1.1.5. Bounds on inference

Let us now look at the relations between deductive and non-deductive reasoning a little more closely with the aim of distinguishing the role of deductive inference and other aspects of deductive logic.

First notice that there is a close tie between the riskiness of an inference and the question whether it 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 this hypothesis goes beyond what the data says, there is a possibility it is wrong even when the data is entirely accurate.

This points to the limits of deductive inference. Still, since the extraction of information can be a first step towards a making a generalization or inferring an explanation. And, by showing how far this first step might take us, deductive logic maps out the territory that we can reach without risking the leap to a generalization or explanatory hypothesis. By distinguishing safe from risky inferences, deductive logic sets a lower bound for inference by marking the range of conclusions that come for free, without risk.

And deductive logic sets bounds for inference also in another respect. One aspect of reasoning is the recognition of tension or incompatibility within collections of sentences, and this, too, has a deductive side. When a incompatibility among sentences is a direct conflict among the claims they make, there is no chance that they 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. In particular, we know that a generalization can never be supported if our data already provides counterexamples to it, and this sort of constraint is also the concern of deductive logic.

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 adds to risk-free conclusions other conclusions that are well supported by the data but go beyond it and are at least somewhat risky. There is large range in the middle of diagram that represents conclusions about which our data tells us nothing. Beyond this, the circle at the right marks the beginning of a region in which we find sentences deductively incompatible with the data. These are claims that are ruled out by the data, that cannot be accurate if the data is accurate. The sentences near this circle but not beyond it are not absolutely incompatible with the data but are in real conflict with it.

The task of deductive logic is to map the sentences within the narrow circle of risk-free conclusions and also to map those that are ruled by our premises. It will turn out that these are not two independent activities: doing one for any substantial range of sentences will involve doing the other.