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The famous problem of verisimilitude flows from this (Musgrave, unpublished):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

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 |

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. |

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. |

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:

- 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.
- 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.
- 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.

- Both of these two claims are true at the same time. They are not incompatible claims. There are two different verisimilitude ordering because there are two notions of verisimilitude.
- Neither answer to the two verisimilitude questions depends on the language. We can use either the standard language or the non-standard language, and we arrive at exactly the same answer to the two questions.
- What we lose is the idea that there is an absolute answer to the question: Is Einstein's theory closer to the truth than Newton's theory? There may exist a set of questions, relative to which Newton's theory happens to closer to the truth than Einstein's theory. We won't attach much importance to this fact, but it is an objective fact nevertheless.
- What we gain is the claim that Einstein's theory is closer to the truth than Newton's theory relative to the set of questions that were actually being asked by scientists. It may be true that this set of questions is subjective in the sense that it is defined by group subjects of science—the scientists. However, it is an objective fact that Newton's theory is closer to the truth than Newton's relative to these questions.

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.

- There is no sense in which the question set is biased towards one theory as opposed to the other. The answer is therefore interesting and informative answer to a question about progress.
- You will want a better scheme for scoring false answers than assumed in the weather example. If theory A and theory B give false answers to a specific question in the set, we don't want to score each answer as a zero. If one answer is numerically closer to the other, then we want to give it a higher score. This is what statistical measures of fit succeed in doing.
- There may be a different set of questions that are of greater interest for a particular
purpose. For example, NASA scientists may be interested in predicting the orbits of space
shuttles, and it may be that Einstein's theory does not receive a higher score than
Newton's for this restricted set of questions. This is important information to them,
because it means that they may choose to use Newton's theory to calculate orbits for
pragmatic reasons, without a great loss in the accuracy of their predictions
*in this domain*. This kind of decision, of when to use Einstein's equations, and when not, is commonplace in physics.