Implication is My Enemy

[nasi112]

#​	p​	q​	p ---> q​
1​	T​	T​	T​
2​	T​	F​	F​
3​	F​	T​	T​
4​	F​	F​	T​

I have a problem understanding implication when I have English sentences.

Let us assume p is (I study well) and q is (I can get A)

According to the truth table, these sentences are:
1
If I study well, then I can get A. True.
2
If I study well, then I cannot get A. False.
3
If I don't study well, then I can get A. True.
4
If I don't study well, I cannot get A. True.

Why the third sentence is True?

 

[topsquark]

I did sentence 3 all the time in my High School Math classes. I generally didn't need to study. You are using the word "can" which is not a definite statement.

What you would do if q were "I will get an A" is beyond me.

-Dan

 

[nasi112]

I did sentence 3 all the time in my High School Math classes. I generally didn't need to study. You are using the word "can" which is not a definite statement.

What you would do if q were "I will get an A" is beyond me.

-Dan

Thanks Dan,

I will get an A or I can get an A is same for me. I am confused about the idea of implication. If studying is a necessity of getting A, how can if I don't study I still get an A.

 

[lex]

The implication does not mean that studying is a necessity for getting A.
[MATH]p \implies q[/MATH] means p is sufficient for q, not that it is necessary for q.
[MATH]q \implies p[/MATH] means p is necessary for q.

 

[nasi112]

Thanks lex.

Let us assume q is i study well and p is I can get an A

if q is False and p is True
q ---> p will be
If I don't study well, I can get an A.

Why getting an A is necessary for not studying?

 

[pka]

Let us assume q is i study well and p is I can get an A
if q is False and p is True
q ---> p will be
If I don't study well, I can get an A.
Why getting an A is necessary for not studying?

Suppose that P implies Q \(P\to Q\). Each of the following is equivalent.
Not Q implies not P, \(\neg Q\to \neg P\).
Not P or Q, \(\neg P\vee Q\).
P is sufficient for Q.
Q is necessary for P.
Remember that each of those five is the same as each of the others.
This is often summed up as:
A false statement implies any statement is true.
A true statement is implied by any statement is true.

 

[lex]

Let us assume q is i study well and p is I can get an A

if q is False and p is True
q ---> p will be
If I don't study well, I can get an A.

Why getting an A is necessary for not studying?

You are confusing the statement with the truth values of the variables.
You tell me that q is i study well and p is I can get an A

(if q is False and p is True)
q ---> p will be
( If I don't study well, I can get an A).

No q ---> p will be just as you defined them:
If I study well then I can get an A

The question you ask concerns the truth value of the statement:
If I study well then I can get an A
Does this statement have value T or F, when q (I study well) is False and p (I can get an A) is True?

That is understanding the question.
Understanding the answer is another matter.

 

[nasi112]

Thanks pka.

When playing with letters, it is easy. I understand what you have written above.

This expression is very important if you want to simplify Boolean expressions [MATH]P \to Q = \neg P \vee Q[/MATH]
But translating English sentences to symbols and vice versa is confusing me.

 

[nasi112]

You are confusing the statement with the truth values of the variables.
You tell me that q is i study well and p is I can get an A



No q ---> p will be just as you defined them:
If I study well then I can get an A

The question you ask concerns the truth value of the statement:
If I study well then I can get an A
Does this statement have value T or F, when q (I study well) is False and p (I can get an A) is True?

That is understanding the question.
Understanding the answer is another matter.

I am confused of all of this . Let us Re-write things again.

p is (I study well)
q is (I can get an A)

Now, look at #[MATH]3[/MATH] in the truth table

p ---> q will be

If I don't study well, I can get an A.

Our argument, why is this sentence true?

 

[nasi112]

I am confused of all of this . Let us Re-write things again.

p is (I study well)
q is (I can get an A)

Now, look at #[MATH]3[/MATH] in the truth table

p ---> q will be

If I don't study well, I can get an A.

Our argument, why is this sentence true?

Why here is studying is not necessary?

 

[pka]

Why here is studying is not necessary?

You are letting the actual meaning get in the way
Consider this statement: If it rains then the grass is wet.
If you look outside is see wet grass can you conclude that it has rained.
No indeed! In the American South we almost every summer morning we wake to a heavy dew.
It did not rain but the grass is wet anyway. That does not make the statement false.

Suppose we find that the grass is not wet can you conclude that it has not rained.
Yes indeed, because we know the given sentence is true.
Had it rained, then the grass would be wet.

Raining is sufficient for the grass to be wet.
The grass being wet is a necessary condition for it to have rained.

 

[lex]

p is (I study well)
q is (I can get an A)

Now, look at #33\displaystyle 3 in the truth table

p ---> q will be

If I don't study well, I can get an A.

You've done the same again!
You need to re-read my previous post and try to understand what it is saying.
We don't change the statement.
(p [MATH]\implies[/MATH] q) is still ((I study well) [MATH]\implies[/MATH] (I can get an A))
irrespective of what values the variables take (whether or not (I study well ) is True/False).

I must adjourn for sleep!

 

[nasi112]

You are letting the actual meaning get in the way
Consider this statement: If it rains then the grass is wet.
If you look outside is see wet grass can you conclude that it has rained.
No indeed! In the American South we almost every summer morning we wake to a heavy dew.
It did not rain but the grass is wet anyway. That does not make the statement false.

Suppose we find that the grass is not wet can you conclude that it has not rained.
Yes indeed, because we know the given sentence is true.
Had it rained, then the grass would be wet.

Raining is sufficient for the grass to be wet.
The grass being wet is a necessary condition for it to have rained.

Thanks pka for the example. It made it clear now.

You've done the same again!
You need to re-read my previous post and try to understand what it is saying.
We don't change the statement.
(p [MATH]\implies[/MATH] q) is still ((I study well) [MATH]\implies[/MATH] (I can get an A))
irrespective of what values the variables take (whether or not (I study well ) is True/False).

I must adjourn for sleep!

Thanks lex again. I got it, but I hope that I can answer the next questions correctly.


I will post another example soon, I will convert the English sentence to symbols, and I hope you see the questions guys.