Shantila's Inside Logic #22

Inferences (Thinking) and Belief Change

So far we've been using the words argument and inference more or less interchangeably, but they can be distinguished. An argument is simply a collection of statements --premises with a conclusion. (Of course, the English word "argument" also is used in a different way to refer to an animated discussion between people who are having an argument.) An inference is a process of mind in which thought "moves" from premises to a conclusion, whereby we "reach" conclusions and form beliefs or reject beliefs.

Given this distinction, we can say that validity and invalidity are primarily about arguments (as collections of statements). That is, being valid (or not) is about the relation between premises as statements and a conclusion as a statement. And, of course (as has been mentioned many times already), an argument is valid just in case the conclusion must be true if the premises are true.

On the other hand, inferences, as thought processes, can be good or bad--but "valid" and "invalid" generally do not apply to inferences. While it can make sense to say that an inference is valid just so long as it conforms to a valid argument, our primary interest in inferences is whether they are good or bad.

The interesting thing is this: just because an argument is valid does not necessarily mean that an inference conforming to that argument is good!!

Suppose, for example, you are wondering whether you should believe in ghosts. There actually are interesting stories one can consider. As you think about this, suddenly you realize that ever since you were 6 years old you have believed that a ghost killed Peter Padusky (an elderly neighbor who died suddenly one summer day). Now obviously, P>G is true: if a ghost killed Peter Padusky, then ghosts exist. So you believe P and you believe P>G, which of course guarantee G. And if you use P and P>G to reason to G (using MP) then your inference conforms to a valid argument.

Does that mean that you therefore should believe G? No, of course it does not necessarily mean that! Certainly P>G, P } G is a valid sequent --but that doesn necessarily mean it is sound --that is, one or more of the premises may be false! Does believing both P>G and P guarantee that you also should believe G? Obviously not! Instead of coming to believe G, you can abandon one of your prior beliefs! You can, for example, stop believing P (that a ghost killed Peter Padusky).You can give up that belief. Reasoning sometimes functions to help us change beliefs. It does not merely lead us to new conclusions based on old beliefs.

Here's another example. Suppose that somebody tells you, and you come to believe, that everyone on the BGSU Dance team is conceited. We can represent this belief using our new symbols as (x)(Dx > Cx), where Dx stands for x is on the BGSU Dance team, and Cx stands for x is conceited. Now suppose you are working on a project with somebody from one of your classes, Tara, let us say, and in making plans she tells you that she won't be able to work on the project this weekend because she will be going to Orlando for the National Dance team competition. What? Tara is on BG's Dance team! We can represent this as Dt. You now believe both (x)(Dx>Cx) and Dt, and it follows that Ct, Tara is conceited. This sequent certainly is valid:

(x)(Dx > Cx), Dt } Ct.

Does this mean that you should believe Ct? It depends. It depends on how good was your reason for coming to believe the universal claim (x)(Dx > Cx) in the first place? Perhaps the reasons you had for that general claim were not really very good, especially compared with the first hand evidence that you might have from actually knowing Tara.

We sometimes have conflicting beliefs in our own minds. And quite often we are not aware of these conflicts until we pay attention to the logical consequences of what we believe.

Logic is important not only to help us get new beliefs (based on the beliefs we already have) but it is equally important in helping us get rid of old beliefs that are not good or that conflict with other beliefs.

The early 20^th century philosophy Bertrand Russell said that the chief merit of proofs is that they tend to engender skepticism about the result proved. This might seem paradoxical since one might think that a "proof" should make us even more confident about a conclusion. However, once we have spelled out the premises that underly a certain belief, we can examine those premises, and ask questions about them to see if they are true, and having done so, this process sometimes will lead us to have doubts about the premises themselves.

Something like that might happen in the example about Tara. Even though you began believing the two premises, you might come to reject the universal claim (x)(Dx > Cx) rather than come to believe that Tara is conceited.

Getting back to ghosts, you might very well ask yourself in the situation we imagined --now precisely what sorts of reasons did I have when I have when I was 6 years old for believing that P (that a ghost killed Padusky)? And reflecting on this question, you might realize that the reasons you had were not very good reasons or that you really had no good reasons at all! For example, suppose that you are able to remember that your mother read you a ghost story the night before your neighbor Padusky died, and the story was on your mind the next day when you overheard your parents talking about him. Being so young, you did not really understand what had happened and so you just blamed it on the "ghosts" about whom your mother had been reading the night before. Reflecting on all of this now, you probably will end up believing ~P and rejecting your old belief that P. And so P won't be part of your "belief system" for use in an inference to G.

Now, changing the story a bit, suppose you have no idea where you got the belief that P about your elderly neighbor. And suppose that you now believe there are no ghosts, ~G. Working backwards from your current belief that ~G, you can use MT with P>G to reach ~P. So you might reject your old belief that P, not because you understand how you got the belief as the result of a flawed process of belief formation, but simply because of your current belief that ~G. You can't remember how you got the belief, but given your current belief that ~G, you might come to accept ~P (instead of using P and P>G to get G).

There is no way to extend our System of Rules to formalize the types of inferences (belief formation and revision) that we just described. You may decide to reject P rather than accept G, on the basis of P>G. Or you might keep your belief that P and then used MP to conclude G. Either way, logic by itself is not going to fault you. Logic by itself is not going to tell you whether to use MP or MT in this situation. Nor is logic by itself going to tell you what to believe about ghosts or dance team members or anything else we might be thinking about.

In this logic course so far we have been focussing primarily on valid forms of arguments. Both MP and MT are valid. Our System of Rules is an elegant way to keep track of some of the primary valid forms of arguments. What logic does not do is to determine the basic beliefs that we use in reasoning. In a way, this is obvious since use of our System to construct proofs always depends upon our making Assumptions by means of our first Rule A. And the system itself doesn't really have anything to say about how good or bad these assumptions may be. In everyday reasoning, of course, the Assumptions we use are simply the basic beliefs that we have about the world. As things happen, within us and around us, we constantly get new beliefs (and reject old beliefs).

Where then do we get our basic beliefs? Are these beliefs completely independent of rational evaluation?

We get beliefs, first, through our physical senses. We see things, we hear and smell and touch and taste things, and we form beliefs on the basis of these perceptions. Sometimes we make mistakes, such as when we "see" a stick in water as crooked when in fact the stick is straight. But usually our perceptions are reliable bases for beliefs. This, at least, is a starting point-- and I think it is fair to say that someone who disagrees needs to explain why we should not start with normal perceptual experiences and the beliefs they give rise to. Where else are we going to start?

Secondly, we form complex beliefs on the basis of our perceptual experiences. Suppose you happen to believe that vampires exist and live among us. Indeed you believe there are two or three who attend logic class every time. I ask you why you believe such an odd thing. Because you saw one once, you say. You start telling me a complicated story about events you witnessed in the Detroit airport -- and then you go, Uh. Wait. Wait a minute. That didn't really happen! That was only a dream I had! After realizing this, do you continue to count your prior "experience" of vampires as a reliable source of information? No, of course not. And unless you have other evidence for vampires, you probably will stop believing in them after you realize that your "evidence" was based on a dream. And in that case, you should stop believing in them, since you do not have a rational basis for believing in them. --There is a big difference between seeing something and dreaming it! --Again some philosophers might disagree, but someone who disagrees with this point would need to explain why.

I have never seen a vampire, and I believe that vampires do not exist now nor have they ever existed. I believe they are merely imagined beings made up in fictional stories and dreams. Nor have I ever once seen a live dinosaur either --but about dinosaurs I do believe that they once existed on earth. Why? Because there are people who have told me so -- I have read about them --and I believe those people are not simply writing fictional stories and I believe that they know what they are talking about. Now why on earth should I believe them? Because I believe they are experts about that sort of thing, and they have good evidence to support what they say. But why do I believe that?

Good question. This process of questioning can go on. There are areas of philosophy like epistemology and philosophy of science that go beyond logic and look carefully into such questions as these. What are the standards that beliefs must meet in order to count as genuine knowledge about our world? What are the standards (other than logical ones) that make our thinking good or bad?

* Practice

22.1 Suppose you believe that Sally lives in Chicago because your friend, Tom, told you so. And of course if she lives in Chicago she lives in Illinois; so the following sequent is valid:

C, C>I } I

Now suppose another friend, Nelly, tells you that Sally does not live in Illinois. And so you come to believe that ~I. Now one might wonder what sort of inference (if any) is reasonable for one to make in this situation. (a) What other information might it be reasonable for you to seek? (b) Suppose you know that Tom tells a lot of lies, whereas Nelly is always reliable. Would that help you decide what sort of inference to make in that case?

22.2 An inference can be bad even though it conforms to a valid sequent, as is discussed above. (a) First use our new rule UI to prove that the following sequent is valid: (x)(Kx>Fx), ~Fa} ~Ka. (b) Now give an example in English of an inference of this form that would be bad.

22.3 The following sequent is not valid. We do not want to prove it (and we cannot prove it).

~(Fa & Ga) } ~Fa &~Ga

Give an English example (a counterexample) to show why this sequent is not valid. Hint: to do this you need to give an example in English where the premise is in fact true but the conclusion is false. Doing this shows that the sequent is not valid, since it shows that the truth of the premise does not guarantee the truth of the conclusion.

22.4 Prove that each of the following sequents is valid. Use any of the Rules A, MP, DN, MT, &I, &E, CP, vI, DS, RA, and UI.

122 (x)(Fx & Kx) } ~(Fc > ~Kc)

123 (x)Fx } ~(~Fc & Ga)

124 (x)~(Fx & ~Kx) } Fc > Kc

125 (x)(Fx > Kx) } ~Fa v Ka

126 (x)(~Fx v ~Kx) } ~(Fa & Ka)

Extra Credit

127 (x)~(Fx > ~Kx) } Fa & Ka