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is clearly not an inductively forceful argument. If the only propositions you knew relevant to the conclusion were PI and P2, then you would be wrong to think that C is probably true.

To say that all rodents have tails is the same as saying: 'Every rodent has a tail', or 'Any rodent has a tail', or 'No rodent has no tail'. To say that most rodents have tails is to say that more than half of all rodents have tails, or that there are more rodents with tails than there are without them. But what about 'Some rodents have tails'? What proportion of rodents must have tails for that to be true? In fact, that is a bad question. Such a statement does not tell you what proportion of rodents have tails. Consider this argument:

P1) Some patients who have been treated with X have developed liver disease.

C) If I am treated with X then I will develop liver disease.

This argument is not inductively forceful. PI could be true even if, say, three patients developed liver disease upon being treated with X, yet thousands were treated with X without developing liver disease.

That much should be fairly obvious. But the word 'some', as ordinarily used, can be rather tricky. Often, when we actually say something of the form 'Some A are B' (like PI), we say it when we believe that more than one A is B, but that not all A are B. Often when using the word this way, we say, 'Only some A are B'. Other times, we say it when we believe that more than one A is B, but do not know whether or not all A are B. For example, a medical researcher who discovers a few patients that have developed liver disease after being treated with X might use the sentence PI to announce the discovery - but this would be to leave open the possibility that perhaps all patients treated with X develop liver disease. He would not be ruling it out.

For the purpose of reconstructing arguments it is most convenient to assume the latter understanding of 'some', whereby 'some A are B' does not rule out that all A are B. It does not say 'Some, but not all A are B'. In this sense, if it is true that some A are B, it might also be true that all A are B.

Further, when using 'some' in our argument-reconstructions, we will take it to mean 'at least one'. That is: if only one A is B, then it will be true that some A are B. On this understanding of 'some', what 'Some rodents have tails' means is simply that it is not the case that no rodents have tails. Similarly, 'Some rodents do not have tails' means the same as 'Not all rodents have tails'. This departs slightly from what ordinary language typically suggests, but it is much more convenient for our purposes. If, when reconstructing an argument, it is clear that by 'some' the arguer means 'at least two', then we can simply make this explicit in the reconstruction, writing 'at least two'.

Inductive soundness

You can probably guess what this is, by analogy with the definition of deductive soundness:

To say that an argument is inductively sound is to say: it is inductively forceful and its premises are (actually) true.

The important thing to note here is that an inductively sound argument, unlike a deductively sound argument, may have a false conclusion. That possibility is precisely what is left open by the definition of inductive force - an inductively forceful argument is the case in which the truth of the premises would probably make the conclusion true, but not necessarily make it true. Look again at the last argument given concerning Fiona, who lives in Inverness. Suppose it is true that Fiona lives in Inverness, and that almost everyone there has some woollen garments. An argument with those two facts as premises, and that Fiona has at least one woollen garment as a conclusion, would be inductively sound - even if, as it happens, Fiona has no woollen garments.

Note that we have not said that one should always be convinced by inductively sound arguments. You could know that an argument is inductively sound, but also know, for independent reasons, that the conclusion is false (review the discussion of inductive soundness if this is surprising). We will return to this point in Chapter 6.

### Probability in the premises

So far in this chapter we have mostly considered premises expressing proportions, such as 'Most Zormons plays chess'. Proportions are expressed using quantifiers such as 'Most', '95 per cent of', and the like. But consider this case:

P1) If Napoleon is not ill then the French will attack. P2) Probably, Napoleon is not ill.

C1) (Probably) The French will attack.

This argument is inductively forceful, but it does not contain any quantifiers. The reason that the truth of the premises would not guarantee that the French will attack, of course, is the presence of the word 'Probably' in P2; it is only said to be probable that Napoleon is not ill. (We would get the same result if in place of P2 we wrote 'Napoleon is probably not ill', or 'It is unlikely that Napoleon is ill', or some other sentence synonymous with P2 as written.)

Note that words such as 'probably' can also occur in the antecedents and consequents of conditionals. For example:

P1) If Napoleon is not ill then, probably, the French will attack. P2) Napoleon is not ill.

C1) (Probably) The French will attack. This too is an inductively forceful argument.4

### Arguments with multiple probabilistic premises

Sometimes, probabilistic elements can occur in more than one premise. Assessing such arguments can be tricky. For example, we may have the following sort of case:

4 Some readers may wonder why we enclose the word 'probably1 in parentheses before the conclusion of an inductive argument, but do not enclose it in parentheses when it occurs in a premise. The reason is somewhat complicated, but what it boils down to is that we want to preserve the simple and common-sense contrast between deductive and inductive arguments according to which the inductive case is that where the premises provide good but not water-tight reasons to accept the conclusion. Thus when we write '(Probably) Such-and-such' beneath the inference-bar, the conclusion is 'Such-and-such1, not '(Probably) Such-and-such'. If the probability-indicator were regarded as part of the conclusion, then the question of how to distinguish deductive arguments from inductive becomes more complicated; we would also have to delve further into the theory of probability than is genuinely needed for our purposes here.

P1) Most people in Glasgow live in council housing. P2) Most council housing is substandard. P3) Ian lives in Glasgow.

C) Ian lives in substandard housing.

This argument is not inductively forceful. 'Most' means 'more than half', or 'more than 50 per cent'. The premises could well be true, then, in the following circumstances: slightly more than half the residents of Glasgow live in council housing; slightly more than half the council housing units in Glasgow are substandard, yet no other housing in Glasgow is substandard. If that were so, then the proportion of substandard housing in Glasgow would be slightly more than half of slightly more than half the total housing units in Glasgow. Since half of a half is 1/2 X 1/2 = 1/4, this means that, in these circumstances, the proportion of housing units in Glasgow that are substandard would be a little bit more than 1/4. So the mere fact that Ian lives in Glasgow, along with the truth of the other two premises, would not make it reasonable to expect that Ian lives in substandard housing.

Of course, if almost everyone in Glasgow lived in council housing, and almost all of it were substandard - then, provided that we knew these facts, Ian's living in Glasgow would be a good reason to infer that he lives in substandard housing.5

The same sort of thing applies when a word such as 'probably' occurs more than once in the premises: inductive force is not 'transmitted' across multiple inferences. Here is a somewhat more difficult case involving an explicitly conditional probability:

P1) Edna will probably come to the party.

P2) If Edna comes to the party, then probably she'll get drunk.

C) Edna will get drunk.

Is this argument inductively forceful? No. All that is required for something to be probable is that its probability be greater than 1/2. Suppose, then, that the chance of Edna coming to the party is exactly 51 per cent,

5 Of course, these premises are not true, and nor is it true that the only substandard housing in Glasgow is council housing!

Logic: inductive force

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