Results

Objections to Stove

There are two ways to object to Stove’s argument—one might deny that our sample is random, and one might deny that the desired conclusion follows even if it is.

First, Stove’s argument assumes that our sample is selected at random.

But why should we think that our samples are selected at random? Why should we think that each member of the population has an equal probability of being in it? This assumption would require a Principle of Indifference (5.5),

THE PROBLEM OF INDUCTION

THE PROBLEM OF INDUCTION ⁸⁵⁸⁵

which would itself be in need of justification.⁷ (This is yet another appearance of the argument at the beginning of 6.3.)

But let’s grant that our samples are randomly selected. Now some inductive inferences are justified, but we have to take care about exactly which ones.⁸ The evidence justifies that all emeralds are green, but still does not justify the hypothesis that the next emerald is green. We can see this by being more specific about the hypotheses. If we take into account the order in which the emeralds are observed, there are not three hypotheses, but four:

GG = Both emeralds are green

GW = The first emerald sampled is green and the other is white WG = The first emerald sampled is white and the other is green WW = Both emeralds are white

P(E|GG) = 1 P(E|GW) = 1 P(E|WG) = 0 P(E|WW ) = 0

E justifies that the next emerald is green if and only if P(E|GG or WG) > P(E).

But we’ve been given no reason to think this inequality holds. Indeed the evidence refutes WG, and it justifies GW just as strongly as it justifies GG, as P(E|GG) = P(E|GW) = 1, and GW says that the next emerald is white!

This can be seen with a picture. The probability that the next emerald is green is represented by the size of the top left and bottom right boxes, GG and WG. E eliminates the bottom row. GG and GW both increase in probability, and remain in the same proportions. Intuitively, why shouldn’t we think that the observed emeralds are one color (green) and the unobserved emeralds another (white)?

Stove might reply that in the long run, a random sample of green emeralds is more likely to come from a population of green emeralds than non-green

Uniform distribution Non-uniform distribution

E

–E

GG GW

WW WG

WW WG

Venn diagram of green and white emeralds.

A CRITICAL

A CRITICAL INTRODUCTION TO FORMAL EPISTEMOLOGYINTRODUCTION TO FORMAL EPISTEMOLOGY 86

86

emeralds. This is true but irrelevant. It isn’t relevant to appeal to the long run because at any given time we only have a finite number of observed emeralds in our sample. However many green emeralds are in our sample, we are interested in whether the unobserved emeralds are green. And the following hypotheses will be equally supported by the evidence:

(a)

(a) There are many green emeralds in my sample, and the rest are green.

(b)

(b) There are many green emeralds in my sample, and the rest are white.

So we may have reason to believe:

Principle of Uniformity-Sampling : Samples resemble the (whole) population.

But this falls short of what we need:

Principle of Uniformity-Sampling : Samples resemble the rest of the population.

The moral is that even if we assume we have random selection, and fix the values of P(E|H), this isn’t enough to justify reasonable beliefs about the unobserved—they also depend on the initial value of P(H).

6.5 Se

6.5 Semantic justificati mantic justification on

An argument that P(H) should have a particular initial value would require an a priori argument, that is, an argument that is independent of any evidence.

Specifically, we need a reasonably high a priori probability that the future will resemble the past. Could there be an a priori argument that nature is uniform?

(Here we return to the question we set aside in 5.3.)

There are two main candidate sources of a priori justification. The first is that we have some faculty that allows us to “see” (metaphorically) that some sentences are probably true. This view is associated with Immanuel Kant, who argued that we have intuitions that allow us a priori justification of various areas, notably mathematics and geometry. For example, Kant (1786/2004) argued that we have a priori justification to believe Newton’s theories.⁹ This is a remarkable claim— it says that we can acquire justification for beliefs about the world just by reflection, with no need for observation.

Thus, it denies empiricism, which says that experience is the only source of justification (about the relevant subject).

Kant’s theory was challenged by Einstein’s theory of relativity, which showed that Newton’s theories are false. Far from it being possible to deduce

THE PROBLEM OF INDUCTION

THE PROBLEM OF INDUCTION ⁸⁷⁸⁷

Newton’s theories a priori , they turned out to be not even true, and faculties of intuition have never really recovered from this blow.¹⁰

The second source of a priori justification is that some sentences can be justified in virtue of understanding the meanings of the words they contain.

Call these sentences analytic .

Analytic sentence = A sentence which can be justified in virtue of understanding the meanings of the words it contains.

We now look at Peter Strawson’s (1952) application of this second source to the problem of induction.

6.5.1 Semantic solution 6.5.1 Semantic solution

Let’s first consider some examples of sentences that can be justified in virtue of understanding the meanings of the words they contain. The easiest cases are when we invent words that are defined in terms of other words. Suppose my nephew invents a new word, “yego,” which means “yellow piece of lego.”

Then it is a priori that if something is yego then it is yellow. And there are plenty of terms in English that function in a similar way. For example, “vixen”

means “female fox,” so it is a priori that if something is a vixen then it is a fox. And it is plausible that understanding meanings is sufficient for justifying belief even in cases of non-made-up words, for example, if something is red, then it is colored.

Now let’s apply this to induction. The idea is that the Principle of Uniformity can be justified a priori , in virtue of its meaning. An example might be: “if someone believes the sun rose yesterday then they have justification to believe it will rise tomorrow.” Could that sentence be analytic?

Strawson (1952, pp. 256–7) argues that it is. Strawson thought that part of the meaning of “the sun rose yesterday” is “there is justification to believe that the sun will rise tomorrow.” Think about this strange claim before reading on. As it stands it doesn’t look very plausible, but Strawson gives an ingenious argument.

He first claims that it is analytic that if you have observed instances of the sun rising, then you have evidence that the sun will rise tomorrow. He then claims that it is analytic that if you have evidence that the sun will rise tomorrow, you have justification to believe the sun will rise tomorrow. Thus the argument runs:

1

1 S has observed instances of the sun rising 2

2 S has evidence that the sun will rise tomorrow 3

3 Therefore S has justification to believe that the sun will rise tomorrow.

A CRITICAL

A CRITICAL INTRODUCTION TO FORMAL EPISTEMOLOGYINTRODUCTION TO FORMAL EPISTEMOLOGY 88

88

Each step is supposed to follow analytically from the last. Schematically, where X is a type of event, the argument has the form:

1

1 S has observed instances of X 2

2 S has evidence that X will occur in the future 3

3 Therefore S has justification to believe X will occur in the future.

We can put the argument in terms of which inferences are licensed. For example, the meaning of “vixen” licenses the inference:

“Foxy is a vixen”^→ “Foxy is a female fox”

Similarly, Strawson is arguing that the meaning of “evidence” licenses both of the following inferences:

“Past instances of X”^→ “Evidence for future X”^→ “Justification to believe in future X”

If Strawson is right, we have a priori justification for the inductive principles we need to solve the problem of induction.

6.5.2 First objection to the semantic solution:

6.5.2 First objection to the semantic solution:

Implausibility Implausibility

The first objection returns to the initial implausibility of the position. Surely

“the sun rose yesterday” is only about what the sun did yesterday. But according to Strawson, it is also partly about what the sun will do tomorrow!

Strawson’s theory says that part of the meaning of “the sun rose yesterday”

is that the sun will probably rise tomorrow. So according to Strawson, words have meanings we didn’t expect. And one might object that our words simply do not mean what Strawson’s theory says they mean.

6.5.3 Second objection to the semantic solution:

6.5.3 Second objection to the semantic solution: