Longer Chains of Reason
Chapter 7, Part I

Previous: Chapter 6, Chains of Reason

To induce means to extract. Induction here consists of extracting conclusions from chains of rules and patterns, one after another, perhaps without stopping or end. Another form or version of inductive reason is concerned with the extraction of patterns from experience and observation. See the last words of the previous chapter.

This chapter explains one version of inductive reason: the recursive or repetitive approach to putting one-way implication rules together, one after another. This chapter ends with a description of the principle of mathematical induction – another method for obtaining conclusions used only in mathematical arguments or computations. There is more to mathematics than just doing arithmetic. 

Recall that rules, which say that when a first situation occurs so should a second, are called implication rules. Implication rules can be linked together, one after another. A ladder-based story illustrates the underlying idea. It is called induction. This story leads to the notion called mathematical induction, a method of reason or logic used in mathematics after arithmetic to get conclusions (or climb ladders). The method is described first with words, a simple story, and then with some shorthand notation.

Romeo and Juliet

Imagine a hero, Romeo, riding a horse towards a tall building (a castle). There is a ladder up the side of the building leading to the room where Juliet lives. The bottom step of the ladder is two meters or more (several feet or more) away from the ground. The ladder is not broken. It is in good condition. A person getting to each step of the ladder can climb to the next. Question: Can an able-bodied individual, Romeo, reach Juliet via the ladder? The answer is yes provided Romeo can get to the first or bottom-most step of the ladder. It is no otherwise. The main logic-related ideas in this brief story are as follows.

1. There is a long ladder to be climbed.
2. When any one step is reached, the next step can be reached. (The ladder must be in good condition for this to hold).
3. The first or bottom-most step can be reached.

This situation implies we (or Romeo) can reach each step of the ladder.

Note that the long ladder may have a finite number of steps, for example 183. Then we (or Romeo) can with enough time and patience, reach the last one, or any step in between.

On the other hand, we can imagine a ladder could have an infinite number of steps. For each step we take, a next is possible. For instance, the whole numbers we use for counting do not stop. Each whole number is followed by another — just add 1.

Now suppose or imagine we have a sequence of steps, a ladder, which goes on and on without stopping. Then with enough time and patience, we can reach anyone you mention. An example is met in counting. We can begin counting with the number 1, then 2, then 3 and so on.

When we begin to count, we may have only a finite number of objects to count. With a long enough life, and enough patience, the count will end. But if we count minutes there will always be one more to count. This minute count will never end. More precisely, each of us counters may end, but the counting of minutes in principle can continue. That is, this minute count can reach any large number you specify in advance with or without you. In principle all minutes after the beginning of the count will be met and counted.

To rephrase the above, on a ladder (or road) with finitely or infinitely many steps, the first step needs to be reachable. And from each step, the next step needs to be reachable. When this occurs, any whole number of steps along the road or ladder in question is reachable.

[2] In practice, if each step takes time, the number of steps reachable will depend on how much time is available.

CAUTION. The conclusion that all steps can be climbed or reached does not follow from the principle of mathematical induction if the ladder is broken, or if the first step is not reachable

or if a tornado comes along, or if you break your ankle, etc.

Check for these nasty situations when you want to use this principle to get a conclusion.

Reading Guide

The principle of mathematical induction stated below describes the above ladder idea in the algebraic shorthand notation favored in mathematics. The last part of this chapter will not make sense to you if you are not familiar with this shorthand notation. If this is the case, you may skip this description of mathematical induction.

Next: Mathematical Induction, Chapter 7, part II

www.whyslopes.com
Volume 1A, Pattern Based Reason

 Chapters 1 to 24

FOREWORD
Three Remarks
1 Introduction
2 Communication
3. Elements of Reason
4 Implication Rules
5. Deception
6 Chains of Reason
7 Longer Chains
For & From Consistency
8. Language Change
9 Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Deductive & Empirical 
     Views of Mathematics
15 Objectivity
16 Origin of Rules
and Patterns
17 Objective Ways
18. Waking up
19. Symbols  & Logic
20. Pronouns or Symbols
21. Truth Tables I.
22. Truth Tables II
22. Biconditional
22. Contrapositive
23. IF-THEN table
24. Indirect Reason Again

To reason often means to persuade someone of the need for an idea or action. That someone could be yourself. So be careful.

1A Logic Postscripts
- online only

+Proof by Absurdity alias proof by contradiction
+How the demand for consistency supports the law of the excluded middle
+Reality versus or with the aid of Imagination
+Links for reason, logic and crtical thinking
+Three Remarks
+History Lost or Missing

There is a difference between
knowing how to spend money,
and having money to spend.

There is likewise a difference
between mastering a skill
and having meeting a situation in which it applies.

 .

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