Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 7.4.1 Sketching Some of the Problems

Unfortunately, predicate logic all by itself, without substantial additional theory and, in some cases, significant structural changes, doesn’t do a good job of capturing the full range of statements that we often make about the world around us.

Consider the following story:

1. It doesn’t rain very often in Austin.

2. Kelly only likes dancing in the rain.

3. Kelly won’t go anywhere unless he can plan a long time in advance and be pretty sure he’ll be able to dance.

4. Judy won’t go anywhere without Kelly unless something really unusual happens.

5. Fran does exactly what Judy does.

6. Judy probably isn’t coming to Austin any time soon. (Because Kelly isn’t.)

7. Fran’s mom isn’t counting on seeing her in Austin any time soon. (Because Judy isn’t coming.)

The derivation of [5] from the premises [1] – [4] seems right. But we can’t do it in our system.

Let’s look at the lines one at a time to see what’s going on:

1. We need to be able to represent statistical truth. What does “very often” mean? Let’s say we could agree that it means that the chances of rain on a given day are less than 5%. How should we represent and reason with even that more concrete fact?

2. This one we can do if we stretch. But we’ll need some way to represent time and place since this sentence is saying that Kelly likes dancing at a particular time, in a particular place, if and only if it’s raining at that time in that place.

3. Again we need statistical reasoning. What does pretty sure mean? And we need to be able to reason from the fact that it doesn’t rain very often to the fact that, a long time in advance, it won’t be possible to know whether it’s going to rain.

4. How can we represent the “unless” clause here? It’s saying that, in the absence of information about some unusual event, we should assume that Judy won’t go if Kelly doesn’t. In other words, we are to take our lack of knowledge as telling us something. But we must be prepared, if suddenly we are told about an unusual event, to undo our reasoning and give up on the conclusion that Judy won’t go.

5. We can represent actions with predicates, such as VisitsAustin(Judy) or Dances(Kelly). Then what we want to say here is something like: P (P(Fran)  P(Judy)). Read this as, for all predicates P, P is true of both Fran and Judy or neither of them. But our logical system doesn’t allow us to quantify over predicates.

6. Again we need statistical reasoning.

7. First, we need to reason that, since Judy isn’t coming to Austin, neither is Fran. In addition, how should we represent not just basic facts (such as it’s not likely that Judy will come to Austin), but also people’s beliefs about those facts?

Summarizing, we’ve recognized the following issues:

1. We need a way to talk about statistical truth.

2. We need a way to reason about time and place.

3. We need a way to reason with default information (assume unless told otherwise).

4. We need a way to reason not just about objects, but also about predicates.

5. We need a way to reason about what we know or believe.

There exist reasoning systems that solve these (and other) problems. We can’t go into them here. But in the next few slides we’ll say a little more about the issues that we’ve just raised.