Valid and Necessary Inferences (Logic)

I teach logic once every year, and find it a joy. For me, there is also a connection between theology and logic: when doing theology, it's so important to reason properly. Doing so entail using valid inferences.

Collins English Dictionary provides this definition: "(countable noun): An inference is a conclusion that you draw about something by using information that you already have about it."

In formal logic, one infers things from statements by moving from the premises to a conclusion. Some logicians say that people infer while statements imply things. In the statement, "If p, then q," q is an implicate of p: it is implied by p. However, not all inferences are the same. I want to focus on inferences that are valid, invalid, and others that are necessary.

Not all logicians define validity the same way, but let's assume that validity denotes "if the premises are true, then the conclusion necessarily follows from the premises." However, note the language of this denotation for validity: it's hypothetical. If the premises are true, then the conclusion necessarily follows. So, there is no guarantee that the premises are true; for all we know, they could be false. But if the conclusion does follow from the premises, we can say the inference is valid because the main concern is formality and rule-following.

As an example, take the statement, "If I live in Atlanta, then I live in the USA." Is the statement valid in the sense that we're talking about here? Yes it is because q (the second part of the utterance) naturally follows from p (the first part of the utterance). Hence, the inference is valid and the conclusion also happens to be true, which is another matter entirely. Yet it illustrates how premises force a conclusion to be valid.

Nevertheless, not all inferences are valid. Would you say that the following inference is valid?

1) If I pass all of my classes, then I will graduate.
2) I will graduate.
3) Therefore, I passed all of my classes.

The set of propositions could be formulated this way:

1) If p, then q
2) q
3) Therefore, p

This formal structure makes it clear that the inferences are not valid, but maybe another example will make this point clear.

1) If my car starts, there is enough gas in the tank.
2) There is enough gas in the tank.
3) Therefore, my car starts.

One cannot rightly infer that just because a car's gas tank is full that the car will start: the starter might not work or the battery could be dead. The inferences are fallacious, the conclusion does not necessarily follow from the premises, and the argument is invalid. Hence, all inferences are not equal.

One final thing to consider here: whether an inference is necessary or not. By "necessary," I mean that the premises must entail the conclusion. One cannot accept the premises of an argument, but reject the conclusion, if the premises force the conclusion to be true. For example:

1) If Sam is a bachelor, then he is unmarried.
2) If Sam is unmarried, then he does not have a wife.
3) Therefore, if Sam is a bachelor, then he does not have a wife.

It should be clear that one cannot accept the premises of the argument and reject the conclusion. The premises force the argument's conclusion, so the inferences are necessarily drawn from the premises, and the argument is valid according to the canons of formal logic. Yet we have seen that not all inferences are valid or necessary. Try to avoid those inferences that are not valid.