Hidden Assumptions, Inductive Reasoning, and Good Arguments

Hidden Assumptions

When people give arguments sometimes certain assumptions are left implicit. Example:

This argument as it stands is not valid. Someone who gives such an argument presumably has in mind the hidden assumption that whatever that is unnatural is wrong. When this assumption is added, the argument does become valid.

But once this is pointed out, we can ask what this assumption really means and whether it is justified. There are plenty of things that are presumably "unnatural" but are not usually regarded as wrong, such as wearing sunglasses or having surgery. So anyone who accepts the argument above will have to either give up the argument, or come up with a different hidden premise. So trying to identify the hidden assumption in an argument can help us think more deeply.

In everyday life, the arguments we normally encounter are often arguments where important assumptions are not made explicit. It is an important part of critical thinking that we should be able to identify such hidden assumptions or implicit assumptions.

So how should we go about identifying hidden assumptions? There are two main steps involved. First, determine whether the argument is valid or not. If the argument is valid, the conclusion does indeed follow from the premises, and so the premises have shown explicitly the assumptions needed to derive the conclusion. There are then no hidden assumptions involved. But if the argument is not valid, you should check carefully what additional premises should be added to the argument that would make it valid. Those would be the hidden assumptions. You can then ask questions such as: (a) what do these assumptions mean? (b) Why would the proponent of the argument accept such assumptions? (c) Should these assumptions be accepted?

This technique of revealing hidden assumptions is also useful in identifying hidden or neglected factors in causal explanations of empirical phenomena. Suppose someone lights a match and there was an explosion. The lighting of the match is an essential part in explaining why there was an explosion, but it is not a causally sufficient condition for the explosion since there are plenty of situations where someone lights a match and there is no explosion. To come up with a more complete explanation, we need to identify factors which together are sufficient for the occurrence of the explosion, or at least show that it has a high probability of happening. This might include factors such as the presence of a high level of oxygen in the environment.

Exercise #1

Source: Joe Lau and Jonathan Chan, https://philosophy.hku.hk/think/arg/hidden.php
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

Inductive Reasoning

§1. What is Induction?

Consider the following argument:

This argument is of course not valid, since Dipsy might be so lucky that he wins the lottery. But this is quite unlikely to happen if the lottery is indeed a fair one. In other words, the premise can be false even when the premise is true. However, even though the argument is not valid, if you believe that the premise is true, you probably will accept the conclusion as well on that basis. In other words, the conclusion is highly likely to be true given that the premise is true.

Here is another example:

Again, it is not logically impossible for Dylan to recover from his coma and join the marathon and finish, but this is unlikely to happen. Again, given that the premise is true, the conclusion is likely to be true also.

Although the two arguments above are not valid, we would normally still regard them as good arguments. What is special about them is that they are inductively strong arguments : the conclusion is highly likely to be true given that the premises are true. With an inductively strong argument, although the premises do not logically entail the conclusion, they provide strong inductive support for it.

There are at least three main differences between an inductively strong argument and a valid argument:

1. As already noted, in a valid argument, the conclusion follows logically from the premises, but this is not the case in an inductively strong argument. It is logically possible for the premises to be true while the conclusion is false.
   
   
2. Deductive validity is not a matter of degree. An argument is either deductively valid, or it is not. But inductive support is a matter of degree, depending on the probability of the conclusion being true given the premises.

   
   For example, consider this slightly modified argument:

   Dipsy bought X tickets in a fair lottery with 10000 tickets.

   So Dipsy is going to win the lottery.

   If we replace X by a small number, say, 5, then the argument is obviously very weak, since it is very unlikely that Dipsy can win by buying so few tickets. However, if we increase X to say 2000, then the inductive strength of the argument will of course increase. If X is 9999, then the argument is even stronger, since it is extremely likely now that Dipsy will win. So you can see that inductive strength is not an all-or-nothing matter.

3. A related point is that inductive strength is defeasible, whereas validity is not. To say that validity is not defeasible is to say that if you have a valid argument, adding new premises will not make it invalid. If it is indeed true that three people have died, then it follows that at least two people died, and this will remain the consequence whatever new information you acquire.
   
   

   However, new information can be added to an inductively strong argument to make it weak. Consider the second lottery argument again, and suppose we add the new premise that Dipsy bought 9999 lottery tickets, but gave them all to Tinky-winky. Obviously this new argument will be a lot weaker than the old one.

Inductive reasoning is very important in ordinary life and science. We believe lots of things on the basis of limited evidence. The evidence might not logically guarantee that the belief is correct, but the belief can still be reasonable. For example, we see dark clouds in the sky and think it is likely to rain so we bring an umbrella. We see mould on our bread and think we will be sick if we eat it.

Good Arguments

§1. What is a good argument?

In this tutorial we shall discuss what a good argument is. The concept of a good argument is of course quite vague. So what we are trying to do here is to give it a somewhat more precise definition. To begin with, make sure that you know what a sound argument is.

§2. Summary

So, here is our final definition of a good argument :

A good argument is an argument that is either valid or strong, and with plausible premises that are true, do not beg the question, and are relevant to the conclusion.

Now that you know what a good argument is, you should be able to explain why these claims are mistaken. Many people who are not good at critical thinking often make these mistakes :

• "The conclusion of this argument is true, so some or all the premises are true."
• "One or more premises of this argument are false, so the conclusion is false."
• "Since the conclusion of the argument is false, all its premises are false."
• "The conclusion of this argument does not follow from the premises. So it must be false."

§3. A technical discussion

This section is a more abstract and difficult. You can skip this if you want.

One interesting but somewhat difficult issue about the definition of a good argument concerns the first requirement that a good argument must have true premises. One might argue that this requirement is too stringent, because we seem to accept many arguments as good arguments, even if we are not completely certain that the premises are true. Or perhaps we had good reasons for the premises, even if it turns out later that we were wrong.

As an example, suppose your friend told you that she is going camping for the whole weekend. She is a trustworthy friend and you have no reason to doubt her. So you accept the following argument as a good argument:

Amie will be camping this weekend. So she will not be able to come to my party.

But suppose the camping trip got cancelled at the last minute, and so Amie came to the party after all. What then should we say about the argument here? Was it a good argument? Surely you were justified in believing the premise, and so someone might argue that it is wrong to require that a good argument must have true premises. It is enough if the premises are highly justified (of course the other conditions must be satisfied as well.)

If we take this position, this implies that when we discover that the camping trip has been cancelled, we are no longer justified in believing the premise, and so at that point the argument ceases to be a good argument.

Here we prefer a different way of describing the situation. We want to say that although in the beginning we had good reasons to think that the argument is a good one, later on we discover that it wasn't a good argument to begin with. In other words, the argument doesn't change from being a good argument to a bad argument. It is just that we change our mind about whether the argument is a good one in light of new information. We think there are are reasons for preferring this way of describing the situation, and it is quite a natural way of speaking.

So there are actually two ways to use the term "good argument". We have adopted one usage here and it is fine if you want to use it differently. We think the ordinary meaning of the term is not precise enough to dictate a particular usage. What is important is to know very clearly how you are using it and what the consequences are as a result.