Chapter3, Chains of Reason

Introduction

This chapter shows how reliable rules and patterns can be directly employed one at a time, or one after another, to get conclusions or further reliable rules and patterns. The question of what rules are reliable is considered in the following chapters.

Rules used to get or suggest conclusions are called implications. Just as there are methods for adding and multiplying numbers carefully, there are also methods for using implication rules by themselves to get conclusions. There are also methods for linking, threading and chaining implication rules together to get more implication rules. This chapter uses examples to explain two basic ideas:

1. how to directly use a single implication rule to get conclusions, and
2. how to link, chain or thread implication rules together to obtain or derive more rules and more conclusions.

The examples are not important (and are perhaps ridiculous) but they illustrate some rule-based methods in reason. Examples which involved real-life situations might distract from mastering these methods. That is, in real-life situations, each of us may have opinions or prejudices about what should occur. That could spoil an explanation of the use and linkage of implication rules. There is a need for neutral examples to illustrate the use of implication rules one at a time or one after another.

Arithmetic, algebra and geometry give many neutral examples for this. The examples below involve no mathematics. Bon Appetite.

Conclusions From a Single Rule
Direct and Indirect Usage

Pretend the following implication rule is never disobeyed.

Each time Suzy the cat is on the ground and Suzy sees a dog, Suzy climbs a tree and stays in it for at least five minutes.

Direct Usage

What can we say for sure when Suzy the cat sees a dog? One possible answer is that Suzy the cat stays in a tree for at least five minutes. Another possible answer is that Suzy the cat climbs a tree. A more complete answer is that Suzy the cat climbs a tree and stays there for at least five minutes. Each of these answers or conclusions is correct. The last conclusion or result is fuller and more complete than the others. It gives more information. Which answer or conclusion is wanted here depends on who is interested in what. When many conclusions are possible, we state only those conclusions of interest to us. We do not have to state the most complete conclusion. The choice is ours.

Indirect Usage

What can you say for sure if Suzy the cat has not climbed nor stayed in any tree for at least five minutes? To check your answer, you might have to remember or revisit the questions in the chapter Implication Rules. But you should do this after you have read the following words.

Linking and Chaining Two Rules Together

The examples below and in the next page show how to chain, link or connect implication rules to get information and conclusions. The examples in themselves are not important. The information in them is silly. But these examples just show how to put implication rules together. So read on, with patience.

• Every time Suzy the cat climbs a tree, it gets stuck in the tree
• Every time Fred the dog visits the park in which Suzy the cat lives, Suzy climbs a tree.

By linking or chaining these implication rules, we can make three conclusions:

1. Whenever Fred the dog visits the park where Suzy the cat lives, Suzy climbs a tree.
2. Whenever Fred the dog visits the park where Suzy the cat lives, Suzy gets stuck in a tree
3. Whenever Fred the dog visits the park where Suzy the cat lives, Suzy climbs a tree and gets stuck.

Each of these conclusions is correct. Each conclusion gives a new implication rule which we could use in our reasoning process. The third implication rule is the most informative. It contains the most information. When we view each correct conclusion as a possible destination for our reasoning process, we may sometimes select our destination.

Putting Several Rules Togethe

We can chain or link not only two but also several implication rules together. This sometimes yields useful, new information. As an exercise, we ask the question: What happens whenever Fred the dog visits the one-tree park? Several answers are possible. Some have more details than others. All are correct. To answer the question, assume or pretend the next five implication rules are never disobeyed. Further, assume that Suzy the cat lives in the one-tree park.

1. When Suzy the cat climbs the tree in the one-tree park, Suzy gets stuck in the tree.
2. Each time Fred the dog visits the one-tree park, Suzy the cat climbs the tree.
3. Every time Charles the human visits the park, Charles sits on a bench for one hour.
4. Whenever a cat climbs the tree in the one-tree park, the five birds living in the tree fly around in the park.
5. Each time birds fly around in the park, sensible worms go underground.

All the information has been stated. We start our reasoning process. That is, we will answer the question: What happens whenever Fred the dog visits the one-tree park?

To answer the question, suppose or assume Fred the dog visits the park. Then from the implication rule (2), we see that Suzy the cat climbs a tree. Next, from the implication rule (1) we see that Suzy the cat gets stuck and from the implication (4) we see that birds fly around the park. Finally from the implication (5), we note sensible worms go underground.

We could list all that occurs when Fred the dog visits the park. Or, we could state only those results of Fred's visit to the park which are of most interest to us. The choice is ours. For instance, one of our possible conclusions follows:

This conclusion is not of interest unless you are a fisherman (or woman) looking for worms, sensible or not, for use as bait. The conclusion selected and stated here hides the reasoning process. That is, it hides the chain of implications leading to it. Our last conclusion does not mention the intermediate events where a cat climbs a tree and birds fly around the park.

The long path by which we get conclusions shows that implication or rule-based thinking can lead to surprising results. These surprising results are true if the initial implications are also true.

In the long path by which we got the conclusions, the information in the third implication (3) about Charles the human is not used. The conclusion we reached is independent of implication (3). In fact, without further information, I see no way of linking the rule about Charles with the other rules. The third rule is extra information. It can be ignored.

In answering questions, we often have extra information. Indeed, you can imagine the five rules given above are stated in random positions among a list of twenty, or hundred and twenty rules. An answer to the question

now depends on finding the rules in the list which can be used. This is a game of hide and seek. So we have to be selective, observant or fussy in deciding or seeing what information leads to our conclusions.

The scenery or route by which a conclusion is reached may contain as much useful information as the conclusion itself. A conclusion may contain a fraction of the information we could have stated or written. Being aware of the route or proof by which a conclusion is attained will sometimes suggest how more conclusions can be reached. This awareness is often more important that any conclusion we state because it allows us to state more conclusions, as needed.

Mathematics students take note. Remembering the route taken in solving a problem is worth more to the development of skills than remembering the solution.

Deductive, Inductive or Empirical Reason

Deductive reason uses or chains together supposedly (or preferably) never-disobeyed implication rules to suggest, to make or to reach conclusions. See the examples above. The implication rules in question may come from assumptions. The assumptions may be tentative.

The phrase inductive reason has one role in mathematics and another outside of mathematics. To induce (or induct) literally means to draw or extract. When you see a rule or pattern that no one has suggested, you are extracting or drawing that pattern from your observations. This process of recognizing rules and patterns that may hold, accidentally or not, is called inductive reasoning. Inductive reason outside of mathematics refers to the identification and recognition of rules and patterns from data and observations. Here rules and patterns may hold accidentally.

Reason which relies on a single or several, experience-found, rules and patterns to arrive at conclusions is called empirical. The underlying problem of inductive, empirical reason is to extract (infer, draw, induct or identify) from experience, in particular, data and observations, rules and patterns not satisfied merely by accident and which appear to be reliable. Self-deception needs to be avoided here.

Inductive reason inside mathematics refers to another process, namely, the extraction or drawing of conclusions from ladder-like chains of reason. See the next chapter for a more precise image or explanation. The rules or assumptions here are usually so certain, that we deliberately ignore the experience-based origins of mathematical reason.

Criteria for the recognition of reliable, non-accidental rules and patterns are described later in chapter 16, Origin of Rules and Patterns .

Selby A, Volume 1A, Pattern Based Reason, 1996.

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