1.1.5. Deductive bounds on reasoning

Let’s look at the relations between deductive and non-deductive reasoning a little more closely with the aim to sketch the role of deductive logic. First notice that there is a close tie between the riskiness of an inference and the question of whether is merely extracts information or does something more. The information extracted from data may be no more reliable than the data it is extracted from, but it certainly will be no less reliable. On the other hand, even the generalization or explanatory hypothesis that is most strongly supported by a body of data must go beyond the data if it is to generalize or explain it. And, if it goes beyond what the data says, there is a possibility it is wrong even when the data is completely accurate.

The extraction of information can be a first step towards generalization or inference to an explanation. And we have seen that extracting information does not merely prepare us to go further: it maps out the territory that we can reach without making the leap to a generalization or explanatory hypothesis. That is, deductive logic serves to distinguish safe from risky inferences. And this sets a lower bound for inference by marking the range of conclusions that come for free.

But deductive logic sets bounds for inference in another respect as well. Another aspect of reasoning is the recognition of tension or incompatibility within collections of sentences; and this, too, has a deductive side when the incompatibility is a direct conflict among the informational content of the sentences and there is no chance that the sentences could be all be accurate. This sets a sort of upper bound for inference by marking the range of conclusions that could not be supported by any amount of further research, such as generalizations to which the data provides counterexamples.

These two bounds are depicted in the following diagram.

Fig. 1.1.5-1. Deductive bounds on inference.

Sentences in the small circle are the conclusions that are the result of deductive reasoning. They merely extract information and are risk-free and always well-supported. Beyond this circle is a somewhat larger circle with fuzzy boundaries that also includes generalizations and inferences to explanations that are well supported by the data but go beyond it and are at least somewhat risky. Beyond the circle at the right are sentences deductively incompatible with the data. These are claims that cannot be accurate if the data is. The sentences near this circle are not absolutely ruled out by the data but are in real conflict with it.