"? Does he lie all the time? Sometimes? With all the vagueness in the sentence, it would seem that it could convey no useful information, but apparently it does as we will discuss further below.

      The problem with vague statements such as the example given is apparent: a continuous parameter is being referred to as if it were fixed. When we refer to a person as being "bald", we are basically saying that on the scale of "full head of hair to no hair", this person falls in the spectrum near the "no hair" end.

      So, statements such as "He is bald" could be reworded into a more lengthy and complex sentence that clearly states that the person's hair count on the scale of zero to 100 percent head of hair, is near the zero end. "He is bald", then is just short hand for a longer sentence that possibly could correctly state the situation. We use shortened statements commonly in our communication which do the job as long as both parties have roughly the same background reference. For the purpose of the message, a vague message can actually be quite adequate. That is, the receiver of the message, "He is bald", may not need to know any more precisely than that his hair count is near the zero end of the spectrum. Of course, this is not always true. Sometimes vague statements are provided where a more precise statement is called far. This is particularly true in the legal field where many vague statements are part of our laws.

      Let us now look at the situation where a statement is not clearly true or clearly false -- the borderline cases.

    3. The Problems with the Borderline
    4. "Someone might seek to obtain precision in the use of words by saying that no word is to be applied in the penumbra, but unfortunately the penumbra is itself not accurately definable, and all the vagueness which apply to the primary use of words apply also when we try to fix a limit to their indubitable applicability.” - Bertrand Russell

      Some folks try to solve the borderline problem by approaching it with a 3 valued logic: 1) It's true, or 2) It's not true, or 3) It's neither true or false (or we just don't know). For example in the case of whether a person is an adult or not (our governments struggle with this one -- we send 18 year olds to war for possible injury or death but we don't let them vote! See "Old Enough to Fight, Old Enough to Drink"). At 45 or more, I think we all agree that the person is an adult and at age 4 years, we agree that he/she is definitely not an adult. But what if the person is 19? Just for fun, try the sorites argument: We agree that at 45 a person is an adult. Would you agree that if she was one month shy of 45 she is still an adult? Would you agree that another downward increment of one month would not change a person's status as an adult? That is, there is nothing about a one month change that would warrant changing the label for an adult to juvenile. Following that argument, we eventually realize that a 19 year old is an adult. But so is a one year old!

      This brings to mind my recent checkout of groceries at our grocery store. There was a sign there that said that all persons under the age of 45 must show an ID in order to purchase alcoholic beverages. Hmm. How can the clerk know if a person is 45? Should they not have to show an ID to prove that? What if the clerk would lose her job if she let someone buy beer that without showing and ID but was only 44 1/2?

      The objects of interest then are separated into 3 regions: true, false, and the penumbra (the location of the borderline cases). But it is already obvious to most of us that we solved nothing but only made the matter worse. For now, we have to decide where the breakpoints are that separate the penumbra from the false and the true! And just as we concluded before that we had a penumbra between true and false, we now have to establish another region of borderline cases between false and the penumbra and also between true and the penumbra. Obviously this process can be carried on to infinity. This appears not to be a solution then.

      I summarize with this quote from “Supervaluationism, Penumbral Connections, and the Nature of Higher-Order Vagueness” by Matt Sayball:

      If the border is thick, so to speak, then there are cases that are in between the extension and the anti-extension rather than within one of them, and the thickness of the borderline containing the indeterminate borderline cases is itself indeterminate. Since thick-border theories admit this ‘in between’ area, they cannot define the range of borderline cases and so admit borderline borderline cases, borderline borderline borderline cases, etc. This phenomenon is known as ‘higher-order vagueness’.

      There is disagreement among philosophers on the issue of "higher-order vagueness" details of which is far beyond the scope of this essay. Two examples, pro and con, hopefully will suffice for now: "VAGUENESS, MEASUREMENT, AND BLURRINESS", by ROY A. SORENSEN and "The Illusion of Higher-Order Vagueness", by CRISPIN WRIGHT. Many other papers on the subject are readily available on the internet.

    5. Vagueness Difficulties and Misuse
    6. There are situations where vague statements seem to be required and where a more precise statement creates unintended or paradoxical results. An example would be legal speed limits. We would like the law to require that people not drive "too fast". That is the speed should be in the range of "safe speeds". But these are vague terms. A policeman would need to make a judgment as to whether the driver was going too fast and a judge would have to judge as to whether the person was guilty if she was arrested. In theory, this is the way it should be. In practice, there are difficulties. Since being arrested is a rather serious matter, and we know that humans are sometimes devious, careless or unsure, we would prefer not to have our lives jeopardized by a vague law. So we insist that the law prescribe a precise value for the speed at which a person would be defined as going "too fast".

      But the breakpoint that is chosen (how it is chosen is discussed below) to define the boundary between "reasonable speed" and "too fast" is arbitrary and cannot be defended logically. There is no speed at which the accident rate takes a sudden jump. The accident rate, in fact, as a function of speed is a continuous curve with no distinct and abrupt changes. That is, if we increase the speed by an infinitesimal amount from any speed, the accident rate only increases an infinitesimal amount.

      Note that a "safe speed" is really a function of many more variables than just speed. In particular, what is safe is actually a function of speed, the capabilities of the driver, the mechanical condition of the auto, the condition of the road, the weather, and other factors. But these other factors complicate the law and are usually not considered. We chose to define the law in terms of speed for convenience. But some legal definitions do not have a handy parameter to put a limit on. A good example is "pornography". Any effort to define this in terms of physically measurable quantities is doomed to failure. For it is the cumulative effect of many parameters that makes an object "pornographic". The term is apparently too difficult to define precisely. So it is left up to the policeman and the judge. Again, in my opinion, this is the way vague objects should be handled -- that is, by judgment. To do otherwise creates even greater unfairness. (Even God has to deal with the problem of vagueness and breakpoints since she has to decide who goes to hell and who goes to heaven. See Theodore Sider's "HELL AND VAGUENESS" for a thorough discussion of vagueness and cutoffs, using the example of God and Hell.)

      A difficulty with terms that are left vague and where no precise cutoff point is assigned, is that their meaning will tend to creep over time. An "A" student is not the same as an "A" student of 50 years ago. What was called pornography 50 years ago is not called pornography today. Governmental services to the citizens that were quite adequate 100 years ago are not adequate today. These objects that have undefined breakpoints tend to reach an equilibrium defined by the conflict of opposing views. This equilibrium will vary over time and locale.

    7. Living with Vagueness
    8. The other side of the fascinating issue of vagueness is that humans seem to be able to communicate quite well using vague terms (I suspect this is true throughout the animal kingdom??). When someone makes statements like "The new Buick is a safe automobile", "The girl is deceitful", "Bono is a small town", "Bono is a very small town", "I will be back in a few moments", "Your job is to separate the large potatoes from the small", "She has a dark complexion", etc., we seem to understand what is meant. At least we understand well enough to make decisions based on the statement. That is, the statement achieves its function in transmitting useful information. In most cases, refining the statement with a definite value ("Bono has 384 residents") would not provide us with any more useful information. We have to conclude that even though a statement is vague does not mean it does not convey useful information.

      However vague communications, while conveying worthwhile information, do result in actions that have only a probability of success. If my wife and I are tending the baby and she says "I will be back in a few moments", and I get a call from the boss saying that he needs me at work immediately, do I dare leave? If "a few moments" means less than 3 minutes, it would be OK. But what if "a few moments" means anything less than 3 days? So, if I do leave I am taking a chance based on a probability distribution curve, in my mind, as to what "a few moments" might mean. My action then is based on a probability rather than definite info as it would be if she said, "I will be back in 2.5 minutes".

  4. Fuzzy Logic
  5. The problem with vagueness must also be faced by the engineers and scientists. Traditional logic that requires that a statement be true or false, a two-valued logic, seems not to be adequate for describing many physical systems that are used by human beings. A third value seems to be needed; "somewhere in between". So, "Fuzzy Logic", a three-level logic (true, false, and "somewhere in between") was invented and is already being utilized in engineering design of appliances, elevators, etc., particularly in Japan.

    While Fuzzy Logic has had some success in the engineering/science field it is not a "solution" to the vagueness problem. However, it is mentioned here as its basic theory provides a bit more insight into the general problem.

    For more information consult the easy reading tutorial on Fuzzy Logic, "FUZZY SYSTEMS - A TUTORIAL" by James Brule'. Another introductory level essay by Erik Horstkotte is here.

  6. How Society Establishes the Cutoff point for Vague functions
  7. The establishment of the cutoff of boundary value for a vague function is often determined by the equilibrium value of opposing forces. An example would be sales taxes (actually taxes in general but sales taxes are simpler). Government employees and politicians have a strong desire to push the sales tax level as high as they can, consistent with maximizing the total revenue. Taxpayers, on the other hand, would like to reduce the tax rate to a minimum consistent with providing basic services in an efficient and honest manner. Depending on the particular state, this will level out to something around 5 to 10 percent. It tends to creep higher over time since the politicians are much more diligent in looking after their interests than the general citizenry is. Note that the equilibrium level has little to do with what is actually needed.

    What we have is a "free market" situation for such functions. That is the equilibrium is established by two opposing forces: the desire for ever more money by the government and the opposition to paying more money by the citizens. Both of these functions vary slightly over time.

    And so it goes with other boundary values for vague terms. "Promiscuity", "indecency", "poverty level", "adequate medical care", "the Federal Deficit", "adequate armed forces strength", "qualification for a PhD", etc. are all determined principally by the balancing of opposing interests. Of course, other factors such as the current state of technology will determine such things as "adequate computer performance". The general wealth of the country will determine what is "adequate" for such things as dress, housing, and food".

    In principle we can set up an equation for the dynamics of the equilibrium point:

    F(x) = O(x)

    where F(x) is the function represents the level of desire for a particular amount, x, and O(x) represents the level of opposition to the x amount. The equilibrium will be the value of x where these two functions are equal. Here is a sketch of an idealized set of functions:

    The value of x at which F(x) and O(x) are equal will determine the breakpoint. Note that this is the equilibrium. During transients, the values may not be equal. In fact, on such issues as tax levels, it make take years for the equilibrium to be establish. Further, x may oscillate a number of times before it settles down.

    A specific example will help at this point. Let x be the sales tax rate and let F(x) be the "desirability function" by a government. In general, the more the better, so the curve slopes upward as x increases. Let O(x) represent the "opposition function" of the taxed citizens. Of course as x increases, the more it is opposed. Note that O(x) is a rather complex function and may vary with time. By use of propaganda the government can change the opposition function. Further, the citizens may feel more receptive to taxes if there is a serious problem such as poor quality schools. So, both F(x) and O(x) will generally vary over time which will cause the equilibrium to be constantly shifting. The functions are at least 3 dimensional, possibly higher, but we will limit them to 2 dimensions for simplicity.

    The problem of defining F() and O() in similar terms so that they can be compared and converting them to numerical values will be left as an exercise for the reader :-).

    With regard to political issues, the important point is not so much the mathematics but the recognition that a cutoff level is established by the equilibrium of these opposing forces; not by determining some idealized proper rate, as the government would like for you to believe. This is particularly evident in the income tax rate structure where the marginal rate varies with income. These marginal rates are not establishes by some sort of fairness doctrine handed down by God but are simply established by the appropriate equilibrium functions. If the voters feel that people who make more than $100,000 per year ought to be taxed more, then that will mean that the "opposition" curve for that level will be lower. That, then, will result in the equilibrium shifting to the right -- which means a higher tax rate for that income bracket.

    We might ought to consider for a moment how the government "feels" the opposition functions. Voting, of course, is one way to put the pressure on government. Consider the taxes and benefits associated with welfare. There are lots of poor people and others that simply don't find work to be all that exciting. These people can and do vote. Their desire for more welfare benefits, expressed through voting, is represented by a lowering of the opposition curve for the welfare portion of taxes. This would tend to increase welfare taxation and, hopefully, greater welfare for the masses. But there are others who vote also and they will be highly opposed to more welfare. Further, there is lobbying which can have a major impact on raising or lowering the welfare opposition curve. Another example is the funding of defence. The poor may not be very supportive, particularly if it means less welfare. But there are massive lobbyists who apply great pressure to increase defence spending. This, of course, results in a lowering of the "opposition to defence spending" curve with the result that defence spending increases. The point is that the opposition that the politicians feel come from many sources and the net result will vary from time to time as the mood of the country varies. Right now defence spending is down a bit and welfare spending is up greatly. That will change with time.

  8. Philosophical Ramifications of Vagueness
  9. I stated in the introduction that a breakpoint established for a vague function or definition cannot be defended. This has serious philosophical ramifications.

    An example will probably get the point across best. Consider a family in which the father has established a rule regarding promiscuousness for his teenage children (one for the son and a different one for the daughter :-) ). He says "Thou shalt not be promiscuous". His son says, "Well, Dad, define what you mean by 'promiscuous". Dad says, "Ok, let us say you are promiscuous if you have sex with more than 5 girls in a year. The son says, "Does it matter how many times I have sex with a particular girl?". Father says, "Well, I suppose it should, but let us forget about that as it clouds the issue". The son says, "Can you give me any good reason why 6 girls is a moral issue but 5 is not?" Father, "I cannot: I had to pick a point". Son, "Oh, so the point is purely arbitrary?". The father, now somewhat befuddled, concedes, " Yes, it is. I see only one solution: If you have sex with even one girl, you are promiscuous. All other definitions are purely arbitrary and I cannot make a case for any one value. I can only make a case for "none" or "any", So "any" is promiscuous. Sorry."

    From this example it should now be clear why you often hear in the news about a such and such "zero-tolerance" policy. Often the incident will involve some really absurb policy of some school. They have no choice, for there is no other distinct place to draw the line.

    Another example would be the speed limit, used as an example several times in these essays. Can you make a case that the speed limit should be exactly 65 mph? You cannot. How can you justify 65 mph over 65.1 mph? You say, "More people are killed at 65.1 than at 65". Well, of course, but there are also more people killed at 65 than at 64.9. So based on that logic, we would go to 64.9. But then the same logic would lead us to 64.8, and so on down to zero mph! There is just no way to justify 65 mph.

    From these examples, it should be clear that there is no way to justify any general rule! This is a very serious claim. If it is true, then all the generalities passed to children by parents, all the many books on ethics, all the sermons on proper living, maintenance handbooks for your car, most legal restrictions, medical and health advice, etc. have no rational justification since the "breakpoints" chosen cannot be justified.

    Consider the massive private and government activity involving "endangered species". Do you realize that there is no sound, logical, precise definition for this term? When is a species endangered? Shall we leave the definition vague? But didn't we decide that we would rather have definitive cutoff points (like 65 mph speed limits) rather than vague definition? So, pick a number. If it is more than one male and one female, both capable of reproduction, you will not be able to defend the value chosen. There is actually another problem with "endangered species" and that is "What is a distinct species?". Another vague definition! Yet people are being thrown in jail as a result of this very vague law.

    If my claim is valid, we are left with a world in which no absolute general rules can be made -- we must be willing to live with vagueness. Further, I would conclude that to try to remove vagueness is pointless and we must learn to live with arbitrary judgments of our fellow human beings. Scary.

    For more on the consequences of the arbitrary breakpoints in vague definitions, see my essay, "Some Consequences of the Arbitrary Breakpoint in Vague Functions".

  10. Associated Terminology and Other Observations
  11. There is an abundance of related observations and terminology associated with vagueness. In daily conversation, you hear such terms as "slippery slope", "gradualism", "incrementalism", etc. usually associated with some government activity or other groups trying to impose their will on the population. I will discuss a couple here.

    1. Slippery Slopes
    2. The concept of "slippery slope" is a recognition of the fact that a particular breakpoint on a vague function cannot be defended. The term is most often used in conjunction with methods of argumentation but it really has wider applicability. Basically, the idea is that for a continuous function, if we start with just a little bit and we agree that a little bit more cannot hurt, then there is no defence against moving the breakpoint to any point. For example, suppose that I convince my girlfriend that if she would let me massage her knee cap, surely no harm can come from it. And if she buys that, next I will get her concurrence that moving up the leg another eighth of inch cannot possibly cause any serious consequences. If she buys that, then the path is clear all the way to . . . -- well, you get the picture. This method of argument is often called, "letting the camel get its nose in the tent", but I am not sure that I should use that analogy here :-)

      For more on this subject, see Douglas Walton's book, Slippery Slope Arguments, Clarendon Press, 1992 (it is unfortunately already out of print).

    3. Incrementalism
    4. Incrementalism is what the government is applying when they gradually go from an income tax rate of 5 percent to 50 percent -- as only one example of many. If you have a vague function -- such as how much tax is "fair" -- that has no definable breakpoint, and you start with some arbitrary value, there is no defense in allowing a change in that value. Using the same argument that initially established the arbitrary value can usually be used for changing that value. The government says that you should not have more than 0.1% alcohol in your blood because -- by definition -- you are drunk and drunk is bad. If I accept that argument, then I cannot defend against a proposed change to 0.08% alcohol. Once all the laws are changed to "0.08%", then the same argument can be used to change to "0.06%".

      The government has used the technique of incrementalism to invade every aspect of our lives. Surely we could not argue against the government's right to not allow us to ingest harmful substances -- right? And if they can do that, should they not also determine when and if smoking is allowed? And if they can do that, prohibiting the eating of fatty foods surely is reasonable? In fact, are not alcohol, beer and wines harmful (in some amount)? Why stop with a prohibition against smoking?

      A recognition of the dangers of incrementalism causes us, as a society, to avoid many activities that might be very beneficial. For example, "assisted suicide" surely is a compassionate, rational action for a situation in which we have a person who has no hope of ever living again without terrible pain and suffering. But, the opposition to this idea says, if you allow that, then what is to stop the hospitals and the government from providing "assisted suicide" to anyone who is simply a big pain in the ass? A good point. I suspect that we detest the idea of using death-row inmates for medical testing more from the fear of incrementalism than compassion -- and rightly so. Obviously avoiding incrementalism comes at a dear price but it is something that is very wise for society to do -- given the history of humanity.

      A short discussion of the application of incrementalism to Civil Rights and Civil Law is given by Kelley D. Ross's essay, "The Corruption of Civil Rights and Civil Law".


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