2.5: Inductive Strength

Consider this argument again:

1. Sam is a line cook.
2. Line cooks generally have good of kitchen skills.
3. So, Sam can probably cook well.

This is a decent argument. The premises do support the conclusion. And yet it might be that both premises are true and the conclusion is false. Sam could be a brand new cook hired because he’s the manager’s son who has never cooked in his life. Many arguments give us good reasons for accepting their conclusions even if their premises being true fails to completely guarantee the truth of the conclusion. This suggests that we need another standard of support for arguments that aim at giving us pretty good but not absolutely compelling grounds for accepting their conclusions. And this standard of support is called inductive strength. Here are two equivalent ways of defining inductive strength:

(I) An inductively strong argument is an argument in which if its premises are true, its conclusion is probably to be true.

(I’) An inductively strong argument is an argument in which it is improbable that its conclusion is false given that its premises are true.

If you look again at the earlier definitions for deductive validity you will find a good deal of similarity. The only difference is in the use of the words "probably" rather than “must be” in the first definition, and “improbable” rather than "impossible" in the second. This is a big difference. As in the case of validity, when we say that an argument is strong, we are not assuming that it’s premises are true. We are only claiming that if the premises are true then the conclusion is likely to be true. Corresponding to the notion of deductive soundness, an inductive argument that is both strong and has true premises is called a cogent inductive argument. Unlike the case if deductively sound arguments, it is possible for an inductively cogent argument to have true premises and a false conclusion.

Lots of good reasons for holding a belief fall short of the standard of deductive validity. The sort of reasoning you were taught as “the scientific method” in secondary school is inductive reasoning. As it is taught in high school, the scientific method consists of formulating a general hypothesis and testing it against a large sampling of data. If the data is consistent with the hypothesis, then the hypothesis is considered confirmed by the data. Here a limited amount of evidence is taken to support a broader more general hypothesis. In the simplest case, inductive reasoning involves inferring that something is generally the case from a pattern observed in a limited number of cases. For instance, if we were to conduct a poll of 1000 Seattle voters and 600 of them claimed to be Democrats, then we could inductively infer that 60% of the voters in Seattle are Democrats. The results of the poll give a pretty good reason to think that around 60% of the voters in Seattle are Democrats. But the results of the poll don’t guarantee this conclusion. It is possible that only 50% of the voters in Seattle are Democrats and Democrats were, just by luck, over represented in the 1000 cases we considered.

When evaluating deductive arguments for validity we ask if it is possible for the premises to be true and the conclusion to be false. This is either possible or it isn’t. Possibility does not admit of degrees. But probability does. The truth of the conclusion of an inductive argument can be probable to a greater or lesser degree. An argument either is or isn’t valid. But inductive arguments can be more or less strong. We can identify a few factors that bear on the degree of strength an inductive argument has. One is how much evidence we have looked at before inductively generalizing. Our inductive argument above would be stronger is we drew our conclusion from a poll of 100,000 Seattle voters, for instance. And it would be much weaker if we had only polled 100. Also, the strength of an inductive argument depends on the degree to which the observed cases represent the makeup of the broader class of cases. So our inductive argument will be stronger if we randomly select our 1000 voters from the Seattle phone book than if they are selected from the Ballard phone book (Ballard being a notably liberal neighborhood within Seattle).

So far, we’ve only discussed inductive generalization, where we identify a pattern in a limited number of cases and draw a more general conclusion about a broader class of cases. Inductive argument comes in other varieties as well. In the example we started with about Sam the line cook, we inductively inferred a prediction about Sam based on a known pattern in a broader class of cases. Argument from analogy is another variety of inductive reasoning that can be quite strong. For instance, I know that my housecat is very similar to cougars in the wild. Knowing that my cat can jump great heights, it would be reasonable to expect that by analogy, or based on this similarity, cougars can jump well too.

There are further varieties of argument that aim at the standard of inductive strength, but we will discuss just one more in detail now. Abduction is inference to the best explanation. Detective work provides a good example of abductive argument. When Holmes discovers Moriarty’s favorite brand of cigar and a bullet of the sort fired by Moriarty’s gun at a murder scene, inference to the best explanation suggests that Moriarty was the killer. That Moriarty committed the murder provides the overall best explanation of the various facts of the case.

The 19th century American pragmatist and logician, Charles Sanders Peirce offers the Surprise Principle as a method for evaluating abductive arguments. According to the surprise principle, we should count one explanation as better than competing explanations if it would render the facts we are trying to explain less surprising than competing explanations. The various clues in the murder case are among the facts we want explained. The presence of the cigar and the bullet casing at the murder scene is much less surprising if Moriarty committed the murder than if the maid did it. Inference to the best explanation aims at strength. So a strong abductive argument in this case needn’t rule out the possibility that the murder was committed by Moriarty’s evil twin who convincingly frames his brother. There might an argument against the death penalty lurking nearby. Inference to the best explanation is worth more attention than if often receives. This kind of reasoning is pervasive in philosophy and science, but seldom gets much notice as an integral part of the methods of rational inquiry.