Contrapositive confusion

[T9909]

I'm almost comfortable with creating contrapositives, but am wondering when we should apply them.

If A is 7 then B is 4 (this is the rule specifically written in the game)
Contrapositive B not 4 then A not 7

When do I apply the contrapositive to the game?

And lastly, there is an example in the most recent edition that states to always use the contrapositive in a double negative situation. Does this mean never use the original rule and apply the contrapositive only in a double negative situation? Or use both the original rule and contrapositive in all games?



Ready to throw my hands in the air with these games!

Thanks for your help, this forum has been fantastic!

[Jonathan Evans]

Hi, T9909,

Thanks for the questions! I concur that it's sometimes difficult to determine when the contrapositive is useful and when it is superfluous or confusing.

Could you please specify which page/edition you are referring to in the Logic Games Bible? In Chapter 5, there is a discussion of contrapositives that concerns how they can be used to form a "Double-Not Arrow."

There is also a discussion (page 284) of different ways the contrapositive could be useful or necessary on games.

As the book explains, a conditional statement and its contrapositive are logically equivalent to one another. While it might not always be necessary to write out the contrapositive explicitly, every conditional statement has a contrapositive.

Let me give you two examples:

• "J is not advertised unless H is advertised in the preceding week."

For this example, we'd have a J [HJ] clue. The contrapositive would be [HJ] J.

This contrapositive does very little to illustrate or elucidate. If we don't have an [HJ] block, then we don't have J? Okay, what does that mean?

This is a pretty good example of a conditional clue for which the contrapositive doesn't add a lot to the discussion.

In contrast, consider:

• If P is not selected, then R is selected.

Here we have the conditional clue P R.

The contrapositive would be R P.

This contrapositive is very useful! It tells us that either R or P (or both) must be selected. It is not possible for both R and P not to be selected. The contrapositive illustrates this scenario.

You might ask how to know whether to bother with a contrapositive. My default rule of thumb is at least to think about what the contrapositive for any conditional would be. Do you need to diagram them all? No. But at least stop for a second to ask what a contrapositive would mean and whether it would add anything to the discussion.

In addition to often being useful on its own, checking to see whether you understand the contrapositive is a quick way to check whether you understand the conditional, since a conditional and its contrapositive are logically equivalent!

Does this answer your question?

[T9909]

Thank you! Very helpful!

So always be aware of the contrapostive, but it doesn't necessarily mean it can always be applied to the game?

For your example, why both? I thought it would only be R or P.

"This contrapositive is very useful! It tells us that either R or P (or both) must be selected"

As for which page I am very confused with is page 287 and the two value system. It states to always use the contrapositive. Does that mean never use the original rule? and because the contrapositive shown on page 287 doesn't mean the same as the original rule, is the reason why take the contrapositive where it is diagrammed as L2--->H2

[Adam Tyson]

"Both" is allowed because there is nothing to prevent it from happening, T9909! That rule about R and P tells you what happens when one of the two variables is out, but it tells you nothing about when one of them is in! To assume that R being in forces P out would be to make a Mistaken Negation of the rule.

As for two-value systems, I'll stress what Jonathan said before, and that is that the original rule and the contrapositive are the same thing. They are logically equivalent, just two different ways of representing the exact same bit of information. My version of the LG Bible is different from yours, probably a little older, so I have a different page number than you, but our discussion of the two value system isn't about avoiding the original rule. It's about avoiding mistaken inferences. If the game has only two groups, labeled 1 and 2, and everyone is in one of those groups, then you have a two-value system. In that situation, a rule that says "if X is in Group 1 then W is in Group 2" does NOT mean that X and W can never be together in the same group. The original rule tells you what happens when X is in Group 1, and the contrapositive tells you what happens when W is in Group 1 (which is the only place he could go if he is not in Group 2). But it would be possible for X and W to both be in Group 2, because in such a case the rule is never triggered. The sufficient condition doesn't occur, so the rule can be ignored!

Use the rule, AND the contrapositive if it helps you to stay clear about things, and avoid mistaken negations and mistaken reversals. I hope that helps!