# Math Help - Understanding truth table of p --> q

1. ## Understanding truth table of p --> q

Hi

I am studying Daniel J. Velleman's How to Prove it. I am various truth tables.
The truth table for $\vee , \wedge , \neg$ . These were simple to
understand with their comparison to usual English language equivalent. But I am having difficult time understanding the truth table of $P\to Q$.
The author gives an example of

"If $x>2$ then $x^2>4$".

I can see why the statement is true when both P and Q is true. I can see the
truth of the statement $P\to Q$ when both $P$ and $Q$ are true. But author then says that $P\to Q$ is true
even when $P$ and $Q$ are both false. I have trouble following
his arguments. We don't think like that in ordinary world. Can people help me ?

Edit: I read on many web sites about the explanations but I didn't find it satisfactory. So I am

2. ## Re: Understanding truth table of p --> q

Originally Posted by issacnewton
difficult time understanding the truth table of $P\to Q$.
truth of the statement $P\to Q$ when both $P$ and $Q$ are true. But author then says that $P\to Q$ is true
even when $P$ and $Q$ are both false. I have trouble following
his arguments. We don't think like that in ordinary world. Can people help me ?
Edit: I read on many web sites about the explanations but I didn't find it satisfactory.
I don't think you are going to satisfied here either.
$\text{If P then Q.}$ simply means if P is true then Q must also be true.
So there is only case where $\text{If P then Q}$ is false: $\text{If T then F}$.
All of the other three are true cases.

Here is a popular saying about implications: A false statement implies any statement; a true statement is implied by any statement.

3. ## Re: Understanding truth table of p --> q

Plato, thats what I have trouble understanding. When was the truth table of the implication invented ? What thought process did go into it ? Do you know any resources (online would be great) where they try to convince the validity of this truth table with lots of examples ?

thanks

4. ## Re: Understanding truth table of p --> q

Originally Posted by issacnewton
Plato, thats what I have trouble understanding. When was the truth table of the implication invented ?
In my view it is incorrect to say "truth table of the implication invented ."

Rather there is a precise meaning for "If...then... " statements.
In the past, I have found that it helps some students to fully realize this equivalence: $\left( {P \Rightarrow Q} \right) \equiv \left( {\neg P \vee Q} \right)$
Here the truth table for the RHS.

5. ## Re: Understanding truth table of p --> q

Plato, yes I saw that equivalence in Velleman's book. He discusses it AFTER introducing the truth table of P --> Q . So I didn't notice it there. But seeing it
now, I think I agree with you that if we take that as the starting point to see the similar logical structure of $\left(\neg P \vee Q \right)$ and
$P\to Q$ . Velleman gives some examples to make this equivalence clear. Do you know any more examples from the net explaining the same point ?
I think when the truth table for the implication was first discovered (or invented), that person probably had $\left(\neg P \vee Q \right)$ in mind.
And later , he/she may have seen that its same thing as $P\to Q$ . am I right ?

6. ## Re: Understanding truth table of p --> q

Originally Posted by issacnewton
If $x>2$ then $x^2>4$
Well, do you find it natural that this statement is true for all x? If so, then it is true for x = 1 even though both 1 > 2 and 1^2 > 4 are false.

In real life, we do assume that "if P, then Q" is true when P is false. For example, the US Constitution says that being at least 35 years old is a necessary condition to become a US president. In other words, for every person x, if x is the president of the US, then x is at least 35 years old. This is a true statement. (I don't know if there were any exceptions in the past, but let's ignore that.) Now take any person x who is not 35 years old; then the premise of the statement above is false, yet the full statement continues to be true.

7. ## Re: Understanding truth table of p --> q

Originally Posted by emakarov

In real life, we do assume that "if P, then Q" is true when P is false. For example, the US Constitution says that being at least 35 years old is a necessary condition to become a US president. In other words, for every person x, if x is the president of the US, then x is at least 35 years old. This is a true statement. (I don't know if there were any exceptions in the past, but let's ignore that.) Now take any person x who is not 35 years old; then the premise of the statement above is false, yet the full statement continues to be true.
so when the person is not 35 years old, the statement will be "If person x is the president of the US , then x is at least 35 years old" .
premise here is P = "Person x is the president of the US " which is False since person is not 35 yet. But why would that make the statement
true ? Has it anything to do with the condition 'if' ?

8. ## Re: Understanding truth table of p --> q

Originally Posted by issacnewton
Do you know any more examples from the net explaining the same point ?
I think when the truth table for the implication was first discovered (or invented), that person probably had $\left(\neg P \vee Q \right)$ in mind.
And later , he/she may have seen that its same thing as $P\to Q$ . am I right ?
That's probably the thought process behind it. I like to think in terms of the contrapositive of "false implies false", which is "true implies true", which is of course true. Since I intuitively understand contrapositives, this is enough to convince me.

9. ## Re: Understanding truth table of p --> q

premise here is P = "Person x is the president of the US " which is False since person is not 35 yet. But why would that make the statement true ?
The statement is true by the formal definition of implication, i.e., its truth table. However, we are now talking about the common-sense rationale for this definition. Do you accept that "for all x, if x is the president of the US, then x is at least 35 years old" is true intuitively?

10. ## Re: Understanding truth table of p --> q

Lob, yes contrapositives are probably good way of approaching it. But sometimes we can come up with statements where there is no causal relation between P and Q.
Consider the following statement.

If $1+1 \neq 2$ then Paris is the capital of France.

so here P is false and Q is true but how do we decide the truth value of the whole
statement ? here we don't have any causal relation between P and Q.

11. ## Re: Understanding truth table of p --> q

Originally Posted by emakarov
However, we are now talking about the common-sense rationale for this definition. Do you accept that "for all x, if x is the president of the US, then x is at least 35 years old" is true intuitively?
Yes , its true.

12. ## Re: Understanding truth table of p --> q

my favorite example is: "whenever the sky is green, i shall be king". such a statement is "vacuously true", because the sky is never green. think of it this way: a statment like: if p, then q is like an instruction, when p is true, we may proceed to q. well, if p is not true, then whether or not q is true, the instruction is still correct. on the other hand if p is indeed true, but q is not, the instruction is incorrect (a bad instruction).

13. ## Re: Understanding truth table of p --> q

Hello, issacnewton!

I am having difficult time understanding the truth table of $P\to Q$.

The author gives an example: .If $x>2$, then $x^2>4$

I can see the truth of the statement $P\to Q$ when both $P$ and $Q$ are true.
But author then says that $P\to Q$ is true even when $P$ and $Q$ are both false.

I have trouble following his arguments. We don't think like that in ordinary world.
Can you people help me ?

I had my doubts, too . . . but finally my professor explained it to me.

Suppose you learn that you have an allergy to, say, asparagus.

Your doctor give you this warning:

. . $\text{If } \underbrace{\text{you eat asparagus,}}_P \; \underbrace{\text{then}}_{\to} \; \underbrace{\text{you will get indigestion}}_Q$

Think of this as a promise of the state of your digestive tract
. . should you be foolish enough to eat asparagus.

Hence, it is impossible for you to eat asparagus and not get indigestion.
. . That is, $P \wedge \sim\!Q$ is not possible.

What if you don't eat asparagus?

It is possible that you do not get indigestion.
. . $\sim\!P\, \wedge \sim\!Q$ is possible.

But it is also possible that you do get indigestion (from another cause).
. . $\sim\!P \wedge Q$ is possible.

The doctor's statement says what will happen if you do eat asparagus.
He made no promises on what happens if you don't eat asparagus.

$\text{And that is why }\,P \to\, \sim\!Q\,\text{ is false, and }\,\begin{Bmatrix}P \to Q \\ \sim\!P\to Q \\ \sim\!P\to\,\sim\!Q\end{Bmatrix}\,\text{ are all true.}$

14. ## Re: Understanding truth table of p --> q

Originally Posted by emakarov
However, we are now talking about the common-sense rationale for this definition. Do you accept that "for all x, if x is the president of the US, then x is at least 35 years old" is true intuitively?
Originally Posted by issacnewton
Yes , its true.
Well, since the universal statement is true, so is each of its instantiations. In particular, one can instantiate x with a person who is not a president, regardless of whether he/she is over 35. Then the premise is false while the conclusion can be either true or false. The whole implication is true as an instantiation of a true universal statement.

15. ## Re: Understanding truth table of p --> q

Deveno, Soroban

Your posts made things crystal clear. This Velleman's book is not THAT clear. Which introductory logic books did you study from ?. Soroban, your example is just
too good. I just have one doubt. Is it possible to use that reasoning even if there is no causal relationship between P and Q.
Do we first construct the truth table with P and Q having some causal relationship , like Soroban's example ,and then we extend the truth table to cases
where there is no apparent causal relationship between P and Q ...... ?

Can you suggest online logic sites where I can do such fun exercises ?

emakarov, thanks for the input.

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