## I need help on figuring out this contrapositive! Please!

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roranoa

Posts: 559
Joined: Sat Aug 22, 2009 11:18 am

### I need help on figuring out this contrapositive! Please!

I don't see how the contrapositive of this sentence works:

You cannot get into YLS only by studying hard.

or

You cannot get into YLS merely by studying hard.

what's the contrapositive of this?

Adding to that if there is a premise that says:

You can obtain X only by doing A

is the following conclusion must also be true?

You cannot obtain X by merely doing B.

I really don't get this. Would someone help me?

It would be helpful also if you can tell me how to diagram this.
Last edited by roranoa on Wed Mar 10, 2010 9:23 pm, edited 1 time in total.

Shrimps

Posts: 269
Joined: Sun Feb 14, 2010 10:04 pm

### Re: I need help on figuring out this contrapositive! Please!

roranoa wrote:I don't see how the contrapositive of this sentence works:

You cannot get into YLS only by studying hard.

or

You cannot get into YLS merely by studying hard.

what's the contrapositive of this?

Where is the example from?

Studying hard is insufficient for getting into YLS, but apparently necessary:

Getting into YLS --> Studying hard (not studying hard --> not getting into YLS)

You can obtain X only by doint A

is the following conclusion must also be true?

You cannot obtain X by merely doing B.

You can earn a million dollars (X) only by starting a successful business (A)
You cannot earn a million dollars (X) merely by starting a successful restaurant chain (B)

If B is at least partly a subet of A, then the conclusion is false. If no B is A (B<-|->A), then the conclusion is true.

roranoa

Posts: 559
Joined: Sat Aug 22, 2009 11:18 am

### Re: I need help on figuring out this contrapositive! Please!

Studying hard is insufficient for getting into YLS, but apparently necessary:

How can it be apprently necessary? the sentence only states that studying hard won't work by itself.

It doesn't mean that studying hard is necessary at all.

autarkh

Posts: 314
Joined: Sat Sep 26, 2009 9:05 pm

### Re: I need help on figuring out this contrapositive! Please!

There's some ambiguity here.

One way to think about it has already been suggested: studying hard, by itself, is insufficient to guarantee admission to YLS.

Implicit in this is that studying hard IS necessary, but must accompany something else. For the sake of argument, let's call "zing" the "something else".

YLS --> Zing & Study Hard; [strike]Zing[/strike] OR [strike]Study Hard[/strike] --> [strike]YLS[/strike] (contrapositive)

Another way to do it is to include the "only" as a part of the condition rather than treating it as a word indicating necessity.

Only Study Hard --> [strike]YLS[/strike]; YLS --> [strike]Only Study Hard[/strike] (contrapositive)

The second method requires no implicit assumptions.

Hope that helps.

BrightLine

Posts: 259
Joined: Mon Mar 08, 2010 10:15 pm

### Re: I need help on figuring out this contrapositive! Please!

autarkh wrote:There's some ambiguity here.

One way to think about it has already been suggested: studying hard, by itself, is insufficient to guarantee admission to YLS.

Implicit in this is that studying hard IS necessary, but must accompany something else. For the sake of argument, let's call "zing" the "something else".

YLS --> Zing & Study Hard; [strike]Zing[/strike] OR [strike]Study Hard[/strike] --> [strike]YLS[/strike] (contrapositive)

Another way to do it is to include the "only" as a part of the condition rather than treating it as a word indicating necessity.

Only Study Hard --> [strike]YLS[/strike]; YLS --> [strike]Only Study Hard[/strike] (contrapositive)

The second method requires no implicit assumptions.

Hope that helps.

EDIT: I rethought it.

Only Study Hard ---> ~ get into Yail
Got into Yail ----> ~only study hard

The difficulty comes because in formal logic, the word "only" has a specific meaning. But the "only" in this case does not refer to the logic. To make is simpler, change "only" to "merely" and it should become more intuitive.

Just to be clear, "~only study hard" means either you studied hard and did something else or did not study hard.
Last edited by BrightLine on Wed Mar 10, 2010 11:52 pm, edited 1 time in total.

BrightLine

Posts: 259
Joined: Mon Mar 08, 2010 10:15 pm

### Re: I need help on figuring out this contrapositive! Please!

roranoa wrote:what's the contrapositive of this?

Adding to that if there is a premise that says:

You can obtain X only by doing A

is the following conclusion must also be true?

You cannot obtain X by merely doing B.

I really don't get this. Would someone help me?

It would be helpful also if you can tell me how to diagram this.

The second part is this

~ A was done----> ~ obtain X
X is obtained----> A was done

The conclusion You cannot obtain X by merely doing B. only follows if the following premise is true "A is not equal to B".

This is because the premise has told you if you dont do A then you cant obtain X. So if you only did B, can you obtain X?

The answer is no so long as A is not the same thing as B. Because if it is then if you did B then you did A as. If you merely did B (which is also A) then you may or not achieve X.

But if A is not B. and you merely did B, that you means you did not do A, which according to the rule means you did not obtain X

I hope i did not make that too complicated, if i did ill rephrase.

autarkh

Posts: 314
Joined: Sat Sep 26, 2009 9:05 pm

### Re: I need help on figuring out this contrapositive! Please!

BrightLine wrote:
autarkh wrote:There's some ambiguity here.

One way to think about it has already been suggested: studying hard, by itself, is insufficient to guarantee admission to YLS.

Implicit in this is that studying hard IS necessary, but must accompany something else. For the sake of argument, let's call "zing" the "something else".

YLS --> Zing & Study Hard; [strike]Zing[/strike] OR [strike]Study Hard[/strike] --> [strike]YLS[/strike] (contrapositive)

Another way to do it is to include the "only" as a part of the condition rather than treating it as a word indicating necessity.

Only Study Hard --> [strike]YLS[/strike]; YLS --> [strike]Only Study Hard[/strike] (contrapositive)

The second method requires no implicit assumptions.

Hope that helps.

You cant force every "rule" into a contra-positive. It can only be done with conditionals.

The only conditional you can actually make is Study hard-----> Get into Yale V ~get into Yale.

But that means nothing.

Agreed as far as "studying hard" not guaranteeing anything -- as you said: (EITHER) get into Yale (OR) NOT get into Yale.

But weren't we told explicitly that "you can't get into Yale only by studying hard"? You don't consider that a conditional?

BrightLine

Posts: 259
Joined: Mon Mar 08, 2010 10:15 pm

### Re: I need help on figuring out this contrapositive! Please!

autarkh wrote:
BrightLine wrote:
autarkh wrote:There's some ambiguity here.

One way to think about it has already been suggested: studying hard, by itself, is insufficient to guarantee admission to YLS.

Implicit in this is that studying hard IS necessary, but must accompany something else. For the sake of argument, let's call "zing" the "something else".

YLS --> Zing & Study Hard; [strike]Zing[/strike] OR [strike]Study Hard[/strike] --> [strike]YLS[/strike] (contrapositive)

Another way to do it is to include the "only" as a part of the condition rather than treating it as a word indicating necessity.

Only Study Hard --> [strike]YLS[/strike]; YLS --> [strike]Only Study Hard[/strike] (contrapositive)

The second method requires no implicit assumptions.

Hope that helps.

You cant force every "rule" into a contra-positive. It can only be done with conditionals.

The only conditional you can actually make is Study hard-----> Get into Yale V ~get into Yale.

But that means nothing.

Agreed as far as "studying hard" not guaranteeing anything -- as you said: (EITHER) get into Yale (OR) NOT get into Yale.

But weren't we told explicitly that "you can't get into Yale only by studying hard"? You don't consider that a conditional?

Yea, i had edited that a couple minutes later.

My mistake was trying to view "only" as part of the logical function and not the condition. It is this

Only Study Hard ---> ~ get into Yail
Got into Yail ----> ~Only study hard

MysticalWheel

Posts: 354
Joined: Sun Jan 17, 2010 8:23 pm

### Re: I need help on figuring out this contrapositive! Please!

Rewrite it into an "if/then" statement:

If you only study hard, you cannot get into Yale.

Thus, the contrapositive is formed by negating the original "then" and setting it as the new "if" and negating the original "if" and setting it as the new "then":

If you got into Yale, you did NOT ONLY study hard.

That seems correct to me. The previous posters stated the same or similar as this but in varying logical symbols which made it a bit opaque (at least for me).

BrightLine

Posts: 259
Joined: Mon Mar 08, 2010 10:15 pm

### Re: I need help on figuring out this contrapositive! Please!

MysticalWheel wrote:Rewrite it into an "if/then" statement:

If you only study hard, you cannot get into Yale.

Thus, the contrapositive is formed by negating the original "then" and setting it as the new "if" and negating the original "if" and setting it as the new "then":

If you got into Yale, you did NOT ONLY study hard.

That seems correct to me. The previous posters stated the same or similar as this but in varying logical symbols which made it a bit opaque (at least for me).

Correct, exactly what I was saying.

The symbols were

---> "Conditional"
v "or"
~ "negation"

Sorry if it was confusing.

MysticalWheel

Posts: 354
Joined: Sun Jan 17, 2010 8:23 pm

### Re: I need help on figuring out this contrapositive! Please!

BrightLine wrote:
Correct, exactly what I was saying.

The symbols were

---> "Conditional"
v "or"
~ "negation"

Sorry if it was confusing.

I should know those anyways, though. Thanks for posting the definitions. The correctness of the idea is the most important component of the exercise; the terms of its expression will naturally have some variation.

Posts: 182
Joined: Thu Jan 28, 2010 6:19 am

### Re: I need help on figuring out this contrapositive! Please!

MysticalWheel wrote:Rewrite it into an "if/then" statement:

If you only study hard, you cannot get into Yale.

Thus, the contrapositive is formed by negating the original "then" and setting it as the new "if" and negating the original "if" and setting it as the new "then":

If you got into Yale, you did NOT ONLY study hard.

That seems correct to me. The previous posters stated the same or similar as this but in varying logical symbols which made it a bit opaque (at least for me).

TITCR

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