First Logic Puzzle

Implication Rules

Explaining the difference between the meaning of If A then B and B if and only A is the purpose of the following two logic puzzles. The questions in the puzzle below are intended to introduce and emphasize the difference.

Teachers: Read the following before, besides or after the two logic puzzles in this chapter, as you like.

Suppose the following:

the local store sells a newspaper if John enters.
the same store sells a newspaper if Jeremy enters.

Then we cannot state

the local store sells a newspaper if and only if John enters it (the local store). .

since the local store also sells a newspaper if Jeremy enters.

So there is a difference in meaning between the two suggestions or statements

the local store sells a newspaper if John enters.
the local store sells a newspaper if and only if John enters it (the local store). .

Seeing the difference in meaning in this simple example is the key to precision reading and writing.

More generally, there is a difference in meaning between the two suggestions or statements

situation B arises if and only if situation A arises

situation B arises if situation A arises.

Here we may say occurs or happens instead of arise, or omit the word arise altogether. That be said and done, we make the convention that the two statements

Situation B if situation A (B if A form)

If situation A then situation B (if A then B)

have the same meaning.

Here we assume that the two following two statements have the same meaning:

B if and only if A,

A if and only B

mean the same.

Understanding there is a difference in meanings is the key to greater precision and exactness in work and study. If you see the difference, you will we hope make an effort to respect the difference and use the difference in following and writing rules or instruction.

The First Puzzle

A One-Way Implication Rule

To help you think and possibly cook more carefully, we look at a very simple puzzle. The puzzle consists of a rule and five questions. The questions test your ability to think carefully and to read exactly what is written. Once you have understood the answers and why they are true, your ability to think carefully and clearly will have advanced. The rule for the puzzle is as follows:

When Aunt Jane visits her nephew Tom's home,
Tom goes out to play.

Try to answer the five questions below. Be careful. The questions may trip you. Answers follow. See if you agree with them.

Five Questions

Answers are given twice

in popup boxes, and
in text below (as in the printed version)

See if you agree with them.²

When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home? This is easy. [Answer]

When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing? Be careful. [Answer]

When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting? Be careful, again. [Answer]

What must happen for the given rule to be disobeyed? This is another easy question. [Answer]

When the rule is not disobeyed, what can you say for sure about Aunt Jane when Tom does not go out to play? See the answer to the fourth question. [Answer]

Hint: The rule provides no information and no reason explaining why Tom goes out to play whenever his Aunt Jane visits. The rule only describes what happens when Aunt Jane visits. We cannot say if he goes out to play to avoid Aunt Jane. We cannot say if he looks forward to her visits. The answers to the above questions only depend on the wording of the question and the given information or rule(s). So control your imagination. Do not assume or imagine too much.

Suggestion: Discuss the questions with your family and friends. Some people will get correct answers immediately. Others require persuasion. Still others will not understand. Talking with people about the questions shows how well they think.

The First Answer

The first question is

When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home?

Its answer is easy: Tom goes out to play.

The Second Answer

The second question is

When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing?

The answer is nothing. The rule only tells what happens when Aunt Jane visits. It does not say what must happen when Tom goes out to play. Tom could go out to play without Aunt Jane visiting. The rule does not say, nor does it suggest that Tom may only play outside when Aunt Jane visits. The rule does not say Aunt Jane must visit when Tom goes out to play.

When the rule is not disobeyed, we cannot say much for sure or certain about Aunt Jane when Tom goes out to play. All we can say for sure is that she may be visiting or she may not be visiting. When she is not visiting, the rule cannot be disobeyed. When she is visiting, the rule is obeyed and so not disobeyed. In either case, the rule is not disobeyed.

The above rule is one way. It says what should happen when Aunt Jane visits without saying that she must be visiting when Tom goes out to play. When Tom goes out to play, the rule is not disobeyed when Aunt Jane is not visiting. It gives no information on her whereabouts. An example of a two-way rule is given later. See the second puzzle.

The Third Answer

The answer to the third question

When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting?

is like that of the second. When Aunt Jane is not visiting, the rule is not disobeyed if Tom goes out, and the rule is not disobeyed if Tom does not go out. When the rule is not disobeyed we can say nothing for certain about Tom when Aunt Jane is not visiting. The rule does not say that the only time Tom can go out to play is when his Aunt Jane visits. Again, the rule is only one-way. When Aunt Jane is not visiting, no information can be extracted from the rule. It says nothing about Tom.

The Fourth Answer

The fourth question is

What must happen for the given rule to be disobeyed?

The rule is disobeyed if Aunt Jane visits and Tom does not go out to play. That is, the situation where Aunt Jane visits and Tom does not go out to play must happen for the rule to be disobeyed.

The Fifth Answer

The fifth question is

When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom does not go out to play?

The rule will be disobeyed when Aunt Jane visits and Tom does not go out to play. To avoid the rule being disobeyed when Tom does not go out to play, Aunt Jane must not be visiting. The fifth answer is Aunt Jane is not visiting.

The contrapositive way of writing the above rule is When Tom not go out to play, Aunt Jane not visit. For this contrapositive rule to be never disobeyed, what can you say for sure when Aunt Jane visits? Answer: Not (Tom Not go out to play), that is, Tom goes out to play. The contrapositive of the contrapositive is the original rule. Later chapters on logic give more information, just a little more, about the contrapositive.

Some Vocabulary

The above rule is called a one-way implication rule. The first aim of this chapter is to show you the difference between one- and two-way implication rules. The meaning and use of the word implication will be talked about later. The five questions should help you see the difference between a one-way and a two-way implication rule. Seeing this difference signals that you can interpret precisely what a rule means.

Three Skills
For
Algebra
Volume 2

Chapter 2.

Implication Rules

The First Puzzle

A One-Way Implication Rule

Five Questions

The First Answer

The Second Answer

The Third Answer

The Fourth Answer

The Fifth Answer

Three Skills For Algebra Volume 2

Chapter 2.

Implication Rules

The First Puzzle

A One-Way Implication Rule

Five Questions

The First Answer

The Second Answer

The Third Answer

The Fourth Answer

The Fifth Answer

Three Skills
For
Algebra
Volume 2