Motivation for definition of conditional?

[lamp23]

Is the following truth table the definition of the conditional?

p      q     p-->q

T      T      T
T      F      F
F      T      T 
F      F      T

What is the motivation for this definition? Is it defined this way just so that the definition for a biconditional works out in a desired way?
When others have tried to explain it to me, they have told me to think of the conditional as a promise. The reasoning makes sense for the two cases where p is true. However a typical line of reasoning when p is false goes like this: "If p is false, q can be true or false and it won't break the promise. Therefore the promise is still true."
1. I'm not sure if thinking of it as a promise is even appropriate. Saying the promise is still true, implies we know it was true before.
2. Just because the promise is not false, would not make it true. Just as we don't have enough information to call it false, we don't have enough information to call it true.

 

[pka]

Suppose that you tell a young child, "If you make an A on the next mathematics test then we will go to the state fair".
What is the only case ​in which you would break that promise to that child?

If she/he makes a B on the test but your family still goes to the state fair, did you break that promise?

If she/he makes a C on the test but your family did not go to the state fair, did you break that promise?

 

[lamp23]

Suppose that you tell a young child, "If you make an A on the next mathematics test then we will go to the state fair".
What is the only case ​in which you would break that promise to that child?

If she/he makes a B on the test but your family still goes to the state fair, did you break that promise?

If she/he makes a C on the test but your family did not go to the state fair, did you break that promise?

Refer to my problem 2.

 

[soroban]

Hello, lamp23!

Consider the "promise" interpretation that pka offered.

Suppose that you tell your son,
"If you make an A on the next mathematics test, then we will go to the state fair".

What is the only case ​in which you would break your promise to your son?

This is the important concept of an implication:
. . the promise was made on what would happen if he got an A.

Nothing was promised on what happens if he did not get an A.


There are four possible outcomes:

\(\displaystyle \begin{array}{cccc}(1) & \text{got A and fair} & \text{Promise kept} \\ (2) & \text{got A and not-fair} & \text{Promise broken} \\ (3) & \text{not-A and fair} & \text{Promise not broken} \\ (4) & \text{not-A and not-fair} & \text{Promise not broken} \end{array}\)

The promise is broken in case (2) only.


If he got an A and went to the fair,
. . he'd be satisfied; you kept your promise.

If he got an A and he did not go to the fair,
. . he has the right to protest "But you promised!"

If he did not get an A and still went to the fair,
. . he'd be pleasantly surprised, but would
. . agree that you did not break your promise.

If he did not get an A and did not go to fair,
. . he'd assume that that was part of the deal.


Hope this makes it clearer . . .

 

[pka]

2. Just because the promise is not false, would not make it true. Just as we don't have enough information to call it false, we don't have enough information to call it true.

Refer to my problem 2.

What you fail to understand is that in formal systems, all statements are either true or false. So if a statement is not false it must be true. Thus it has nothing to do with having enough information. In a formal setting statements are taken at face value.

Suppose we are given that "If P then Q" is a true statement.
If we also know that P is true then is must be the case that Q is also true. On the other hand, if we know that P is false the Q could be either true or false.

Likewise if we know Q is false then P must also be false. And if we know that Q is true then P can be either true or false.

It might be useful to review the following.
\(\displaystyle \text{1) If P then Q.}\)
\(\displaystyle \text{2) Not P or Q.}\)
\(\displaystyle \text{3) If not Q then not P.}\)
\(\displaystyle \text{4) P only if Q.}\)
\(\displaystyle \text{5) P is sufficient for Q.}\)
\(\displaystyle \text{6) Q is necessary for P.}\)
All six of those statements are equivalent.

 

[lamp23]

x=1 --> x+1 = 2 Is this the type of conditional we are talking about?

if so, then

x \(\displaystyle \neq\) 1 --> x+1=2 ?

 

[Someone2841]

x=1 --> x+1 = 2 Is this the type of conditional we are talking about?

if so, then

x \(\displaystyle \neq\) 1 --> x+1=2 ?

Also, it can be shown that \(\displaystyle x \neq 1 \to x+1=2\) is false because when \(\displaystyle x\neq1\) is true, \(\displaystyle x+1=2\) is false. Therefore, \(\displaystyle x\neq1 \to x+1=2\) is a false statement.

All \(\displaystyle p \to q\) means is that q is true whenever p is true. If this condition is not met for all p, \(\displaystyle p \to q\) is a false. It should be noted that p and q themselves can be any sort of values (e.g. binary or numerical). Let's take another example:


\(\displaystyle x = 1 \to x^2 = 1\) is a true statement because when \(\displaystyle x=1\), it can be shown that \(\displaystyle x^2=1\) is true. Let's take a couple other values of x. When \(\displaystyle x=-1\), \(\displaystyle x^2=1\) is also true. In this case, we have a \(\displaystyle \text{False} \to \text{True}\) situation (i.e. \(\displaystyle x=1\) is false since \(\displaystyle x=-1\) but \(\displaystyle x^2=1\) is true because \(\displaystyle (-1)^2 = 1\)) in a conditional statement that has been shown to be true. Any other value of x will result in \(\displaystyle \text{False} \to \text{False}\) (you can check this for any other value of x).


The truth table is not the definition of the conditional statement \(\displaystyle p \to q\); rather, it simply gives us some basic properties of the such a relationship between some p and q . It does not allow us to make wild claims like \(\displaystyle \text{Grass is blue}\to\text{Sky is green}\) (this can be proven with a simple thought experament: if we were to find blue grass, would the sky change color to green?) Rather, this truth table follows from the concept (or more rigorously, some applied definition) of \(\displaystyle p \to q\).

Again: \(\displaystyle p \to q\) simply defines a relationship between some p and q so that, for each and every value of p, q is necessarily true. It does not make any claims about q when whatever condition given by p is false.


Hope this helps.