Negation/Contrapositives of "Unless" statements

[zuize411]

Hi, could someone help me out with clearly understanding "unless" statements? I just realized that I don't fully grasp this concept after reviewing some drills in Lesson 2 and Lesson 5. How do you know when the logical opposite (same as contrapositive, correct?) of a statement is "might be" vs "will not be"? For example:

Lesson 1 has a statement that says: We cannot win the war unless we receive additional supplies.

Diagram: Win->Receive additional supplies
Contrapositive: do NOT receive additional supplies-> do NOT win

Lesson 5 has a statement that says: Happiness is impossible unless we profess a commitment to freedom.

I would diagram this statement like this: Happiness NOT impossible->profess a commitment to freedom
Contrapositive: do NOT profess a commitment->happiness impossible
The answer though, says: Even if we do not profess a commitment to freedom, happiness may still be possible.

I think what I'm missing is that the second statement builds on the knowledge in Lesson 2, so technically, the correct answer for the first statement would include could be true terms as well? Or am I totally missing the mark?

-Maria

[Dave Killoran]

Hi Maria,

Thanks for the question! I think your issues may go back to this comment you made:

How do you know when the logical opposite (same as contrapositive, correct?) of a statement is "might be" vs "will not be"?

The thing that appears to be tripping you up is thinking these two thing are identical (contrapositive and logical opposite) when they are not!

The contrapositive is the same statement in a different form (I sometimes say it's like two sides of the same coin, they look different but have the same value). It means the same thing but expressed in a different fashion. Physically, that proves is one where the two terms are reversed and both are negated. Both the contrapositives you posted above are correct.

A logical opposite is actually the opposite of the statement in question and so it does NOT have the same meaning. With a conditional statement, that means negating only the necessary condition, and not reversing anything.

The difference in ideas is why you see different results in those lessons. Lesson One is working with the contrapositive whereas Lesson Five is working with logical opposites. So, the outcomes will be very different!

Does that make sense? Let me know because I think that's where the problem is but we want to make sure it makes sense to you

Thanks!

[2020//Vision]

Hi Dave! I was hoping I could ask you a follow up question to this!

My confusion is with understanding the difference between the following:

• Negation in conditional logic.
• Negating statements.

For conditional logic, "will" negated becomes "will not"
For negating statements, "will" negated becomes "not necessarily" (ora term with similar logical force).

Am I understanding this correctly? If so, would it be accurate for me to write in my notes that when negating conditional statements we use polar opposites, but when negating non-conditional statements, we use the opposition construct?

[Robert Carroll]

2020//Vision,

Negating always involves logical opposites. That's definitional - the logical opposite of something is its negation. Conditionals are statements just like anything else, so there is no different kind of negation used. Can you give me some context or a particular example from a test question or homework drill that is possibly unclear? Thanks!

Robert Carroll