Test your reasoning skills

[Civbert]

Try this basic reasoning test.

Suppose there are four double-sided cards laid out next to each other on a table, like this: (|...| indicates the border of each card and has nothing to do with the content of each card.)

| I | | T | | 8 | | 5 |

Now consider the following rule:

(R) If there is a vowel on one side, there is an even number on the other.

-Objective:

Flipping over as few cards as possible, which cards would you have to turn over to see if the rule was true of these four cards?

I guess someone already got it but I'll post before I look at all the answers.


You need to flip two cards.

(R) is "If V, then E".

You need to flip the | I | card.
Justification: to test (R).

You need to flip the |5| card.
Justification: (R) implies "if not-E, then not-V" by modus tollens.
Since 5 is an odd number (or not-E), then the flip side must be a consonant (or not-V).

You do NOT need to flip | T |
Justification: (R) does not imply "if not-V, then not-E".

You do NOT need to flip | 8 |
Justification: (R) does not imply "if E, then V".


*************************

Correction: I was assuming that on every card there is a letter on one side, and a number on the other. But since this is not the case, then you would need to also flip | T | since it is also "not-E" and the flip side must be "not-V" (a consonant or a number).

So my answer then is three cards: | I | | 5 | | T |, since 'I' is a vowel, and '5' and 'T' are not even numbers. Since '8' is an even number, anything can be on the reverse side and not violate the rule.

 

[Civbert]

...
-edit- I realized that my PS is wrong b/c I wouldn't have to turn over the eight. If it has a vowel well and good, but if not -even if it has another numeral- it doesn't disprove the rule as stated. Did you post the answer in the thread or do you U2U it or will it be a couple days (in which case I'm going to be thinking about this in my sleep) or what?

I think you got it.

 

[BrianLanier]

I will post the answer tomorrow; I want to get more people a shot.

 

[Civbert]

I will post the answer tomorrow; I want to get more people a shot.

Brian,

One thing that you did not mention in the problem was if each card will have a number on one side and a letter the other, or if a card may have a letter or number on both sides. I believe that would effect the answer. What say you?

 

[BrianLanier]

I will post the answer tomorrow; I want to get more people a shot.

Brian,

One thing that you did not mention in the problem was if each card will have a number on one side and a letter the other, or if a card may have a letter or number on both sides. I believe that would effect the answer. What say you?

You are correct, I should have mentioned that! I will change the original post. (Although, most people probably assumed it anyway).

 

[BrianLanier]

But the problem is that we seem to be able to use (1) in my example all the time. We can imagine saying (1). Should we not make statements like (1) anymore?

I wonder if this has to do with the content (or semantics). Just like,

This sweater is green,
So, this sweater is not red,

seems to be an inference that we make all of time, but its form is not valid (p, so ~q). The form of the argument is invalid, but because of the content or semantics of the argument, it seems to be an inference we should be willing to accept.

Or, if not, then possibly one could say that the form of your argument is a 'default' form but doesn't perserve truth in *all* instances. But just because it is not truth-perserving in *all* cases, it does not follow that we should not make statements using its form anymore--for the same reason that just because sense experience isn't always reliable, it doesn't follow that we shouldn't generally rely on it.

But my argument was valid, so your first response was disanalogous. Furthermore, I'd actually say that your first argument was an enthymeme. It's invalidity is due to missing premises.

Then, do we really want to deny the universality of M.T.?

The argument of the 'sweater' was not meant to be analogous or a counter-example to your valid argument. It was mearly intended to show the effect that 'content' and 'semantics' play in argument forms. (A counter-example to the sweater arugment would be: I have brown hair, so I don't wear glasses.) The form was invalid, but because of the semantics of the argument, it still has a true conclusion. I was just wondering whether or not that had something to do with your argument. (Not saying that it does, it was just a thought that popped into my head!)

The 'sweater' argument is not an enthymeme, since adding more premises to the argument, while maintaining its *form*, would not then render it valid.

As far a M.T. goes, what would be wrong with denying its universality? If your example is a valid argument and is not truth perserving, then it seems to follow that M.T. is not *universally* truth-perserving. I think that this is because of the problems that have been noted with the conditional in general (since the first premise of M.T. is a conditional). These problems have been generally recognized within the literature (so far as I can tell). The same goes for L.E.M. (Hence the development of non-classical logics such as, fuzzy-logic, paraconsistent logics, multi-valued logics, modal logics, etc.) That is why I said that maybe it is best to maintain classical logic (since it seems to be the stongest) and think of these 'laws' as default functions, while noting the problems that we run into. I am by no means convinced that this *should* be the position to take, just that it seems to be a plausible one.

 

[Croghanite]

you may have to flip over all four cards to prove the rule is true.
I would start with the card that has a vowel on it. Then the card with the even number. If the rule proved to be true after flipping those cards in that order, then you must flip over the rest to make sure the rule is true with the four cards that are on the table.

The rule can be proved wrong with flipping as little as one card. You will have to flip them all to prove the rule is true.

 

[Me Died Blue]

I say two: the I and the 5.

The original statement (R1, "If there is a vowel on one side, there is an even number on the other") is equivalent to its contrapositive, which in this case is, "If there is not an even number on one side, there is not a vowel on the other." And since every number has to be either even or odd, every letter has to be a consonant or a vowel, that is equivalent to saying, "If there is an odd number on one side, there is a consonant on the other" (R2).

Of course one can prove the rule wrong in as little as one flip, depending on if it violates the rule or not. But if it is right, it would take exactly two flips to prove so:

|I|: To prove the rule (in the form of R1), one needs to ensure the other side is an even number.

|T|: Regardless of what it on the other side of this card, it can only violate the converse or the inverse (logically equivalent) of the original statement (R1/R2).

|8|: Same as T.

|5|: To prove the rule (in the form of R2), one needs to ensure the other side is a consonant.

 

[Dan....]

I would say three cards: "I" "T" and "5"

For the "I", there must be an even number on the opposite side.
For the "T" there must not be a vowel on the opposite side.
For the "5" there must not be a vowel on the opposite side.

Whatever is on the opposite side of the "8" is irrelevant because "if X then Y" does not prove "if Y then X."

 

[Mushroom]

This is new:

each with a number on one side and a letter on the other

Since the original post has changed, I will return to my second(!) answer, 2 cards. Since it is now necessary that all the cards have one letter and one number, then it is unnecessay to flip the T. The rule does not specify that a consonant have either an even or odd number on the reverse, so the number on the other side is immaterial, and the same applies to the 8, since the rule does not stipulate a requirement for the reverse of an even number. Only the I and the 5 need to be flipped.

Prior to the change of the OP, I believe my third (!!) answer would have been correct, since it was possible that the T had a vowel on the reverse, which would have disproven the rule.

Of course, my first answer was wrong no matter what, so I don't win the door prize.

 

[Croghanite]

What if the other side of the T card was an even number?

This is new:

each with a number on one side and a letter on the other

Since the original post has changed, I will return to my second(!) answer, 2 cards. Since it is now necessary that all the cards have one letter and one number, then it is unnecessay to flip the T. The rule does not specify that a consonant have either an even or odd number on the reverse, so the number on the other side is immaterial, and the same applies to the 8, since the rule does not stipulate a requirement for the reverse of an even number. Only the I and the 5 need to be flipped.

Prior to the change of the OP, I believe my third (!!) answer would have been correct, since it was possible that the T had a vowel on the reverse, which would have disproven the rule.





Of course, my first answer was wrong no matter what, so I don't win the door prize.

 

[Me Died Blue]

What if the other side of the T card was an even number?

That would not be a problem, for the rule only said, "If there is a vowel on one side, there is an even number on the other." That is not logically equivalent to saying, "If there is an even number on one side, there is a vowel on the other." It only means that all vowel-cards have to also have even numbers on them - but there also could be even numbers on other (non-vowel) cards as well. A good example one of my philosophy professors used to illustrate the difference is that it's a true statement (by definition) to say, "If someone is a bachelor, that person is a male." But we know that it's not true to say, "If someone is a male, that person is a bachelor."

 

[Cheshire Cat]

I read the first post only, and here is my answer: 2 Cards. The first ‘I’ card, and the last ‘5’ card.

‘I’ is the only vowel. T is not a vowel, so that card isn’t relevant.

Now the rule is “if there is a vowel on one side, there is an even number on the other. It doesn’t say one way or the other whether or not a non-vowel could have an even number on its other side, so the 8 card doesn’t have to be flipped in order to verify the rule. The 5 card is the tricky one for me. I think the 5 card needs to be flipped in order to see if there is a vowel on the other side. Because if there is, then the rule would be falsified.

Now its time to read the thread and see where I messed up haha.

edit: After reading the thread I'm still not sure if I'm right

 

[Croghanite]

your correct.
You would still need to flip the T card to see if a vowel is on the other side.

What if the other side of the T card was an even number?

That would not be a problem, for the rule only said, "If there is a vowel on one side, there is an even number on the other." That is not logically equivalent to saying, "If there is an even number on one side, there is a vowel on the other." It only means that all vowel-cards have to also have even numbers on them - but there also could be even numbers on other (non-vowel) cards as well. A good example one of my philosophy professors used to illustrate the difference is that it's a true statement (by definition) to say, "If someone is a bachelor, that person is a male." But we know that it's not true to say, "If someone is a male, that person is a bachelor."

 

[Jim Johnston]

I wonder if this has to do with the content (or semantics). Just like,

This sweater is green,
So, this sweater is not red,

seems to be an inference that we make all of time, but its form is not valid (p, so ~q). The form of the argument is invalid, but because of the content or semantics of the argument, it seems to be an inference we should be willing to accept.

Or, if not, then possibly one could say that the form of your argument is a 'default' form but doesn't perserve truth in *all* instances. But just because it is not truth-perserving in *all* cases, it does not follow that we should not make statements using its form anymore--for the same reason that just because sense experience isn't always reliable, it doesn't follow that we shouldn't generally rely on it.

But my argument was valid, so your first response was disanalogous. Furthermore, I'd actually say that your first argument was an enthymeme. It's invalidity is due to missing premises.

Then, do we really want to deny the universality of M.T.?

The argument of the 'sweater' was not meant to be analogous or a counter-example to your valid argument. It was mearly intended to show the effect that 'content' and 'semantics' play in argument forms. (A counter-example to the sweater arugment would be: I have brown hair, so I don't wear glasses.) The form was invalid, but because of the semantics of the argument, it still has a true conclusion. I was just wondering whether or not that had something to do with your argument. (Not saying that it does, it was just a thought that popped into my head!)

The 'sweater' argument is not an enthymeme, since adding more premises to the argument, while maintaining its *form*, would not then render it valid.

As far a M.T. goes, what would be wrong with denying its universality? If your example is a valid argument and is not truth perserving, then it seems to follow that M.T. is not *universally* truth-perserving. I think that this is because of the problems that have been noted with the conditional in general (since the first premise of M.T. is a conditional). These problems have been generally recognized within the literature (so far as I can tell). The same goes for L.E.M. (Hence the development of non-classical logics such as, fuzzy-logic, paraconsistent logics, multi-valued logics, modal logics, etc.) That is why I said that maybe it is best to maintain classical logic (since it seems to be the stongest) and think of these 'laws' as default functions, while noting the problems that we run into. I am by no means convinced that this *should* be the position to take, just that it seems to be a plausible one.

The sweater argument could have been an enthymeme, I think. You just left out the first premise:

Unstated Premise. If my sweater is green, then it is not red.

1. My sweater is green.

2. Therefore, my sweater is not red.

Moving on....

The M.P. was a *valid* argument. It was not *sound.* P2 negated P1. So, we have no problems on that score which woulld make us challenege the universality (*other issues* with conditionals not withstanding). So, we're safe.

As far as the challeneges to material implication (and other challenges to conditionals), I suggest these entries by Valicella:

Maverick Philosopher Clearing up Confusion about Material Implication

There's about 9 there.

It's at least the "other side" of the argument.

As far as the rest, I'd begin by saying your comments may be similar to those who argue that "induction is a fallacy" by holding it to the standards of deductive logic. So, conditionals in porppositional logic must be used properly.

 

[Me Died Blue]

your correct.
You would still need to flip the T card to see if a vowel is on the other side.

With the original wording of the question...

 

[sotzo]

I and 5.

"I" could have an odd # and negate the rule.
"T" doesn't matter because it wouldn't negate the rule whether an odd or even #.
"8" doesn't matter because it wouldn't negate the rule if it were a consonant.
"5" could have a vowel and negate the rule.

By the way, I think the Trinity Broadcasting Network website has a similar challenge on their boards...something to do with working out how many of their programs you have to watch before you turn into a Precious Moments character.

 

[B.J.]

I scrolled down as fast as I could so that I did not see any answers. My guess is that card |I| and |8| are the only ones that need to be verified because they are the only ones that have a vowel and even number. The other two cards are irrelavent given the rule. So go ahead nd tell me I am wrong. Whats the trick?

 

[sotzo]

BTW, did I miss the answer in the thread somewhere?

What is the official solution to this epistemic dilemma?

 

[Croghanite]

your correct.
You would still need to flip the T card to see if a vowel is on the other side.

With the original wording of the question...

(R) If there is a vowel on one side, there is an even number on the other.

If you flip the T card and find a vowel, that would break the rule. You will have to flip all cards to prove the rule to be true. You may only need to flip as little as one card to prove the rule to be false.

 

[Jaymin Allen]

I'm going to say just 2 cards.

Step 1:
Flip over | I |, and check if there is an even number on the other side.

Step 2:
Flip over | 8 |, and check if there is a vowel on the other side.

The rules apply only to vowels and even numbers. Nothing was said about consonants and odd numbers. If I flip over | T |, the number can be odd or even. If I flip over | 5 |, the letter can be a vowel or a consonant. It doesn't matter.

 

[swilson]

I haven't read all the posts, because i dont have time....but here is my answer:

Zero

A rule is a rule whether followed or not...the rule is still true of the four cards even if it is not followed by the four cards....but, I am an absolutist.

 

[Civbert]

By the way, I think the Trinity Broadcasting Network website has a similar challenge on their boards...something to do with working out how many of their programs you have to watch before you turn into a Precious Moments character.

 

[Mushroom]

If you flip the T card and find a vowel, that would break the rule.

Nay, friend, for this has been added:

Suppose there are four cards, each with a number on one side and a letter on the other

Therefore the T must have a number on the other side, and whether an odd or even is immaterial to the rule.

 

[Croghanite]

If you flip the T card and find a vowel, that would break the rule.

Nay, friend, for this has been added:

Suppose there are four cards, each with a number on one side and a letter on the other

Therefore the T must have a number on the other side, and whether an odd or even is immaterial to the rule.

How many times has the question changed ?!

The new question answer is 2 cards. The I and the 5 card. The I card must have an even number and you must check to see if a vowel is on the other side of the 5 card.
Final answer, lock it in....... When will the official answer be given ?

 

[Croghanite]

What if the table the cards are on is made of glass ? you wouldn't have to flip any of the cards to see if the rule is true.

 

[Jim Johnston]

brian,

If we can deny the universality of M.T., why was my answer automatically wrong??

 

[BrianLanier]

brian,

If we can deny the universality of M.T., why was my answer automatically wrong??

Paul,

If we can deny the universality of our sense experience, then why is my seeing pink fairies right now automatically wrong.

 

[Jim Johnston]

brian,

If we can deny the universality of M.T., why was my answer automatically wrong??

Paul,

If we can deny the universality of our sense experience, then why is my seeing pink fairies right now automatically wrong.

Is that a fallacious shifting the burden of proof on your part?

In any event, if you mean that you are being appeared to in a pink-fairy way, then it's not.

If you mean that there are actual extra-mental objects that are pink and in ther fairy category, it's not *automatically* wrong either. Now perhaps *I* wouldn't be warranted in believing that said entities exist, but since I don't deny the possibilty that they exist, then I don't *automatically* conclude that you're wrong - if we grant the setting of the argument is that you're being honest about how you are being appeared to - in the sense of saying that you most certainly are not. Now, I may say that I *think* you are mistaken, and so for your claim to be analogius, you'd have to edit your answer on page one that "one *is not* the answer." In fact, if M.T. isn't a universally valid way of reasoning, then how don't see how you could say that my answer was wrong. At best you'd have to say something like, "I don't think it is correct, but it could be." (I'm just trying to save some face here, brother. if I can get you to say that my answer may also have been right, then I'll save the reputation of paedobaptists everywhere! )

Likewise, if what you said about M.T. above is correct, then I don't see how my answer was wrong; given those assumtions.

 

[BrianLanier]

I was obviously kidding in my last post.

I was just reading "Conditionals" by Dorothy Edgington in The Blackwell Guide to Philosophical Logic, and as usually, things are very much complicated. However, the extended discussion there still agrees with much of what Jennifer Fisher says in On the Philosophy of Logic that I was telling you about. The consensus still seems to remain, namely, that there does exist certain problem cases with the conditional, whether or not you take it as truth-functional (hook, or horseshoe), non-truth functional (arrow), or Probabilistically. I am so, so far from forming a confident belief in the matter.