There are many steps we take to plant a tree. When we plant a tree we roughly go through the same steps. Now this process has certain characteristics and they will be different from most of the characteristics of, say, making a kite. To plant a tree we have to dig a hole and put the seed in. No one will expect that when we make a kite we will have to dig a hole and then put some kind of seed in. The two processes are different. There is no reason why they should have the same characteristics.
True, a tree grown large gives us joy, so does a kite finished; still this will induce no one to think that because of this the two different processes will have the same characteristics or roughly the same characteristics. That the two processes lead to results that we would sometimes describe using the same terms, such as 'joy-giving', is not a good reason for expecting that this will happen.
What characteristics a process will have is an empirical matter not subject to our wishes. We can dig a hole and put some kind of seed in but by doing this we will not produce a kite however hard we wish. We can change the size of the hole and vary the number of seeds; still this will not help. If we want to make a kite we will have to find out empirically what process or processes will produce kites. This may require a certain amount of trial and error but this is how we usually discover or invent this kind processes.
In this paper I draw attention to two processes and their two different sets of characteristics. The two processes are 1) (valid) deductive reasoning, and 2)following clues. The latter I also call the theseological process since following clues is like Theseus following his ball of thread (needless to say 'theseological' should not be confused with 'theological'). These two processes are different and therefore most of their characteristics will be different.
Why am I interested in contrasting these two processes in this way? This interest has been with me for some times and comes about because I have noticed something strange, something most of us do at least sometimes, myself included. We know that the two processes—(valid) deductive reasoning and following clues—are different. By following clues we can discover things we originally did not know. This is certainly different from deductive reasoning. Deductive reasoning can only tell us what is true if something else is true. No one can fail to see this difference; yet we all expect at one time or another that the characteristics of these two processes are close to each other. How close? That is, how big is this mistake that we frequently make? This can be measured by how surprised readers will be after they have finished reading this paper.
All lawyers are good athletes.
John Lennon is a lawyer.
Therefore John Lennon is a good athlete.
To explain how deductive reasoning works we usually start with an example resembling the above. We point out that although it is false that all lawyers are good athletes, and also false that John Lennon is a lawyer, it would be true that John Lennon is a good athlete if the two premises were true. Now because this is a simple example most people after a time will see that this is indeed the case, affording us an opportunity to explain how deductive reasoning works.
In my writings and lectures I have been trying to explain why by following clues we can discover things that we originally did not know. What example is there that I can use for this purpose?
I can find nothing that is in any way comparable to the John Lennon example above. Nothing that simple. And this is understandable, for if we want to explain to a person why by following clues we can discover things we originally did not know the example we use will really have to be one in which that person by herself can discover by following clues something that that person originally did not know. We cannot tell that person that she has to take our word for it that she has made such a discovery. She has to see it for herself. She has to be able to say to herself what I have found is really true; I don't need any one to tell me. If we do not proceed in this fashion the person will never understand why by following clues we can discover things we originally did not know.
What example can I use? Can we ever make a discovery all by ourselves in two or three lines? Or even ten or twenty? I can find no such. The simplest I have found is to ask my audiences to solve a simple cipher, such as the following:
SBR SBCTU DBCKERVS FCGG WTTCXR SFH FRRJD YTHE SHUWI
Now the interesting thing is, with a simple cipher like this, after it is solved most people can tell on their own that they have found the hidden message, which message they originally did not know.
Deductive reasoning can be illustrated with a simple example. To show what it is like to make discoveries by following clues the simplest example I can find is already quite complicated.
Of course there is no reason why if deductive reasoning can be illustrated by a simple example the clue-following process can be also. If we expect the latter we have no one to blame but ourselves. The two processes are different. What goes for one does not have to go for the other.
In deductive reasoning there are always premises and conclusions. If we have premises only and no conclusion at all we don't yet have deductive reasoning. Premises and conclusions are the main ingredients in deductive reasoning.
In the theseological process we of course need clues. In response to clues we form hypotheses. In our cipher example above SB occurs twice, at the beginning of the first two words. This could be a clue, in response to which we could form two hypotheses: 1)S stands for T, and 2)B stands for H.
Where do we find the clues? Where do they reside?
Clues reside in the evidence. They are those parts of the evidence which give us some inkling of what is behind the evidence. In our cipher example the cryptogram (the message in its cryptic form) is the evidence. In it we find clues. With the aid of these clues we arrive at a set of hypotheses telling us what each of the letters in the cryptogram stands for, so that eventually we will be able to decipher the evidecne / cryptogram and find out what means.
In following clues we are interested in what the evidence means. We think there is something behind this evidence and we want to get at this something. In our cipher example the evidence is a string of symbols; we want to know what message is behind it. In solving a crime part of the evidence could be a weapon; the question we have is whether it is the murder weapon. Scientists observe spots of light in the night sky. What are they? they ask; perhaps they are material bodies obeying the same laws that billiard balls obey?
In the theseological process there are three main ingredients. They are 1)evidence (which contains clues, if there are any), 2)hypotheses (proposed in answer to clues), and 3)the meaning of evidence (obtained by applying hypotheses to evidence). We do not find these ingredients in deductive reasoning. The ingredients in deductive reasoning are premises and conclusions. The ingredients in the clue-following or theseological process are different from the ingredients in deductive reasoning. This is as it should be since the two processes are different.
In deductive reasoning we distinguish between premises and conclusions. Premises are propositions, so are conclusions.
In following clues we have to interpret evidence. Suppose that at the scene of a crime we find a credit card receipt. Basing ourselves on theories we have advanced we say this receipt is left behind by the murderer. That this receipt belongs to the murderer is our interpretation of this particular piece of evidence. This interpretation is a proposition, capable of being true or false. If the receipt indeed belongs to the murderer, it is true; if not, it is false.
But the receipt itself is not a proposition. It is a physical object. It will be picked up, stored away carefully, and presented one day at court as evidence.
In solving a crime detectives often have to rely on the testimonies of eye-witnesses. These testimonies are a part of the evidence, and they are made up of propositions—for example, ‘I heard a gunshot at about 12:45 a.m..’ So the evidence in an investigation sometimes contains propositions. But it can also contain physical objects, as in the case of the credit card receipt.
In following clues we look for evidence and we interpret it, basing ourselves on theories. But not all these three ingredients in an investigation—evidence, its interpretation and theory—are always propositional in form. Interpretation of evidence is propositional; so is theory; but evidence is often non-propositional.
In deductive reasoning the main ingredients—premises and conclusions—are all propositional. Should all the main ingredients in an investigation be propositional as well, as in deductive reasoning? There is no reason why they should. The theseological process on the one hand, and deductive reasoning on the other, are two different process. As such they have different characteristics.
In deductive reasoning sometimes it takes many steps to reach the conclusion. But it is not unusual to have a deductive argument which consists of just one single step. The argument about John Lennon is one such. In just one step we have concluded that he is a good athlete.
In our cipher example, if we manage to find only one single clue and this one single clue does not lead to any others it, this single, isolated clue, is next to useless. We may think it means this, we may think it means that, but we will have no way of coming to a decision: there are far too many possibilities. Indeed the clue may not even be a clue at all.
We cannot make discoveries with just one single clue. We need more than one, usually quite a number. Clues are the characteristics structures, disguised. The English language has a structure. It is a characteristic of the English language that in it many words begin with TH. This is the reason why SB occurring at the beginning of two words could be a clue. Now when we follow clues what we are trying to do is to re-construct those structures that give rise to them. In trying to decipher the SBR cryptogram we are trying to reconstruct those English words and phrases that make up the secret message. Now to reconstruct a structure the more characteristics of this structure we know the easier this reconstruction. If we know only one characteristic we will not be able to distinguish the structure we want from some neighbouring structure. If the only thing I know about the word I want to decipher is that it begins with A I will not be able to distinguish it from all the other words that begin with A. This is why we cannot make discoveries with just one single clue. And since in following clues we interpret the clues one by one we necessarily have to take many steps. We cannot make a discovery in one single step.
In deductive reasoning it is possible to arrive at a conclusion in just one single step; there is no requirement that it be reached only by taking many. In following clues, if we are to arrive at results to which we can attached any confidence we need many steps; we cannot do it in just one. The theseological process is different from deductive reasoning.
In deductive reasoning, if the premises are true and the reasoning valid the conclusions are, we can say without hesitation, necessarily true. In deductive reasoning we can give definite verdicts; we can tell clearly and unambiguously that we have reached true conclusions; it is just a matter of making sure we have done what is required.
When following clues we determine whether results are right by seeing whether they are prompted by clues and whether they lead to more and more new clues. At the beginning of a theseological process we usually will not have many clues.1 If we do not develop new clues we will not have enough to determine that we have successfully reconstructed those structures we want to know. Clues are the characteristics of structures. If the number of characteristics we know is too few we cannot reconstruct these structures. And of course if we do not follow clues at all there is no chance that our results will be anywhere near the truth.
So in the theseological process we have to follow clues and develop new clues from old. But to develop new clues we have to have interpreted the old clues correctly. Old clues incorrectly deciphered will not lead to new clues. Take as illustration a simple example. If we have a three-letter word and we decipher the first two letters any old way we will not thereby have a clue to the third. For example suppose we have deciphered the first two letters as KJ. This can give us no idea what the third letter is. KJ? is not a clue. But if we have deciphered the first two letters correctly as TW we will then have a good idea what the third letter could be from the clue TW?.
In the theseological process to develop new clues we have to have interpreted the old clues correctly. However, sometimes old clues wrongly interpreted will lead to a false clue. But false clues peter out; which is to say they cannot lead to more and more new clues. This means if a clue leads to more and more new clues it is likely to have been correctly interpreted. The more generations of new clues it leads to, the more likely this is the case. Since this is the way in which we evaluate results when following clues, that is, by seeing whether they lead to more and more new clues we can never be absolutely, definitively certain of these results; we can only be more and more certain of them.
In deductive reasoning when the premises are true and the reasoning valid the conclusions are necessarily true. While this is the case with deductive reasoning the situation is quite different when following clues. Verdicts arrived at in the theseological process are always uncertain.
In deductive reasoning there are rules telling us when arguments are valid and when not. These rules are iron-clad, allowing no exceptions. Modus Ponens, for example, is always valid; affirming the consequent, on the other hand, is always a fallacy.
If we want to reconstruct hidden structures—such as secret messages, or the structure of the universe—we have to follow clues and develop new clues from old. If we do not follow clues; if we only make wild guesses; we have no hope of success. Similarly if we do not develop new clues from old (as we have already explained). So when we are following clues it is useful to remind ourselves of these two important tasks: 1)follow clues (that is, don't make wild guesses); and 2)develop new clues from old. Now (1) can be disobeyed occasionally—for example, when we run out of clues and are desperate. Of course when we disobey (1) and start making wild guesses the chance that we will hit upon a right hypothesis is extremely small. But once in a blue moon we are that lucky—and luck, as is well known, has a place in making discoveries.2
In deductive reasoning the rules are iron-clad; there are no exceptions. When following clues we can occasionally depart from what we normally should do and still achieve results.
Deductive reasoning is ‘mechanical’: it is governed by rules so clear and unambiguous that even machines can follow. And when we follow these rules, so long as we start with premises known to be true, the conclusions we draw from them will be true. We know they will be true without direct verification. Suppose that the conclusion is that Socrates is mortal. If we know that this conclusion is drawn from true premises, we know then that he is indeed mortal. We can be so certain that we do not have to be present, pin in hand, at his funeral.
The theseological process is not mechanical. In this process we have to follow clues. But to follow clues we first have to detect them and then find out what they mean. Both detecting clues and finding out what they mean require ingenuity, and although machines sometimes can be brought in to help they cannot be done completely mechanically.
However, although the theseological process is not mechanical, if we manage to follow clues and develop new clues from old we will uncover knowledge we originally did not possess. When we have, we can tell that we have while remaining in the dark; there is no need for direct verification either. To tell that we have broken a cipher we do not have to go cap in hand to our enemies and beg them to check our results for us. We can tell that our results are (likely to be) correct on our own. We can tell because we have done what is required, even though what we have done is not mechanical.
In deductive reasoning even when the premises are false an infinite number of conclusions can be drawn. That the premises are false does not prevent us drawing from them conclusion after conclusion. This is to say in deductive reasoning we can be led down the garden path and not know it.
In following clues, if we make serious mistakes at any point we will not be able to make much progress afterwards, if any.3 For in this process we have to follow clues and develop new clues from old. If we have made mistakes, serious mistakes, we cannot develop new clues. Without new clues progress will be impossible once the old clues are used up. But this means in following clues we cannot be led down the garden path, at least, not too far down. Once serious mistakes are made the 'system' after a point will shut down. It will not come to life again until the mistakes are corrected.
Most deductive reasoning is carried out intuitively, that is, we do not check every single step we take against some explicitly stated rule before we move on to the next. Of course even when reasoning this way we will try to be as careful as we can. But sometimes despite the care we take we still make mistakes. And when we do there is no mechanism internal to the process which on its own will alert us to the presence of this kind of mistakes, that is, mistakes in reasoning. The only exception is the appearance of contradictions. When contradictions appear we know mistakes have been made. If we start only with true premises and our reasoning is valid there should not be any contradictions anywhere. But if contradictions do not appear; that is, if we do not notice them; it is still possible that mistakes in reasoning have occurred, except that we are not aware of them.
In following clues when mistakes are made, serious mistakes, the process will come to a halt (because clues will have dried up). Knowing this when we find progress becoming more and more difficult, or even impossible, we suspect the presence of serious mistakes. This is to say in the theseological process there is a mechanism internal to the process itself which can alert us to the possible presence of serious mistakes. When serious mistakes have occurred a red flag will come up, the red flag being difficulty in making progress or lack of progress altogether.
In deductive reasoning, if we want to determine whether we have made mistakes in reasoning we can check our steps against the rules of logic. But if we have made mistakes concerning the premises; if we have let falsehood crept in in the premises while trying to arrive at true conclusions; deductive reasoning affords no remedy. Deductive reasoning leads to true conclusions only if we start with true premises. If the premises are false; if there are errors in them; deductive reasoning cannot correct these premises for us. It is not its business. It is not the business of deductive reasoning to correct premises. It is our business. It is our business to make sure the premises are true if we want the conclusion to be true.
In following clues we can correct our own mistakes. This is one of the interesting characteristics of the theseological process. To people not familiar with this process it is hard to believe. We follow clues when we are in the dark as to what the truth is. But if we do not know the truth, equally we should not be able to tell what departs from it. And if we do not even know what departs from it, that is, what constitutes a mistake, how can we correct mistakes?!
In following clues when the mistakes are minor they can be corrected from context as we find out more. If we have succeeded in deciphering large parts of a long message context will tell us where the mistakes are and how to correct them. In this respect correcting minor mistakes during the theseological process is not unlike correcting typographical errors during proof-reading.
What about major errors? How do we correct major departures from the truth when we don't know what the truth is?
First we have to know that major errors have occurred; at least, we have to be able to suspect that they might have occurred. But this we can do. In following clues, if serious mistakes have been made progress in the investigation is impossible.
How do we know these major mistakes have been corrected?
We know when we are able to move forward again. So this is what we do when we want to correct major mistakes in the theseological processes: We depend on trial and error. We re-do some of the steps we have taken and through trial and error see if we can resume our advance. If we can, we will have both located the mistakes and corrected them.
We can correct mistakes without knowing what the truth is in the theseological process. Inasmuch as this is the case we say that this process is self-correcting. Deductive reasoning is not self-correcting: by itself it cannot correct mistakes in the premises.
In deductive reasoning we cannot afford to make any mistakes. If we want the conclusions to be true the premises have to be true and the reasoning valid. If a mistake should occur, whether it be in the premises or in the reasoning, the conclusions can no longer be trusted. Deductive reasoning is delicate to the extreme: it works only when everything is exactly right.4
In following clues we can detect mistakes and correct them on our own. Since this is the case in the theseological process we do not have to get everything exactly right the first time. In practice we never can. But this, of course, has not prevented us from finding out things that we originally did not know. The theseological process is robust. Like the family car it can take a lot of abuse and still works.
Robust machines can work even in non-ideal situations. For delicate machines to work everything has to be exactly right (or very close to being exactly right). Compare for this purpose a racing car and the family sedan. But while robust machines can work in non-ideal situations the results they produce tend to be mediocre. The theseological process is robust and the results it produces are not perfect. But it has one saving grace: they, the results, are improvable, at least sometimes. As the context of the investigation enlarges; as more is known; the results obtained can be fine-tuned.5
In deductive reasoning we want to be sure that every step we take is right before we move on to the next. We do this because we know a false step anywhere will render all subsequent steps suspect.
In the theseological process we follow clues. But clues are often vague: they could have more than one meaning. When we meet with a vague clue, of the many possible interpretations how do we know which is correct? To answer this question we often have to wait: we have to see which interpretation will lead to new clues later on. This means in the theseological process it is normal to resort to trial and error. We try one interpretation of a clue first. If it does not lead to new clues we try another. As the saying goes in the clue-following process we sometimes have to ‘plunge ahead’, and retrace our steps later on if necessary.
In deductive reasoning it is bad policy to plunge ahead. In deductive reasoning we want to make sure every step we take is correct before moving on to the next.
In deductive reasoning we need true premises to arrive at true conclusions. If we want true conclusions we have to be careful where we start. We cannot start with false premises and still arrive at conclusions which we know will be true.
We use the theseological process to home in on the truth. To do this we do not have to start with truths already known, truths of which we are absolutely certain. In the theseological process we follow clues. To detect clues we have to make assumptions or rely on the results of earlier investigations. None of these we can be absolutely certain of. In practice they often turn out to be wrong. But this will not prevent us from uncovering knowledge we originally did not possess. For the theseological process is robust; it allows us to correct past mistakes. Of course the more mistakes we make the more difficult it is to correct them. Still the process is such that we do not need perfect starting points.
In deductive reasoning, because false premises can sometimes lead to true conclusions, when we see a true conclusion we cannot infer that it must have been derived from true premises. If we do so infer we will have committed the fallacy called affirming the consequent. The following is an example of a valid deductive argument with false premises and a true conclusion:
All lawyers are good musicians.
John Lennon is a lawyer.
Therefore John Lennon is a good musician.
In following clues, if we arrive at result after result then all the assumptions and hypotheses that lead up to these results are likely to be correct. The older they are, the more likely to be right they are.6 If they had been wrong they could not have produced clue after clue. Without clues there could not have been the results that were obtained. This is to say when following clues we use what comes later in the process to confirm what comes earlier.
But things are very different in deductive reasoning (as we should expect). In deductive reasoning conclusions are drawn from premises but in deductive reasoning we cannot use the truth of the conclusion as evidence for the truth of the premises.
In deductive reasoning the same term cannot have more than one meaning; the meaning of a term should remain exactly the same from beginning to end; we cannot change the meaning of a term midstream. In deductive reasoning we have to guard against ambiguity because ambiguity will lead to mistakes.
We often use approximations when following clues. For a clue can be looked upon as a small gap. For example in a ten-letter word in which nine of the letters are already known we have a small gap; this gap is a clue. Since we know nine of the letters already we can fill in this small gap left by the tenth. When following clues we do not find all the clues first and then answer them all at once. We find some first, answer them and then find more. In the process we often develop new clues, clues that would not have been there if we had not answer some of the clues first. Now an approximation is a small gap away from the truth. Since in an investigation we do not close all the gaps at once anyway there is no harm if we introduce a few more if by doing so we develop clues that we otherwise would not have. Indeed this will be to our advantage. With extra clues it is easier to advance. When we have closed more and more of the gaps even the ones we have introduced will be closed eventually because of the increased context. Thus we should not mind using approximations in investigations, so long as these approximations are not too far away from the truth.
Now this ability to profit from approximations applies to language as well. It is not hard to see that if we can decipher (through cryptanalysis) a message in standard English we should be able to decipher messages written in non-standard English also, even forms of non-standard English that we originally were not acquainted with. We do it this way: We first decipher the message on the assumption that it is in standard English. Since standard English approximates non-standard English this assumption—that the plaintext is in standard English, though false, should nevertheless enable us to decipher large parts of the message. Once this is done the rest can be deciphered by relying on context as clue.
Now some words in non-standard English will not have the same meaning as in standard English. When we decipher a message by first assuming it is in standard English and then realising it is not, our understanding of some of the words in the message will have changed. Now there is nothing wrong in this. Indeed this is one of the merits of the theseological process: it allows us to correct our own mistakes.
In deductive reasoning we cannot change the meaning of words midstream. When following clues we often do, with profit.
In deductive reasoning we fix on a language at the beginning. The language should not change as the reasoning continues.
When we follow clues we can change our language mid-stream, as we have shown in the last section: we can decipher a message in a language we do not know if this language approximates to one we already know. This is to say in the cryptanalytic process we can start with a language we already know and then switch to one that we do not but which approximates the one that we do. Now since we can do this, starting with one language, by a long series of approximations, we can come to be able to decipher languages radically different from the one we started with. This will take time of course, and much effort, but it is, both in theory and in practice, possible.
In many investigations we cannot tell in advance what the appropriate language is for describing the matter under investigation. The discovery of the appropriate language in these cases will have to be part of the investigation and may have to be made through a series of approximations.
Of course to start the investigation we have to use some language. Of course, too, we cannot use just any language. If we are eventually to arrive at a language suitable for the matter under investigation the language we start with has to reveal sufficient clues for new clues to develop.
In following clues we can change language mid-stream. When we find that we have to do so what results is a language more suited to the matter under investigation. In other words in the theseological process it is often the case that we improve on language as we proceed; we do not have to be in possession of the perfect language right from the start.
In deductive reasoning we fix our language at the beginning; we do not change it in the course of the reasoning. When we follow clues we should not fix our language in advance; we should allow it to change, to develop, to improve as needed.
In deductive reasoning, if everything goes well, with every step we take we should be closer to the conclusion. There is no reason to turn back or go around in circles, unless we make mistakes.
In following clues we depend on feedback loops. They help us in this difficult process in many ways. Here are some examples:
In following clues we can make use of approximations to help us advance (as we have earlier explained). With their aid we can find out more, after which we can turn back and bring the approximations closer to the truth. This kind of fine-tuning is therefore the result of a feedback loop. The feedback is positive: approximations make possible advance in the investigation; advance in the investigations makes possible improvements on the approximations.
In following clues we often need more evidence than we initially can find. To develop additional evidence we depend on (positive) feedback loops. We start with clues in the evidence we already have. By following these clues we discover more and more about the unknown. As we know more it will be easier for us to detect those clues that will lead us to the evidence we are looking for.
In following clues what we know at the beginning is usually very little and highly uncertain. But as we find out more and more what we know becomes more and more certain. So here again that there is a feedback loop. Because of this feedback loop a small amount of highly uncertain knowledge can in time become a large amount of not-so-uncertain knowledge.
In deductive reasoning we frown on circles. In the theseological process circles in the form of feedback loops are common and we welcome them. This should not surprise us. The two process are different. They each have their own characteristics.
In deductive reasoning one frequently finds that a complicated truth can be deduced from a few simple truths. This leads to the idea that perhaps from a few simple truths, or even from a single one, all other truths can be deduced. Deductive reasoning suggests the possibility that one could know everything once one knows a few simple things.
There are many kinds of ciphers and new ones are constantly being invented. There is no such thing as a universal cipher from which all other ciphers can be deduced.
There are ciphers which are unbreakable in theory (because they leave behind no clues) and there are ciphers which, though breakable in theory, are unbreakable in practice because breaking them requires too much time (say, in the order of thousands of years even when using the fastest computers). Ciphers which are unbreakable in practice now could be breakable in practice later on as technology advances. However, as technology advances, new ciphers can be designed which are unbreakable in practice using technology available at that time.
Those who follow clues are aware that there are things we do not at this moment know and there could be things we will never know. It is not likely that the day will come when we will know everything.
We often expect the two processes—deductive reasoning on the one hand and following clues on the other—to have characteristics close to each other. In fact their characteristics are very different, as we have seen. This is as it should be. The two processes are different. Different processes have different characteristics.
1 And usually not much evidence either.
2 How do we know whether a wild guess is right? We tell by seeing whether it leads to more and more new clues.
3 Minor mistakes will not hold up an investigation; they can be corrected as the investigation advances.
4 We often convey this point by comparing deductive reasoning to a chain. A chain is only as strong as its weakest link.
5 If the theseological process were delicate like deductive reasoning, we all should have been overcome by that extreme form of scepticism which leads to total inaction. For in looking for knowledge how do we know we have not made, and will never make, even a tiny mistake? But the theseological process, instead of being delicate, is robust; we could make mistakes and still, in time, gradually close in on the truth. We may not reach it in every case, and we can never be absolutely certain, but progress is better than inaction.
6 By ‘old’ we mean old in terms of the investigation, not in terms of time. If much has been discovered in a short time, a result old in terms of the investigation could be relatively new in terms of time. Also, we should be aware that sometimes an old result might not have participated in the production of new clues. Such a clue has a greater chance of being wrong than one that has.