Contextually Weak or Strong Relevance

As it turns out, the determination of weak and strong relevance may also depend on context - which means that additional information about the statements p and q may affect whether p is strongly or weakly relevant to q. To illustrate this consider the following example:

Let p be the statement, "Smoking doubles your risk of getting lung cancer and 90% of the inhabitants of Warren County Kentucky have smoked since young adulthood" and let q be the statement, " About 90% of the inhabitants of Warren County will contract lung cancer".

Suppose it is known that only 1 in 10,000 people get lung cancer (this is a figure which is entirely made up). Then doubling your risk means 2 in 10,000 people get lung cancer, which is still a small chance. Knowing this means that p does not support q, but rather supports its negation (90% of the inhabitants of Warren Country will probably not contract lung cancer). On the other hand, suppose that the risk of getting lung cancer were 1 out of 2 people. Then doubling that chance (by smoking) results in 1 out of 1 people contracting lung cancer. Since 90% of the inhabitants of Warren County smoke, this mean about 90% will get lung cancer, so in this case p is relevant to q.

The above example is just one example of what are often termed statistical fallacies. We will treat fallacies in more detail later, but at this point a fallacy can be thought of as a case where p and q are weakly related but they are wrongly thought to be strongly related. Thus statistical fallacies are errors in reasoning related to misuse or misunderstanding of probabilistic and statistical statements.

What contextual relevance tells us in general is that when a relation between two statements is not one of implication new information can change the strength or weakness of an inference. While not obvious at this point we will note that if a relation between two statements is one of implication new information can not change the nature of the inference.