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FROM THE FINITE TO THE UNIVERSAL

A Puzzling Question

There is a lot of empirical evidence supporting the proposition 'All metals expand when heated';. For example on many, many occasions we have observed metals expanding when heated. Because of this large amount of evidence we take the proposition to be true (or highly likely to be true).

The proposition 'All metals expand when heated'; is called a universal proposition. A universal proposition mentions all things of a certain kind. The proposition 'All metals expand when heated'; is about all metals, not just some of them; it is about all pieces of metal, not just some pieces. Any piece of metal, any kind of metal, according to this proposition, will expand when heated. Now as we have said there is a great deal of evidence supporting this universal proposition; it is not a proposition that many would doubt. But however large the body of evidence supporting this proposition, and whatever the variety, this body of evidence is finite. Our experience is finite, even our collective experience. But the proposition is universal: it refers to all metals, not just this bit or that bit, nor this kind or that kind. Any bit of metal at all, whatever kind, will expand when heated. This is what the proposition says. The proposition does not just say, only those bits of metal that we have observed, expand when heated. Those bits we have observed, they expand when heated; but the proposition also says, even those bits that we have not observed expand when heated. This is what the 'all'; means; this is why we call the proposition universal.

The evidence which supports 'All metals expand when heated'; is finite. The proposition which draws support from this evidence is universal. How can finite evidence support a universal proposition? A universal proposition, even when supported by evidence, goes well beyond evidence. There are infinitely many bits of metal. Compared to the many bits of metal in the universe the bits we have observed is only a tiny fraction. How can a tiny fraction support such a large number?

When we have no evidence for a proposition we would not take the proposition to be true. When we have only a little evidence for a proposition we still would not take it to be true. If we are to take a proposition to be true, or likely to be true, we require a large amount of evidence. But the amount of evidence we have for the proposition 'All metals expand when heated' is never really large, whatever we may ordinarily think. 'Small' and 'large'—these are comparative terms. We ordinarily think the amount of evidence for the proposition 'All metals expand when heated' is large. It is large only when compared to the evidence we have for some other propositions, but compared to what the proposition says the amount of evidence is never large. It is not even small; it is tiny; it is close to zero! The amount of evidence for a universal proposition is always finite. The universal proposition is about the infinite. The finite compared to the infinite is next to nothing.

A Once-Popular Answer

It appears strange that finite evidence can support universal propositions. We have used metals expanding as an example but the same question can be raised in other cases. Much of the knowledge we have are represented by universal propositions; but the evidence we have for them can only be finite. How can finite evidence support universal propositions?

Not so long ago a popular answer to this question has it that finite evidence can support universal propositions because besides deductive logic there is also an ampliative logic (also called inductive), a logic that allows us to arrive at conclusions which contain more information than is present in the premises. If there is such a logic, we can use propositions describing the evidence as premises and see if these premises validly lead to some universal proposition as conclusion. If they do, the universal proposition receives logical support. For example we can use propositions reporting actual instances of metals expanding as premises and see if these premises validly lead to the conclusion that all metals expand when heated. If they do, then these premises logically support the conclusion that all metals expand when heated.

Why speak here of an ampliative logic? How is ampliative logic different from deductive logic?

A logic is a set of rules telling us when reasoning is valid. In deductive logic we have a set of rules telling us when deductive reasoning is valid. In valid deductive reasoning, there is never more information in the conclusion than is present in the premises. An ampliative logic is a set of rules telling us when ampliative reasoning is valid. In valid ampliative reasoning there is more information in the conclusion than is present in the premises. We need an ampliative logic if finite evidence is to provide logical support to universal propositions because deductive logic cannot explain why such support is possible. Indeed, according to deductive logic, if we reason from finite evidence to universal propositions our reasoning will necessarily be invalid. It will necessarily be invalid because the universal propositions go beyond the evidence (where the information they carry is concerned). In our example about metals expanding the universal proposition is about an infinite number of bits of metal whereas the evidence gives us information only about a finite number. In valid deductive reasoning the conclusion never goes beyond the premises. Since deductive logic cannot explain why finite evidence can support universal propositions, if we want such an explanation, so the argument goes, we need a different logic: an ampliative logic, a logic which allows an inference to be valid even when the conclusion goes beyond the premises.

Why does an inference have to be valid? What do we mean by validity? Clearly, this is important. If we are to explain why finite evidence can logically support universal propositions the evidence has to lead to the universal proposition validly; otherwise the universal proposition will not receive logical support. In reasoning premises do not validly lead to just any conclusion; they only validly lead to some and not to others. From a set of premises we can, if we want to, draw any conclusion, but the conclusion we draw will not necessarily be valid. From the premise that I have one dollar in my bank account I can, if I want to, draw the conclusion that I have one hundred dollars in my bank account but this conclusion will be invalid. To be valid my inference has to follow general rules which allow of no exceptions, rules of such a kind that whenever the premises are true the conclusion will be true. There is no general rule which allows me to draw from the premise that I have one dollar in my bank account the conclusion that I have one hundred dollars in my bank account. When we have general rules of the kind mentioned, simply from knowing that we have followed these rules and that our premises are true, we will know that the conclusion is true; nothing more is needed. Validity in reasoning has to do with general rules, rules detailing the forms of reasoning in which if the premises are true, the conclusion will also be true. It is for this reason that valid inference is useful.

It is important to notice that in valid reasoning it is not sufficient that the premises and conclusions happen to be true. Suppose it is true that I have one dollar in my bank account and also true that I have one hundred dollars in my wallet. From the premise 'I have one dollar in my bank account' I cannot validly infer the conclusion 'I have one hundred dollars in my wallet'. I cannot validly do this because there is no general rule that allows me to do so, even though in this case both premise and conclusion happen to be true. But it is not sufficient that they happen to be true. For the inference to be valid the truth of the conclusion has to be guaranteed to follow from the truth of the premises by some general rule.

Are there general rules guaranteeing that the truth of the conclusion will follow from the truth of the premises in ampliative reasoning? If there are such rules; if there is an ampliative logic; finite evidence can support universal propositions.

Until recently, as I have said, it is popular to think there is an ampliative logic; it is just that we do not know it yet. It, this ampliative logic, has yet to be found, according to this popular belief.

The search for an ampliative logic started a long time ago but despite much effort none has been discovered. As a result this once popular answer—that finite evidence supports universal propositions logically—is not so popular any more.

Is it likely that there is an ampliative logic still waiting to be found? That none has been found so far does not mean that one cannot be found later on.

It seems not likely that there should be an ampliative logic. If there were, then simply by reasoning—that is, by treating propositions in a certain way—we should arrive at knowledge not contained in the evidence. An ampliative logic is supposed to lead to conclusions containing more information than is present in the premises. If we know the premises to be true, by following such a logic we should arrive at knowledge that goes beyond the knowledge present in the premises. The excess results simply from this passage from premise to conclusion. The knowledge present in the premises could have resulted from experience but the excess present in the conclusion does not; it is simply produced by the ampliative logic. It is not likely that there is a logic this powerful.

In the absence of an ampliative logic, Karl Popper [1959, 1963] says we can never claim that a universal proposition is true, or close to being true, or even probably true. We may hope that it is true but whatever the amount of evidence there can be no support that it is. Evidence is always finite. Finite evidence, Popper says, cannot support universal propositions. Finite evidence can be used to refute a universal proposition but never support it. If there is one piece of metal which does not expand when heated the proposition that all metals expand when heated is false. But no amount of evidence about metals expanding can support the proposition that all metals expand when heated, according to Popper. According to Popper we do not know that all metals expand when heated. That all metals expand when heated is just an as-yet unrefuted theory.

Support other than logical

Is it true that if universal propositions can be supported by finite evidence, the only way is by using an ampliative logic? If it is, and if there is no ampliative logic, then universal propositions cannot be supported by finite evidence.

But how do we know that this is true? How do we know that the only kind of support for universal propositions is logical? Is it beyond imagination that finite evidence can support universal proposition in some other way? When a statue is about to fall down we can prop it up in many ways: we can use wood, or metal, or masonry, or balloons, …. For a statue there is no reason why support should be limited to just one way. If we use our imagination we can think of many. Why should the support provided by finite evidence to universal propositions be different? Why should this support be limited to just one way—which, moreover, does not work. Is our imagination that impoverished?

In fact do we have to depend on imagination to find a way by which finite evidence can support universal propositions? Have we not seen in the example in cryptanalysis we have been using in earlier essays in this book finite evidence supporting universal propositions?

Example Revisited

In our example we are given the following cryptogram to solve.


SBR SBCTU DBCKERVS FCGG WTTCXR SFH FRRJD YTHE SHUWI


What are some of the results we have obtained? They are, put slightly differently, the following.


All ciphertext letters S stand for cleartext letter T.

All ciphertext letters B stand for cleartext letter H.

All ciphertext letters R stand for cleartext letter E.

...


We see here that all these are universal propositions!

What evidence do we have for these universal propositions? What evidence do we have that supports them?

We have finite evidence! The evidence we have for arriving at these universal propositions—which we think are likely to be true—is the cryptogram that we have been given to solve, the cryptogram that begins with SBR and ends with SHUWI, a cryptogram that we can squeeze into a single line, a cryptogram that contains only 43 letters.

Why do we think the universal propositions we have obtained are likely to be true? Is it because of a non-existent logic?

Clearly no. We crack ciphers by following clues. When following clues we can tell that our results are likely to be right even though we cannot compare them to originals. We can tell because there is a method embedded in the clue-following process: the theseological method. The theseological method tells us that we should follow clues and not make wild guesses. Not only this but we should also develop new clues from old. When old clues lead to more and more new clues they, the old clues, are likely to have been correctly interpreted. The more (generations of) new clues they lead to, the more likely this is the case.

In cracking our cipher we arrive at a set of universal propositions by using the cryptogram as evidence, a cryptogram which is finite in length. From this example we see, finite evidence supports universal propositions not because of some non-existent logic, but because of a method: theseological method. The support that universal propositions receive from finite evidence is not logical but methodological.

Perhaps the theseological method is a kind of logic? This it cannot be. A logic tells us what conclusions to draw from what premises, but the theseological method is not about premises and conclusions at all; it is about clues and new clues.

Structures

Why is it that by following the theseological method we can arrive at universal propositions from finite evidence? How is it that the theseological method can bridge the gap between the finite and the universal? This gap is huge; it is hard to imagine how it can be filled. Compared to the infinite, the finite is close to nothing.

If we look carefully into our understanding of the theseological method this apparent mystery—that the finite can support the infinite—disappears. The first important thing to notice in this connection is that the universal propositions (all S is T, etc.) are not obtained from the evidence by some kind of inference; they are, instead, suggested by clues. The clues are in the evidence. In answer to these clues we put forward the universal propositions as hypotheses. When these hypotheses lead to new clues we know then they are likely to be correct. But when the hypotheses are correct, what do they tell us? They tell us how the cipher works, the cipher we have broken. Why is the cipher breakable? It is breakable because it has a structure, the reason why there are clues. What describes the structure of this cipher we have broken? The universal propositions that we have found to be correct!

So this is how finite evidence leads to universal propositions. It does so by providing those clues which eventually lead to the correct description of a structure by a set of universal propositions. There is no gap between the finite and the infinite. There is such a gap if we keep thinking that the universal propositions are inferred from propositions describing the evidence. The evidence is made up of particular things that we have experienced (for example, the SBR cryptogram). When you have a finite set of premises describing particular things and you infer from this set of premises a universal proposition, of course you will have a huge gap, the gap between the finite and the infinite. But when we follow the theseological method the universal propositions are not inferred from the evidence. The theseological method is about clues and new clues. The evidence contains clues which, being the characteristics of structures, allow us to recognise those structures hidden behind the evidence. Of course the more clues we have the easier it is to pin down these structures, but there is no requirement that we should have an infinite number of clues. Since there is no need for an infinite number of clues, there is no need for an infinite amount of evidence. Finite evidence can provide us with enough clues to be reasonably certain what the hidden structures are, as in our SBR example. Now some of the structures we uncover this way we describe by universal propositions. The cipher in our example we describe by a set of universal propositions. In this cipher, all S is T and all B is H. If we change any of these universal propositions, we will have a different cipher. Now it does not matter how many times a cipher is used; we will still describe it by using universal propositions. The cipher in our example could have been used only once or it could have been used a million times; in all cases all S is T. In all cases this universal proposition is hidden in the cryptogram generated by the cipher. Now all this we can understand; there is no mystery here, no gap between the finite and the infinite, no such gap because the infinite is not inferred from the finite.

Of course, if we revert to thinking that the universal propositions are inferred from the evidence, forgetting all that we have said about clues and structures and the way some structures are described, the gap between the finite and the infinite reappears. But why should we revert to that way of thinking? Is there any one who prefer insoluble mysteries to rational explanations?

Investigations vs. Generalisations

The way metals behave shows a pattern: whenever they are heated they expand. In addition to this bit of evidence, we also have the universal proposition 'All metals expand when heated', which proposition most people believe to be true. Now if we put these two pieces of information together, does it mean that whenever we see a pattern repeating itself we can form a generalisation from it? For example, suppose I have found from past records the stock market always rise the day before a full moon, can I infer that it will always rise the day before a full moon, not just in the past but also in the future?

From what we have done so far we can see that we cannot validly make such an inference. This inference is ampliative. As far as we know, there is no ampliative logic. It is true that in practice many people do make this kind of inference but that many do does not make this kind of inference valid. The proper way to ascertain the truth of a universal proposition like this one is not through ampliative reasoning but by carrying out an investigation, that is, by following clues. Those who understand this clue-following or theseological process will not generalise indiscriminately: they know that not all patterns that have repeated themselves in the past will do so indefinitely in the future. A bank robber casing a bank several days in a row will not keep on doing so for ever; one of these days he will stop and rob the bank. How a repeating pattern will behave depends on the reason behind the pattern. The way to catch hold of this reason is to follow clues, and to develop new clues from old. Using observations about metals expanding as a clue, and developing from it other clues, we have found there is a structural reason why metals always expand when heated: it has to do with the vibration of the molecules. We have no structural reasons to believe that the stock market will always rise the day before a full moon.

What is included in the evidence

When following the theseological method finite evidence can support universal propositions. We have the proposition 'All metals expand when heated' which most people believe to be true. What evidence is there that supports it? Is the only evidence the fact that in the past all metals expand when heated?

From what we have said in the last section we can see that the fact that in the past all metals expand when heated cannot be the only evidence supporting 'All metals expand when heated'. That a pattern has repeated itself in the past is not sufficient evidence that it will repeat itself in the future. Some patterns that has repeated itself in the past will repeat itself indefinitely but some will not: it depends on the reason for the pattern.

If the fact that metals expand when heated in the past is not sufficient evidence that all metals expand when heated, what other evidence is needed?

From what we have understood of the theseological process we can see that the evidence needed in addition to the pattern itself is evidence in the presence of which the hypothesis that all metals expand when heated has led to new clues. There is a large body of such evidence except that we are not in the habit of citing it. In practice there is no need. In most investigations we usually make use of a large number of third party results, for which we do not cite the evidence. As long as our investigation is able to advance, we take these results to be correct.

Not Isolated

When looking for knowledge of things hidden we should guard against looking at universal propositions in isolation. We describe the structure of a cipher by a set of universal propositions, not just a universal proposition. Similarly we describe the structures we find in this world by sets of universal propositions. The proposition 'All metals expand when heated'; is one member of one of these sets. When we go after hidden structures by following clues we do not go after a single, isolated universal proposition; we go after the sets by which these structures are described. Because the proposition 'All metals expand when heated'; is one of the universal propositions with which we describe a structure, it can be supported by finite evidence. This means also, as we have pointed out, the evidence supporting the proposition 'All metals expand when heated'; will have to include more than just those instances in which we observe metals expanding when heated. In our example in cryptanalysis we do not uncover the universal proposition S = T in isolation; we uncover other universal propositions as well. Together these universal propositions describe the cipher. But we use the same cryptogram as evidence. The same evidence that supports S = T supports other universal propositions at the same time, such as B = H and R = E and so on.

A plaintext attack is possible only sometimes

The evidence supporting the universal proposition that all metals expand when heated includes actual instances of pieces of metal expanding when heated. Without these actual instances, could we have arrived at the same universal proposition?

Some may think we could not. Without actually seeing pieces of metals expanding when heated, they will ask, how could we arrive at the proposition that all metals expand when heated?

From a (purely) logical standpoint, that we could arrive at the proposition that all metals expand when heated without seeing a single piece of metal expanding when heated, appears strange. But we could have by following clues. In our SBR example to uncover the proposition that all S is T we did not use as evidence even a single instance where we see an S being a T. We see S but not T. We see SB in the cryptogram. We think SB stands for TH. But we do not see TH. TH is in the plaintext which is hidden from view. In this case therefore, without seeing one single case of an S standing for a T, we arrive at the universal proposition that all S is T. To arrive at a universal proposition when following the theseological method we sometimes do not need to have experienced any of those instances included under the universal proposition.

Why is it that sometimes we can experience such instances and sometimes not? Are there situations in cryptanalysis in which we can experience instances included under the universal proposition?

There are. This kind of situation in cryptanalysis is called a plaintext attack. Let me explain.

In our SBR example we were required to find the cipher and the plaintext when given only the cryptogram. This of course makes the cryptanalysis difficult. If we had been given the plaintext and the cryptogram at the same time, and were asked merely to find the cipher, our task would have been a lot simpler. Solving a cipher when given both the cryptogram and plaintext is called a plaintext attack. If we had been able to employ a plaintext attack in solving our cryptogram, we would have seen many instances of Ss being Ts when we put the plaintext and the cryptogram side by side.


THE THIRD SHIPMENT WILL ARRIVE TWO WEEKS FROM TODAY

SBR SBCTU DBCKERVS FCGG WTTCXR SFH FRRJD YTHE SHUWI


Both in cryptanalysis and elsewhere, to arrive at a universal proposition by following clues we sometimes could have experienced instances included in the universal proposition. But this is only sometimes. At other times we could have arrived at a universal proposition without having experienced any instances included in the universal proposition. In science text books we find in plain view many universal propositions about things unobserved, for example, universal propositions about the electron. No one has ever seen an electron. We can see the effects of an electron but not the electron itself. How can such universal propositions be possible? There is no mystery from the theseological standpoint. In following clues we can legitimately arrive at universal propositions about the unobserved.

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