Math Problem, Arguement with the teacher. Easy Logic.
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You're talking about material implications, colloquially refered to as "if" statements. They're a formal construct with particular truth conditions (true if and only if both the antecedent and the consequent are true or if the antecedent is false). They are not the definition of "if", though many bad teachers often equate the two. An astounding amount of metaphorical blood has been shed in both philosophy and linguistics trying to figure out how to describe what "if" actually means formally (spoiler alert: this is still a very open, very difficult question, which is why so many posts here are arguing about it with no real conclusion), so there is actually no such thing as an actual "if statement" in the overwhelming majority of logics.Singletap said:
For more on material implications, you could take a look at this: http://en.wikipedia.org/wiki/Material_implication
Especially the "Philosophical problems with material conditional", which is badly named (there aren't any real problems with it, there are just problems with how people keep misapplying it).
When we learned "If, Then" statements, we were given the options of Always, Sometimes, or Never. Things work like that in all branches of math (Algebra, Geometry, Calculus, etc). To say that there are only two absolutes (True/False, Always/Never) is a failure on the teacher's part. There are almost always exceptions to the rule.
Actually, it's a great example to use, because it gets to the heart of the matter--the truth of a statement, in the sense that we use "truth" in logic, isn't based on whether the statement is useful or whether we agree with it. In the logic realm, "true" and "false" are really just based on whether or not the logic is sound.Plurralbles said:
GIVEN: All turtles have wings, and Speedy is a turtle.
The statment "Speedy the turtle has wings" is TRUE. Not because turtles have wings in the real world, but because the conclusion follows from the premises. The statement is logically sound.
Getting students to separate normal ideas of true/false, and to become aware of their assumptions about logic, is the most important thing a teacher can do when covering logic problems like this. You may only assume the information that is given. Everything else must be arrived at through logical operations on the given, regardless of what you think about the information.
So you were taking "If you take your medicine then you will feel better" as the true statement, correct? Then you needed to determine if "case 3 and 4" were true or false, based on the first true statement. Am I right in this explanation of the problems?Singletap said:
If I am right (hopefully, otehrwise this post is meaningless...) case 3 and 4 ("If you don't take your medicine then you'll feel better"; "If you don't take your medicine then you won't feel better") are indeterminable as true or false. The teacher wanted you to say 3 is false (which it is) and 4 is true, but given the first statement, you cannot tell with case 4.
It's actually a common logical fallacy. I forgot what the name was but it goes like this:
If A implies B; (If you take your medicine then you will feel better)
Then not A does not neccessarily imply not B. (If you don't take your medicine then you won't feel better)
So your "case 4" is inteterminable as true or false, given the premise. Hope my explanation helps...
Basic if then says "if A, then B". Which, unless proven otherwise, you are to generally to assume is a true statement. The trick with if then is it doesn't work backwards.
In your case, I'm guessing each question is stand alone, not related to each other. Which would make both true by default. If the 4 cases are related then one has to be false.
In your case, I'm guessing each question is stand alone, not related to each other. Which would make both true by default. If the 4 cases are related then one has to be false.
Because in common parlance 'Truth' is the opposite of 'False'. It's quicker to say. Simple really. I'd have thought that you'd have been taught what it really means in your classes. Perhaps that's coming later after challenging you with this or something.Singletap said:
Saying you'll feel better if you take your medicine does NOT prove that you won't coincidentally feel better if you don't, or that you will feel like shit if you don't. Meaning those cannot be declared false, but they also cannot be declared true.Singletap said:
Your teacher is wrong, you are not allowed to assume ANYTHING. That's the most important part of logic, I just did a unit on boolean algebra in a Discrete Math course, she's full of shit, find a reputable link on line showing an example that is different than what she said, print it and bring it to class.
There is no information placing them into one category of the other, you don't just say something is true because you can't prove it false, I just did an extensive unit on boolean algebra and logic in a Discrete Mathematics course, she's full of shit, the whole point is to PROVE these things.Danny Ocean said:
Im going to have nightmares tonight, thanks to this thread. Instead of give my view (since it is the same as atleast half of the comments in this thread), ill just say I agree with your teacher.
I am no math teacher so I may be wrong but, math almost always has to be true unless you do it wrong. With math you should never have to guess, there is always a formula or way to do a something. Also if you do not have enough info to complete a problem you can almost never do it. Even with variables you are not guessing you have enough information to solve it. With conditional statements A has to lead to B (if it rains we do not play the game) sorry that is just how it goes. The only thing I do not understand is "if you do not take the medicine then you will get better" maybe eventually you will get better. Correct me if I am wrong please.Singletap said:
The way I see it is this:Singletap said:
Let's rephrase the problem trying to use a less ambiguous choice of words:
1. You are ill (Cpt. Obvious)
2. Your illness has no cure except for the above mentioned medicine
3. The aforementioned medicine certainly cures the illness (in other words it cannot fail in curing you)
4. You have the choice of either taking or not taking the medicine.
If #2 is true, then:
a.If you take the medicine, you'll feel better. (true, as per #3)
b.If you take the medicine, you won't feel better. (false, as per #3)
c.If you don't take the medicine, you will feel better. (false, as per #2)
d.If you don't take the medicine, you won't feel better. (true, as per #2)
If #2 is not true:
a'.If you take the medicine, you'll feel better. (true, as per #3)
b'.If you take the medicine, you won't feel better. (false, as per #3)
c'.If you don't take the medicine, you'll feel better. (undefined, but true, per missing #2)
d'.If you don't take the medicine, you won't feel better. (same as above)
To try to define the 2 undefined cases you must rephrase them in more mathematically-precise derivatives:
Assuming a set S containing an infinite number of cases (think of it as parallel universes) on whether you take this medicine (or not) causing your being well (or not):
c'^. Each and every case in the set S has the effect of curing the illness without taking the medicine.
d'^. Each and every case in the set S has the effect of not curing the illness without taking the medicine.
Using this wording, then they're both false or undefined respectively whether you can demonstrate even a single event that doesn't match those statements or not.
Instead, rephrasing differently:
c'*. There exists a case in the set S that has the effect of curing the illness without taking the medicine.
d'*. There exists a case in the set S that has the effect of not curing the illness without taking the medicine.
In this case, to prove these statements true, you only have to demonstrate that there is a single case in which the course of events matches the statement, else the cases are undefined.
Now, you have narrowed and stated the undefined cases.
If you want to put this in binary logic instead of ternary logic, you'd have to have a postulate saying that
5. All cases that aren't definitely provable and therefore undefined are to be considered as false.
(a postulate saying
5'. All cases that aren't definitely provable and therefore undefined are to be considered as true.
works equally well)
By taking then any of the aforementioned alternative rephrasings you'd have a certain true/false statement.
Now, if you look at the (34)'* rephrasings, with the ternary-binary narrowing condition (also called defaulting condition) 5', then the cases are always both true therefore explaining what your teacher had intended to say.
I hope this was not too long and please excuse my mathematical/logical fallacies (should there be any) as I'm just a computer engineering student and not a mathematician.
These are all shitty examples.
All you need to know about geometry-level logic:
"If A, then B." Is the original statement, which you assume to be true.
Then "If B, then A." Would be the converse, which you can't assume to be true.
Furthermore, "If not A, then not B." Would be the inverse, which you also can't assume to be true.
Finally, "If not B, then not A." Would be the contrapositive, which you can logically deduce is true.
Your teacher phails at geometry-level logic by providing statements that are none of the above. To use your statement,
"If take your medicine, then you'll feel better." - Original Statement (Assumed true)
"If you feel better, then you took your medicine." - Converse (Not always true)
"If you don't take your medicine, then you won't feel better." - Inverse (Not always true)
"If you don't feel better, then you didn't take your medicine." - Contrapositive (Definitely true)
Therefore, not only does your teacher introduce a statement that's none of the above ("If you don't take your medicine, then you'll feel better."), but she gets one of the statements wrong ("If you don't take your medicine, then you won't feel better." Isn't always true, since it's an inverse).
All you need to know about geometry-level logic:
"If A, then B." Is the original statement, which you assume to be true.
Then "If B, then A." Would be the converse, which you can't assume to be true.
Furthermore, "If not A, then not B." Would be the inverse, which you also can't assume to be true.
Finally, "If not B, then not A." Would be the contrapositive, which you can logically deduce is true.
Your teacher phails at geometry-level logic by providing statements that are none of the above. To use your statement,
"If take your medicine, then you'll feel better." - Original Statement (Assumed true)
"If you feel better, then you took your medicine." - Converse (Not always true)
"If you don't take your medicine, then you won't feel better." - Inverse (Not always true)
"If you don't feel better, then you didn't take your medicine." - Contrapositive (Definitely true)
Therefore, not only does your teacher introduce a statement that's none of the above ("If you don't take your medicine, then you'll feel better."), but she gets one of the statements wrong ("If you don't take your medicine, then you won't feel better." Isn't always true, since it's an inverse).
While I may not know the specifics of proofs I can give some advice about student/teacher relations. If you have a disagreement with what is being taught, it's best to say these things in a more solo or private setting, instead of in front of the class.Singletap said:
Well, I'd say your sentiments are quite common; not in that you didn't question how geometry was taught but that you aren't interested in it. Math in general is a rather unpopular subject, and I'd wager that how it's taught has something to do with that. Students generally might find it a lot more interesting if they had such an understanding of it's basis, rather than treating it as merely something that has to be memorized for school. But anyway, sorry for getting off topic.Sonic Doctor said:
Well, Humans invented Maths, and so, as humans are imperfect, we just have to assume that whatever calculations we're using are right, otherwise nothing gets done.Singletap said:
Besides, you CAN get worse if you take the message. I never said you couldn't. I was just rushed for time and barely managed anything.
Think about it. Medicine helps recovery, but isn't 100% effective. People still die. I'm not arguing that.
But, if you got better if you took your medicine, then that doesn't rule out the fact that you can get better if you DIDN'T. They're not mutually exclusive, as both would play a part in it.
What I'm saying is, you seem to be intentionally causing trouble here, as it's a maths lesson, not a philosophy one, and the teacher could have been right.
Hell, you just assumed there was a 50/50 chance of getting better if you did take the medicine. You have no proof to back that up there, so you just did the same thing as the teacher, and assumed.
I don't see the problem here. It's a simple question which has two statements that aren't mutually exclusive. I mean, if I said Red is a colour and Blue is a colour, then you wouldn't argue right? Not even that because Red is a colour then how on EARTH could Blue be a colour AS WELL? I'll tell you. Because they're not mutually exclusive. (Red and Blue can both be colours at the same time)
Also, as a side note, I know Mutually Exclusive term doesn't work too well on the Medicine one, but I couldn't remember the one that did.
The first statement gives us a condition: that we take medication. It gives us a result: that you feel better. This is a coherent and logical statement (we have established a cause and an effect). The second statement does the same thing.Singletap said:
How you interpret these statements will vary greatly. There are three basic interpretations:
1) Each statement is a part of the "rules" governing the logical system. In this case each statement is necessarily true within that rule set.
2) The one statement or the other is part of the rules governing the logical system. In this case, given one rule the second rule is not necessarily true.
3) Neither statement is a rule of the logical system in which case the reader cannot assume either statement is true.
In the first case it is easy enough. These arbitrary rules can be defined as true because the logical statement being made is simply "if A then B". One might be tempted to reason that there is no proof inherent but no such proof is required when one is simply stating an arbitrary connection such as this.
The second case is a hair more complex. Because the statement that is made is "if A then B", and we are given no other information it does not stand to reason that if we do not have A then we would also not have B. A simple change can of course give us this meaning. If we said "You will feel better if and only if you take your medicine" then we would know that if we did not take our medicine we would not feel better as it has been established that the only way to feel better is to take the medicine (the power of the "if and only if" part).
The third case is the way most people would view the statements unless otherwise directed. You establish a connection between the two without any supporting evidence and as such neither statement has value. If I were to simply make the assertion "If you take Viagra you will gain the hidden power of flight", people would expect some supporting evidence.