Negating Unless/Without statements

[thomas33]

Hi I just want to confirm that I understand how to negate unless/without statements:

1) I will not go to the store unless Jane comes with me
Translation: Store --> Jane
Negated statement: I will go to the store even if Jane does NOT come with me

2) The economy will grow, unless interest rates increase
Translation: /economy grow --> interest rates increase
Negated statement: The economy will grow even if interest rate increase

I assume the same rules apply to without and maybe also except??

Are the above statements correct? Especially is the negation of statement 2) correct?

[Dana D]

Hey Thomas,

Your initial diagramming of the statements is correct, but I'm not sure what the negated statements are supposed to mean. I listed below here the contrapositives, which are the logical opposites to the statements listed. Your negated statements read almost as entirely new logical statements rather than the logical opposites of the conditional reasoning. The written out version of the contrapositive is what I think you were looking for with the 'negated statement', so I included that as well.

Store Jane comes
Contrapositive: Jane comes store
Written contrapositive: If Jane does not come, then I am not going to the store.


For the second statement, it should be:

Economy grows interest rates increase
Contrapositive: interest rates increase economy grows
Written contrapositive: If interest rates do not increase, the economy will grow.

Hope that helps!

[thomas33]

Thanks, Dana. I ask because I sometimes struggle with negating unless statements on Necessary Assumption questions (using the negation technique). I have read this previous article: viewtopic.php?t=8457

Where I am a bit confused by your answer is that I thought to create a logical opposite of a conditional statement, we need to negate the necessary condition. For unless statements this is tricky for me because I like to use the Unless Equation, which requires negating what is in the sufficient condition first and then I think through negating that statement.

To create a contrapositive, we need to negate both the necessary and sufficient conditions.

[Jeff Wren]

Hi Thomas,

I think it may have been unclear from your initial question why you were writing the negated statements of your conditional statements because that is not something that we usually do (except in the specific context of using the Assumption Negation Technique). Your follow up question clarified this, so this makes a lot more sense now. Contrapositives, on the other hand, are always something that you can use as they are logically equivalent and are used in many different LR questions with conditional reasoning.

As far as your questions, the first step is to use the Unless Equation to properly diagram the sentence.

Once you have correctly diagrammed the sentence, you no longer need to think of the sentence using "unless."

For example,

1) I will not go to the store unless Jane comes with me
Translation: Store --> Jane

Once you've properly diagrammed this, you can now think of this sentence going forward as:

If I go to the store, then Jane comes with me.

(This can be easier to work with than trying to deal with the tricky nature of "unless.")

As to your negation, it's almost right except for one word.

It should be:

I may (rather than will) go to the store even if Jane does NOT come with me.

(If you go back and look at Nikki's post about negations that you linked, you'll see the word "may" used.)

The reason is that what we are showing with the negation is that the sufficient condition may occur without the necessary, not that the sufficient condition will certainly occur without the necessary.

For your second example, the diagram looks good.

2) The economy will grow, unless interest rates increase
Translation: /economy grow --> interest rates increase

This would be worded "if the economy will not grow, then interest rates increase."

Unfortunately, your negation here is incorrect. It should be:

The economy may not grow even if interest rates do not increase.