Conditional Statement Problem

[Jason76]

This might go in "Advanced Math" not sure.

Conditional Statement

"If you study 4 or more hours, you will get an A on the exam."

Truth Table

I didn't study study 4 or more hours, but I got an A on the exam. (if, then is T) How is that so?

The others I understand.

I studied 4 or more hours, and I got an A. (if, then = T)

I didn't study 4 or more hours, and I didn't get an A (if, then = T)

I studied 4 or more hours, and I didn't get an A (if, then = F)

This has something to do with the idea of hypothesis.

 

[pka]

This might go in "Advanced Math" not sure. Conditional Statement
"If you study 4 or more hours, you will get an A on the exam."
Truth Table
I didn't study study 4 or more hours, but I got an A on the exam. (if, then is T) How is that so?

The others I understand.
I studied 4 or more hours, and I got an A. (if, then = T)
I didn't study 4 or more hours, and I didn't get an A (if, then = T)
I studied 4 or more hours, and I didn't get an A (if, then = F)
This has something to do with the idea of hypothesis.

The statement that "If 1=2 then you got an A on the exam" is true.
"If 1=2 then you did not get an A on the exam" is also true.

"If 1=2 then P" is true, regardless of what P is.


The are two popular saying.
1) "A false statement implies any statement"

2) "A true statement is implied by any statement".

Those two sentences together sum up the whole idea of implication

 

[soroban]

Hello, Jason76!

If you study 4 or more hours, you will get an A on the exam.

Think of an implication as a promise.

Your teacher promises, "If you study 4 or more hours, you will get an A on the exam."

Note that he tells you what will happen if you study 4 or more hours.
. . He does not say what will happen if you do not study 4 or more hours.

Under what circumstances will he have kept his promise?
Under what circumstances does he break his promise?

[1] You studied 4 or more hours and you got an A.
. . . He kept his promise.

[2] You studied 4 or more hours and you did not get an A.
. . . He broke his promise.

[3] You did not study 4 or more hours and you got an A.
. . . This is possible. .He did not break his promise.
. . . He kept his promise.

[4] You did not study 4 or more hours and you did not get an A.
. . . This is possible. .He did not break his promise.
. . . He kept his promise.

Do you follow?

 

[JeffM]

A different way to think about this issue is this.

A conditional statement "If A then B" is OBVIOUSLY true whenever A and B are both true.

A conditional statement "If A then B" is OBVIOUSLY false whenever A is true and B is false.

But what do we say when A is false. In ordinary speech, we may say that "If A then B" is false if A is false. In more sophisticated speech, we say that if "A then B" is meaningless when A is false. But the logicians want everything to be true or false. Now how would I prove "If A then B" to be false? The statement says NOTHING about what happens when A is false. Consequently, nothing can prove it to be untrue when A is false. But that means that the statement is not false. By logical definition then, the statement is true.

An analogy is this. In American law, a jury must render a verdict of guilty or not guilty. In Scottish law, a jury can render three verdicts, guilty, not guilty, or not proven. Logic has decided that it is more useful to define "true" and "false" as exhaustive and mutually exclusive possiblities. If I say "If I had been Prime Minister of England in 1938, there would have been no Second World War", you cannot prove that I am lying so the logicians say I must be telling the truth. The point I am making is that the truth or falsity of conditional statements when the antecedent (A in our example) is false is a matter of definition. The implied definitions of ordinary discourse are not the same as the definitions of logic and mathematics. I was not Prime Minister in 1938 (in fact I was not even alive). So some people might say that the statement about my preventing war is a lie, but most people on reflection would say that the statement is just silly. Ordinary people, like the law of Scotland, normally think of three options, true, false, and meaningless. Logic, like American law, thinks about two options, and meaningless gets included with true.

You can't argue against definitions. All you can do is understand that different definitions lead to different results. At least in the realm of mathematics and logic, the experts have decided that it is most useful to have truth and falsity form a strict dichotomy. In most situations in real life, I personally think it is more useful to add the third category of meaningless to true and false. To remember the implications of the definitions in logic, both pka and soroban have given you very succinct explanations of what those implications are.

 

[Mrspi]

Hello, Jason76!


Think of an implication as a promise.

Your teacher promises, "If you study 4 or more hours, you will get an A on the exam."

Note that he tells you what will happen if you study 4 or more hours.
. . He does not say what will happen if you do not study 4 or more hours.

Under what circumstances will he have kept his promise?
Under what circumstances does he break his promise?

[1] You studied 4 or more hours and you got an A.
. . . He kept his promise.

[2] You studied 4 or more hours and you did not get an A.
. . . He broke his promise.

[3] You did not study 4 or more hours and you got an A.
. . . This is possible. .He did not break his promise.
. . . He kept his promise.

[4] You did not study 4 or more hours and you did not get an A.
. . . This is possible. .He did not break his promise.
. . . He kept his promise.

Do you follow?

Hooray, Soroban! This is the exact explanation that I used for YEARS as a geometry teacher (all other methods I'd tried previously didn't "work" with my students). So your answer validates my thinking! Thanks.

 

[dsk2]

This condition has to be satisfied at any cost: "If you study 4 or more hours, you will get an A on the exam." This is a rule written in stone.
If this rule is obeyed, the student will pass. No more, no less. It means a 1 year baby studies for 4 hours or more and takes the exam, the baby WILL pass. No questions. Why? Because the baby followed the rule.


The 4th statement is the only one that truly defies the condition and hence it is 'F'.

If it doesn't relate directly to the condition above, the outcome can be anything. In other words, if you dont study for four hours or more, the rule does not apply and the outcome can be anything. You have to accept the outcome. Eg. A nobel laureate in physics studies only for 3 hours 59 minutes and takes this test (lets say this test is an elementary 1st grade physics test) and does his/her best in the exam and can fail or pass. Both answers are 'T' => possible. If only he had studied for 2 more minutes, he would have surely passed no matter what only because now the rule kicks in.

In your example, the 1st statement does not directly defy the condition, hence the outcome can be anything. Both are possible outcomes, and you have to accept (acceptable means T) both outcomes.

Think about it!

Summary: 'T' means acceptable or possible outcome. It does not mean always true. 'F' means the rule has been broken and hence it is impossible or never possible. Read above explanation for clarification.

Cheers,
Sai.

P.S: Conditions are used a lot in programming to tell machines what to do, and unfortunately, all that the machines know is the conditions; they cannot see the surrounding circumstances or use general knowledge let alone use basic intuition to produce the result. In order to prevent the case where the outcome 'can be anything' as described before, in programming, you make sure you cover ALL possible scenarios and define what the outcome should be. In general, bad programs are ones where lets say a remove possibility has not been considered and the program crashes if the outcome is unpredictable and if you are lucky it may not crash but it is surely unstable.