The Problem of Verisimilitude

The Problem of Progress

Popper (1963) noted that there is no general agreement on the answers to two very basic questions:

(A) Can we specify what scientific progress consists of?

(B) Can we show that science has actually made progress?

One quick answer to (A) is that

(1) Science aims at true theories,

and that progress consists of the fulfillment of this aim. In answer to (B), we should add:

(2) Science has made progress in meeting this aim.

The problem now arises when we add a third plausible statement to the first two:

(3) Scientific theories have all been false.

In the history of planetary astronomy for example, Ptolemy’s geocentric theory is false, Copernicus’s version of the heliocentric theory is false, Kepler’s laws are false, Newtonian gravitational theory is false. It would be naive to suppose that Einstein’s general theory of relativity is true. This conflict is "the problem of progress."

The Problem of Verisimilitude

The famous problem of verisimilitude flows from this (Musgrave, unpublished):

Realists . . . seem forced to give up either their belief in progress or their belief in the falsehood of all extant scientific theory. I say ‘seemed forced’ because Popper is a realist who wants to give up neither of them. Popper has the radical idea that the conflict between (1), (2), and (3) is only an apparent one, that progress with respect to truth is possible through a succession of falsehoods because one false theory can be closer to the truth than another. In this way Popper discovered the (two-fold) problem of verisimilitude;

(A*) Can we explain how one theory can be closer to the truth, or has greater verisimilitude than another?

(B*) Can we show that scientific change has sometimes led to theories which are closer to the truth than their predecessors?

Note: Closeness to the truth is not the same as the probability of truth. A simple example shows this: A: The time on this stopped watch is correct to within one minute. B: The time of my watch (which is 2 minutes fast) is accurate to within one minute. Both hypotheses are false. But B is closer to the truth than A. But A is more probable than B, because the probability of A being true, though small, is non-zero. But the probability of my watch being accurate to within one minute, given what we know about my watch, is zero.

Popper's Definition of Verisimilitude

So, how should we define verisimilitude? Popper (1963) defined verisimilitude as follows:

DEFINITION: Theory A is closer to the truth than theory B if and only if (i) all the true consequences of B are true consequences of A, (ii) all the false consequences of A are consequences of B, and (iii) either and some true consequences of A are not consequences of B or some false consequences of B are not consequences of A.

Note: Popper’s definition allows that there are false theories A and B such that neither is closer to the truth than the other.

A fatal flaw in this definition was detected independently by Tichư (1974) and Miller (1974). They showed that, according to Popper’s definition, for any false theories A and B neither is closer to the truth than any other. This is a fatal flaw because the philosophical motivation behind Popper’s definition was to solve the problem of progress by showing that it is possible that some false theories are closer to the truth than other false theories.

The Weather Example

Suppose that the weather outside is hot (h) or cold (~h), rainy (r) or dry (~r), windy (w) or calm (~w). Suppose that the truth is that it’s hot, rainy, and windy (h & r & w).

Now consider two competing theories about the weather. A says that its cold, rainy and windy (~h &r & w) and B says that it’s cold, dry and calm (~h &~r & ~w). Both theories are false, but intuitively, one might think that A is closer to the truth than B. But Popper’s definition does not yield that result.

First, B has some true consequences that A does not have. For example, B implies that A is false, but A does not A does not imply that A is false.

Second, A has some false consequences that B does not have. For example, A implies A, which is false, while B does not imply A. More generally, it is possible to prove that no false theory is closer to the truth than any other false theory by Popper’s definition.

Therefore Popper’s definition cannot solve the problem of progress because it requires that some false theories are closer to the truth than other false theories.

Nobody noticed the defects in Popper’s definition for over 10 years, and so it is not a trivial result. The best way I know of "seeing" the result clearly is to represent the theories in terms of possible world diagrams.

Possible World Diagrams

Figure 1: Each point inside the rectangle represents a possible world. There are 8 kinds of possible worlds that can be described in terms of h, r, and w. Any proposition expressible in the language is represented by the set of possible worlds in which the proposition is true.
Note that the three simple propositions, h, r, and w are each true within four numbered regions. The conjunction of any two of them is true within two numbered regions, and the conjunction of three of them is true within a single numbered region.
weather2.gif (2377 bytes) Figure 2: The theories T, A, and B are represented by the set of possible worlds for which those propositions are true. Since they are conjunctions of three simple propositions, they are each represented by the single numbered regions 1, 2, and 8, respectively.
Logical deduction, or logical entailment, is represented by the subset relation, because, by definition, one statement entails another if and only if the truth of the first guarantees the truth of the second. Therefore, for example, T ̃ h, T ̃ r, and T ̃ w, as we would expect.
weather3.gif (3394 bytes) Figure 3: The proposition that the weather is minnesotan is represented by the shaded region. Notice that it is the combination of four numbered regions, 1, 4, 7, and 8. Each of h, r, and w is also represented by four numbered regions.
Remember that logical deduction, or logical entailment, is represented by the subset relation. Thus, B ̃ m because if the actual world were in region 8, then it would also be in the shaded region representing m. Note also that T ̃ m, but A does not entail m
weather4.gif (2755 bytes) Figure 4: Suppose that we also define a (the weather is arizonan) by the regions 1, 3, 6, and 8. If we reorganize the numbered regions according to whether they fall inside the areas h, m, and a, we are still able to express exactly the same propositions as before. That is, the set of simple propositions {h, m, a} forms an alternative language, which is just as powerful as the original language (based on {h, r, w}). In fact, they are entirely commensurable, because we can translate any statement in one language into a statement in the other.
weather5.gif (2713 bytes) Figure 5: In the new language, based on{h, m, a}, the proposition m is expressed in a very simple way. But its meaning is the same as before, because it is true in exactly the same set of possible worlds. This is evident when you compare the shaded region in this figure with the shaded region in Fig. 3.
weather6.gif (2408 bytes) Figure 6: Likewise, the theories T, A, and B may be re-expressed in the new language, without there being any meaning change. Notice that the representation is slightly different, since B is now next to T. Notice also that A entails m and a, while B entails neither of these propositions.
The truth or falsity of T, A, and B are the same in either language. We might also expect their closeness to the truth to be the same in either language, but this is not the case for Tichư’s definition of verisimilitude.

Harris’s Results

The intuitive idea behind Popper’s definition is that A is closer to the truth than B if A has more truths and fewer falsehoods in its set of deductive consequences than B. Why not think of this statement purely in terms of the number of true and false consequences of A and B rather than requiring that A includes all the true consequences of B. That is, why don’t we allow that A and B quite different sets of consequences.

Unfortunately, this does not solve the problem, as is clearly shown by our possible world diagrams. For in the weather example, A and B have exactly the same numbers of true consequences, and exactly the same number of false consequences. Harris (1974) generalized this result to show that the idea does not work.

Two Commensurable Languages

The second fact we learn from the possible world diagrams is that we can express any fact about the weather entirely in terms of the weather predicates {h, m, a}. To show this, it is sufficient to be able to defined r and w in terms of {h, m, a}. The following equivalencies establish the inter-translatability between {h, r, w} and {h, m, a}:

r if and only if either h & m or ~ h & ~m

w if and only if either h & a or ~ h & ~a

Tichư’s Definition of Verisimilitude

Tichư (1974) presented an alternative way of defining the verisimilitude ordering of propositions in a first order logical language. His idea is adequately illustrated in terms of the weather example. Tichư says that A is closer to the truth than B because A makes one mistake but gets two things right, while B is wrong on all three counts. One mistake is better than three mistakes, so A has greater verisimilitude or truthlikeness. Now it is possible for one false theory to be closer to the truth than another.

Language Variance?

Truth is not language variant. That is, Newton’s theory is true when expressed in English if and only if it is also true when expressed in French, or any other language. We should also expect that closeness to the truth is also language variant.

Unfortunately, Miller (1974) shows that this ordering may be reversed under simple changes of language. Suppose one says that the weather is Minnesotan (m) if and only if it is either hot and rainy or cold and dry, and Arizonan (a) if and only if it is either hot and windy or cold and calm. We may now re-express all theories according to whether they say the weather is hot, Minnesotan, and Arizonan. That is, we may replace the three weather predicates {h, r, w} with the set {h, m, a} and retain exactly the same expressive power. Any hypothesis that can be stated in terms of {h, r, w} can be restated in an equivalent form using {h, m, a}. The true theory is re-expressed as h & m & a. A is re-expressed as ~h & ~m & ~a, while B is re-expressed as ~h & m & a. Now the A makes three mistakes and B makes only one mistake. The order is reversed!

There are three possible responses to this objection:

  1. Abandon the thesis that one false theory can be closer to the truth than another. Give up on any objective sense of verisimilitude, and therefore give up on any objective notion of progress.
  2. Embrace the idea that verisimilitude ordering depends on our language, and therefore embrace a form of relativism. This is a weaker form of relativism than saying that truth is relative, but it is still as bad as option 1 because it still implies that there is no objective sense of progress.
  3. Explain why we should regard one particular representation as "privileged." That is, insist the set of questions {Is it hot? Is it rainy? Is it windy} is objectively the correct set of questions to ask of nature because the properties of being hot, being rainy, and being windy are the properties that really exist in nature. The weather properties of being Minnesotan and being Arizonan are concocted, and artificial, and do not capture the true "essence" of the world.

Miller’s Objection to Option 3: This third option, according to Miller, must appeal to the fallacious and outmoded doctrine of essentialism, which is the doctrine that says that one set of weather predicates role in capturing the essential or fundamental properties of the world. Not only is it controversial to assume that any particular set of predicate predicates has such a privileged ontological status, but it also raises a problem that we would have to know what this status is before we could measure verisimilitude. That is, problem (B), the task of saying whether science has actually made progress towards the truth, becomes a lot harder than it appears at first.

Cultural Chauvinism?

The third option avoids relativism. The verisimilitude ordering is the same no matter which language we use because when we translate the three questions {Is it hot? Is it rainy? Is it windy?} into the new language, they received exactly the same answers. Thus, Tichư’s proposal avoids relativism. But it does this at the price of introducing a kind of cultural chauvinism.

Suppose we come across an alien culture living in a valley where they grow two kinds of corn: Minnesotan corn and Arizonan corn. Minnesotan corn grows in the downstream part of the valley, while Arizonan corn grows in the upstream part of the valley. Their daily decisions do not concern whether to wear warm clothes, take an umbrella, or wear a windbreaker. Their decisions are more important. Each day they need to know whether they need to tend their Minnesotan corn or their Arizonan corn (or both). They cannot afford the time and energy to travel to the lower valley or the high valley when it is unnecessary. They need to tend to the Minnesotan corn if and only if the weather is Minnesotan, while they need to tend to the Arizonan corn if and only if the weather is Arizonan. Fortunately, their science, though not perfect, has progressed has progressed considerably in recent decades. Once they believed theory A, which was unsuccessful at predicting whether it was hot, whether it is Minnesotan, and whether it is Arizonan. Now theory A has been superceded by theory B, which is still bad at predicting temperature, it is now successful at predicting the two most important facets of the weather.

Should we not say that their science is progressive? Should we say that their achievement was not a genuine and objective scientific achievement simply because the questions that concerned them the most happened not to be about the "true objective essences" of the world? Wouldn’t that be an unsavory kind of cultural chauvinism?

The Fourth Solution

Verisimilitude is not an absolute. The is no such thing as the verisimilitude of a theory. Verisimilitude is relative to a particular set of questions and a formula for weighing the relative importance of those questions. There is not one truthlikeness relation, as solution (iii) above assumes. There are many truthlikenesses, depending on the questions asked. So, for example, one set of questions is Set 1 = {Is it hot? Is it rainy? Is it windy?}, where it is understood that the answers to each question are given equal weight. Another set of questions is Set 2 = {Is it hot? Is the weather Minnesotan? Is the weather Arizonan?}. There is verisimilitude relative to Set 1. Theory A is closer to the truth than theory B relative to Set 1. But relative to Set 2, theory B is closer to the truth than theory A.

What we have learnt

What we have learnt from the problem of verisimilitude is that we cannot measure the progress of science in terms of a single measure, which we call absolute verisimilitude. Rather, progress has many aspects, even if we restrict our attention to the question of progress with respect to epistemic goals. Progress is a multifaceted property of science. Why should this be surprising?

An Analogy: The IQ myth. It is a bit like measuring intelligence. We once held out the hope that there was a single number that would measure, called IQ, that would measure all facets of human intelligence. The hope was a high IQ score would correctly predict high math skills, high language skill, and high analytical skills simultaneously. It is an empirical fact that this aptitudes do not go together. People with high math skills can have poor language skills, and vice versa. People with poor language skills can have extremely good analytical skills, and so on. So, we now measure intelligence in a multi-dimensional way, by recording independent scores for these aptitudes (even that is limited measure of course). There is no attempt to summarize the three scores in a single score (an average say). Epistemic progress in science must go the same way. In the weather example, there are at least two scores. One with respect to Question Set 1 and the other with respect to Question Set 2. For some purposes we may be more interested in one as opposed to the other, but both scores are objective measures of different kinds of progress.

Final Remark: The weather example is a bit of an odd example, because we made Question Set 1 appear to be somewhat arbitrary by showing that a different question set may look natural in a different language. But in practice, our research interests are defined in terms of a single language. In the case of competing theories, there exists an observation language that is neutral between them (it may be theory-laden, but it is not laden with the theories in question). For example, in planetary astronomy, we could define a question set along the following lines: {Where is planet 1 at time 1? Where is planet 1 at time 2? Where is planet 2 at time 1? Where is planet 2 at time 2? …}. There is an objective measure of verisimilitude relative to this set of questions.