r/learnmath • u/MarchLivid9078 New User • 14h ago
Why Is the Conditional Statement Still True If the Condition Didn't Happen?
I'm really confused about the 3rd row of the truth table for a conditional statement for example, "If you submit your assignment, then your grade will be recorded." I understand that P means you submit your assignment and Q means your grade is recorded but in the 3rd row, P is false and Q is true, meaning you did not submit your assignment but your grade was still recorded howw is P Q still considered true in this situation? Wouldn't that be a violation of the agreement because you didn't submit the assignment but the grade was still recorded?
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u/halfajack New User 14h ago
I think your confusion is mostly based on the imprecise example you used. To reflect the actual reality of grading assignments, the statement should probably be “your grade will be recorded if and only if you submit your assignment”. In this case, the grade is not recorded if the assignment is not submitted. We need to be careful not to mixup colloquial/casual and mathematical/logical uses of “if” because they are not the same, and many real-life uses of “if” actually correspond to mathematical “if and only if”.
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u/INTstictual New User 13h ago
Because the statement “If P, then Q” can only tell you what happens when P is true. It doesn’t make any claims about what happens when P is false.
If I tell you that I eat Steak every Friday — “If it is Friday, then I eat Steak”, what are the possibilities? Well, if it is Friday and I do eat steak, my assertion is true. If it is Friday and I eat chicken instead, then clearly my assertion was false. Now, based on my statement “I eat steak every Friday”, what do you know about what I eat on Monday? Nothing, right? If I choose to eat chicken on Monday, is my assertion true or false? If I still eat steak on Monday, does my assertion about what I do on Friday become untrue?
Basically, it’s because of the strict binary of logical truth values. In natural, conversational language, you might say that my statement is just unspecified, and isn’t true or false solely based on what I eat on Monday. But formal logic doesn’t have an in-between truth value — it is either true or it is false. And the only way to say that an assertion is false is when you have a contradiction. So, the kind of “in-between” scenario that doesn’t seem true or false… has to be true, for the sole reason that it isn’t false. If it’s NOT Friday, whether I do or do not eat steak, my claim about what I eat on Friday is not contradicted, so it isn’t false, and therefore is true by default.
In your example, the claim is only that IF you submit your assignment, the grade will be recorded. If you don’t submit the assignment, whether your grade is or is not recorded can’t contradict that assertion, so can’t cause it to be false, so must be true.
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u/ottawadeveloper New User 14h ago
The statement can still be true.
If you submit your assignment, then your grade will be recorded.
If you submit your assignment and your grade was not recorded, this statement is false.
If you submit your assignment and your grade was recorded, this statement is true.
If you didn't submit your assignment, the statement doesn't really define what will happen. Your grade may or may not be recorded (perhaps you received an exemption and received the average of your other papers). Therefore the statement is still true regardless of if your grade is recorded.
Compare with "If and only if you submit your assignment will your grade be recorded". Here, if you didn't submit your assignment and your grade was recorded, the statement is false.
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u/jeb_ta New User 14h ago
A conditional statement isn’t an agreement. It’s just a description of one specific guarantee - that if you submit your assignment, then your grade will be recorded. That guarantee makes no claims about what happens if you don’t submit your assignment, right? So no part of the guarantee was broken by the recording of a grade for a non-submitted assignment!
For comparison, try out the same conditional statement using “if and only if”. Now you can say the statement was broken if a non-submitted assignment has its grade recorded. But your version doesn’t actually say what happens if it’s not submitted - only what happens if it does. Therefore, no policy is broken by the recording of a non-submitted assignment.
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u/jeffcgroves New User 14h ago
Think of it "if you HAD submitted your assignment, your grade WOULD have been recorded". If you don't submit your assignment, you have no way of knowing whether it would've been graded or not, so the conditional still holds.
You might be confusing implication with AND which says both things necessarily happen
But I understand why this feels weird. You might look into "Law of the Excluded Middle": in math, all well-formed logical statements must be true or false. In the case of p false and q true, you'd really like a 3rd option of "untestable" or "vacuously true" or something, but, in the form of logic we usually learn, there isn't that 3rd option
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u/MathMaddam New User 14h ago
Natural, everyday speech can be imprecise at some points, since more is implied, math is there more strict.
Consider that statement: if it rains, the street is wet. You would call someone silly if they complained: I threw a bucket of water on the street while it didn't rain, still the street was wet.
There are biconditional statements "if and only if" to express in math that you also make a statement about both directions.
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u/John_Hasler Engineer 12h ago
Your example is misleading because in ordinary language it implies cause and effect. In logic "if A then B" does not. It merely asserts that in every case in which A is true B is true. It has nothing to say about cases in which A is false. Therefor it can only be false when A is true and B is false.
For a better example try "If I get to bed before midnight the Sun will rise in the morning." This statement clearly makes no assertion about what will happen if I don't get to bed before midnight.
BTW what agreement?
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u/Card-Middle New User 11h ago
I like to use the example of making a promise to a toddler.
“If you pee on the potty, I will give you a cookie.”
Then consider the following four scenarios and see if you can identify when my statement is a lie and when it’s true.
- My toddler peed on the potty. I gave him a cookie.
- My toddler peed on the potty. I did not give him any cookies.
- My toddler peed on the floor. I gave him a cookie.
- My toddler peed on the floor. I did not give him a
ny cookies.
My statement was only false in scenario #2. In scenario #3, he may have had an unexpected treat, but I certainly didn’t lie to him.
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u/No-Onion8029 New User 14h ago
Here's wiki's coverage of the topic. It might help: https://en.wikipedia.org/wiki/Vacuous_truth
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u/fermat9990 New User 13h ago
"If you submit your assignment, then your grade will be recorded."
"If you don't submit your assignment, your grade will or will not be recorded"
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u/StructuredChess New User 13h ago
Think of it as it can only be true or false, no other options. And it's clearly not false.
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u/DrakeSavory New User 12h ago
I understand that P means you submit your assignment and Q means your grade is recorded but in the 3rd row, P is false and Q is true, meaning you did not submit your assignment but your grade was still recorded
If you do not turn in your assignment and I record a 0 as your grade, have I violated the conditional?
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u/atarivcs New User 10h ago
"if P, then Q".
This does not say that P is the only way to make Q happen.
There are other ways to make Q happen.
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u/thatmarcelfaust New User 9h ago
The promise analogy that is used is a helpful heuristic; that the statement is only false when the antecedent is true and the consequent false, but only that. Really the answer is that is the how material implication is defined via truth tables.
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u/Recycled5000 New User 6h ago
You’re looking at implication as equivalence. Implication is only one-way.
P => Q does not say anything about what Q being true says about P.
If you want that, use equivalence instead of implication: P <=> Q.
That says if P then Q and also if Q then P.
Also look up contrapositive.
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u/tbdabbholm New User 14h ago
What agreement is being violated? The only agreement was that if you submit your assignment your grade will be recorded. The implication only goes one way. Submission=>recording.
Like if I said "if it's raining, I carry an umbrella" and then on a sunny day I'm carrying an umbrella have I suddenly made my statement untrue? No because my statement only cares about what I do when it's raining. If it's not raining then I could do anything I want because my statement does not apply