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JoshG May 14, 2020

Breaking Apart Compound Statements

Hi There, I'm looking for some clarity in regards to when it is and is not acceptable to break apart conditional statements. I believe there are four possible combinations here: 1.) AND IN SUFFICIENT -- Could be broken apart. Carrots and Turnips are vegetables. C & T --> V This is logically equivalent to: Carrots are vegetables. Turnips are also vegetables. C --> V T --> V 2.) OR IN SUFFICIENT -- Could be broken apart. Carrots or turnips are vegetables. C or T --> V This is logically equivalent to: Carrots are vegetables. Alternatively, carrots are vegetables. C --> V T --> V 3.) AND IN NECESSARY -- Could be broken apart (needed for choice E). Dogs are cute and smelly. D --> C & S This is logically equivalent to: Dogs are cute. Dogs are also smelly. D --> C D --> S 4.) OR IN NECESSARY -- Cannot be broken apart. Dogs are cute or smelly. D --> C or S This would not be logically equivalent to Dogs are cute. Alternatively, dogs are smelly. D --> C D --> S Is this a correct analysis? Am I missing any alternatives? The contrapositives of these statements would be covered under these rules, as well, correct? Thank you.

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JoshG May 14, 2020

Can you please also address how the presence of quantifiers impacts the breakability of and/or statements, if at all?

Brett-Lindsay July 18, 2020

Hi @JoshG,

I've been wrestling with this question for about 2 hours now, reading every message and diagramming constantly. Thanks to some of the messages from @Sam and @Ravi, among others, I think (hope) I've finally got it.

All of your statements are correct except for the first one.

Carrots and Turnips are vegetables.
C & T --> V

I think that we would only use an "and" to combine sufficient statements when they are dependent on one another. Carrots and Turnips are independent variables, so they shouldn't be combined with "and."

If we did diagram C & T --> V, it would mean that if an item were Both a Carrot And a Turnip, it must be a Vegetable. Or, as one of the instructors (Sam, I think) wrote, "All Carrots that are Turnips are Vegetables"

I'm not sure, but maybe it should be diagrammed as
C or T --> V
"If it's a Carrot, or If it's a Turnip, it must be a Vegetable"

If I've made any mistakes, please correct me. This is really hard for me, too.

shunhe July 23, 2020

Hi @JoshG and @Brett-Lindsay,

Thanks for the question! Just wanted to confirm what @Brett-Lindsay said; all of those statements are correct except the first. Basically, we can break up “and” when it’s in the necessary condition, and we can break up “or” when it’s in the sufficient condition. When we have something that says

A & B —> C

That means that if something has BOTH properties A and B, then we can infer C. But if it just has one, then that might not be true, which is why we can’t break it up. For example, let’s say I will only go outside if the two conditions hold: it’s not raining, and a friend comes with me. Both of these conditions need to hold. So we diagram this

Friend & ~Rain —> Go outside

Now if we break this up, we would have

Friend —> Go outside
~Rain —> Go outside

But if we take those two conditionals separately, that means “friend” by itself would be sufficient for me going outside, whether or not it was raining. Or “not rain” would be sufficient for me going outside, whether or not I had a friend. But I want both conditions to be true before I go outside. So we can’t split up “and” in the sufficient like this.
?So it’s a bit tricky because when we say things like “carrots and turnips are vegetables,” we really mean “if something is a carrot, or something is a turnip, it’s a vegetable,” since both carrots and turnips are vegetables. So that would be diagrammed as

C v T —> V

The phrase

C & T —> V

Actually means that if something is both a turnip and a carrot, it’s a vegetable. It has to have both conditions be true. So something that was just a turnip wouldn’t be a vegetable, under this definition; it would also have to be a carrot.

Hope this helps! Feel free to ask any other questions that you might have.

Brett-Lindsay July 26, 2020

Thanks @shunhe.

Great explanation! Understanding this stuff is taking a while, but it's coming.

Krystina April 4, 2022

"We can do this when there is an OR statement in the sufficient condition (as in the first sentence) or an AND statement in the necessary condition (as in the second)."