Formal Logic

Formal Logic

In formal logic, often thought of as a foundation for or a relative of Critical Thinking, we distinguish deductive from non-deductive arguments. Deductive arguments, we say, are those whose reasons to think a conclusion guarantee the truth of that conclusion. Non-deductive arguments are those whose reasons to think a conclusion only make the conclusion very likely to be true. In this, we appeal to structure, truth, and judgements about likelihood.

The pieces of evidence we deal with in formal logic are called premises. We infer from premises to conclusions. In this process we apply rules to help determine whether the move from the premise to a conclusion is justifiable, as well as assess the truth of the premises. We judge that the steps follow one another, and that they lead to the conclusion.

Generally, in formal logic we separate the structure of an argument from its content. When the premises of a deductive argument, if they were true, ensure the truth of the conclusion, we call the argument “valid”. When we evaluate that the premises of a valid argument are actually true, and therefore that the conclusion is true, we call the argument “sound”.

However, not all arguments are best rendered in the terms of formal logic. In Critical Thinking we approach these other sorts of arguments differently. Moreover, formal logic does not explicitly accommodate the fallibility of our judgements. Here we create a system to do just that. Here we create a system that allows for correction and revision of reasons and conclusions, which allows us to strengthen or weaken our arguments as we discover new facts and evidence.

In informal settings, such as critical thinking, we concern ourselves not with truth but with reasonableness and best explanation. When the reasons given in an inference are good, it is “reasonable” to accept an explanation as “best” or to accept a recommendation as “reliable” (terms that we will continue to refine in the course of this discussion).

This distinction between “reasonable” and “true” can make critical thinking seem a subjective version of formal logic. However, our judgements of informal arguments are not capricious. There are standards against which we can judge, and by and large, this process resembles judgements we make in our ordinary lives quite effortlessly.

For example, I might judge (conclude, assess, think, say, determine) “it is raining”. We might choose to represent my thought process as a deductive argument, perhaps something like this:

A Valid Argument

Premise 1: When students enter a lecture hall with wet umbrellas, it is raining.
Premise 2: Students are entering the lecture hall with wet umbrellas.

Conclusion: It is raining.

In formal logic, we would investigate the structure of this argument to determine whether it is valid. That is, we would consider, hypothetically, if the premises were true, would the conclusion follow from those premises. If so, the argument is valid.

Then, if it is valid, and this sample argument is, we would investigate the truth of the premises to determine whether it is “sound”. A valid argument with true premises is “sound”. If we found that the argument was not sound (and this argument is not sound, because Premise 1 is not “true”), then we could not for certain say that the conclusion is true. In formal logic, the exercise would end here; our assessment of the argument would be complete.

Consider this example, of an invalid argument with a true conclusion:

An Invalid Argument with a True Conclusion

Premise 1: All Presidents of the United States have been male.
Premise 2: Obama is male.

Conclusion: Obama has been President of the United States.

When we assess this argument, we see that the conclusion does not follow from the premises (lots of males are not President). This does not mean the conclusion is false. It simply means the proposed path does not lead to the conclusion. End of story.

We might say that in formal logic, the primary interest is in arguments and their structures. In Inference to the Best Explanation, the primary interest is in the conclusions themselves, and their relationships to evidence and circumstances. This is to say that, here, our investigation into the reason for wet umbrellas, for example, continues.

As noted, in the language of Formal Logic, Premise 1 in the umbrella example is false. There are lots of reasons why people might carry wet umbrellas, so there is something dissatisfying about simply determining the argument un-sound; we still wonder “what’s with all the wet umbrellas?” Since in the form above it is difficult to see how to proceed, here we introduce a variation of how we might come to the judgement “it is raining”. Informally we might write the argument as follows.

Example 1.1
It is raining”
Why are people carrying damp umbrellas? It could be the sprinklers are on. It could be there’s a movie crew outside and they have a rain machine. It could be that it is raining. Without evidence of sprinklers or a film set nearby, the best explanation is that it is raining.

In short, a comparison of the two expressions of a simple argument shows us how, in formal logic, a conclusion might follow from a set of premises. We can see that the premises and the conclusion are related in the right sorts of ways that the relationships between the premises, assuming the premises are true, licenses us to assert that the conclusion is true.

On the other hand, when we release ourselves from the burden of truth we allow ourselves to assess nuances and alternatives. So, here our task here is to come up with a systematic way of representing Best Explanations that licenses us to assert that one-among-the-many answers to an investigative question is the best. Lurking behind both deductive and informal cases are acts of judgement.

Last modified: Wednesday, 7 February 2018, 9:21 PM