[Guest]

You are suddenly faced with four questions. They read thus:

What are the four answers to these questions? Technically, you need only solve the first three, but it is no easy feat! Think very carefully before you assume you have the answer; you may have to think about this problem in a rather novel way (unless you've faced this sort of problem before).

Edited October 27, 2010 by tartolb

[Guest]

Sorry, folks. There is a slight error in premise C. The last part should read "...or otherwise identical to that to question D."

However, don't think that that small error made the problem harder. Quite the contrary!

Edited October 27, 2010 by tartolb

[Guest]

Here. Let's formalise the statements so that they're more condensed.

A) If C = D, then A = B, else A = C.

B) B =/= C.

C) If A = B, then C = A, else C = D.

D) D = riff-raff.

This doesn't help uncover the solution, so it isn't a "hint".

Edited October 27, 2010 by tartolb

[EventHorizon]

Here. Let's formalise the statements so that they're more condensed.

A) If C = D, then A = B, else A = C.

B) B =/= C.

C) If A = B, then C = A, else C = D.

D) D = riff-raff.

This doesn't help uncover the solution, so it isn't a "hint".

not mutually satisfiable.

First, assume c=d. then a=b (From eq A). Since a=b, c=a (eq C). Since a=b and c=a, c=b. But we are told b != c (eq B). Contradiction!

So c!=d. This means a=c (eq A). We know b!=c (eq B), so a!=b. Since a!=b, c=d (eq C). So c=d and c!=d.... another contradiction.

[EventHorizon]

I'll rewrite them as implications...

1. (c=d) => (a=b)    (eq A)

2. !(c=d) => (a=c)   (eq A)

3. !(c=b)            (eq B)

4. (a=b) => (a=c)    (eq C)

5. !(a=b) => (c=d)   (eq C)

6. (c=d) => (a=c)    (1&4)

7. (a=c)             (2&6)

8. !(a=b)            (3&7)

9. (c=d)             (5&8)

10. (a=b)            (1&9)

11. Contradiction    (8&10)

[araver]

Here. Let's formalise the statements so that they're more condensed.

A) If C = D, then A = B, else A = C.

B) B =/= C.

C) If A = B, then C = A, else C = D.

D) D = riff-raff.

This doesn't help uncover the solution, so it isn't a "hint".

First, statement C is not true.

Assume C is true then:

- if B is true then B=/=C but they're both true - contradiction

- if B is false then B=C which means B is actually true - contradiction.

So C is false.

C actually means (A=B AND C=A) OR (A=/=B AND C=D). C is false so nonC is true.

Applying DeMorgan laws:

nonC = non((A=B AND C=A) OR (A=/=B AND C=D))

=non(A=B AND C=A) AND non(A=/=B AND C=D)

=(non(A=B)OR non(C=A)) AND (non(A=/=B)AND non(C=D))

=(A=/=B OR C=/=A) AND (A=B OR C=/=D)

Assume A =/= B.

........Then C=/=D (because the second part of non-C must be true).

........If A is true, then from A we get A = C (else statement) which means A is false contradiction.

........Therefore A must be false.

........Since A = (C=D AND A=B) OR (C=/=D AND A=C) is false

........non A = (C=/=D OR A=/=B) AND (C=D OR A=/=C) must be true.

........But since C=/=D and A=C (they're both false) the second part (C=D OR A=/=C) cannot be true. Another contradiction.

Therefore A=B.

........Then C=/=A is true (first part of nonC must be true) which means A and B are both true.

........Looking at A (which we know is true) we see that if C=/=D then A=C impossible.

........Therefore C=D. Which means D is False (the answer is not riff-raff it's FALSE ).

Answers:

A - True

B - True

C - False

D - False

Edited October 28, 2010 by araver

[araver]

not mutually satisfiable.

First, assume c=d. then a=b (From eq A). Since a=b, c=a (eq C). Since a=b and c=a, c=b. But we are told b != c (eq B). Contradiction!

So c!=d. This means a=c (eq A). We know b!=c (eq B), so a!=b. Since a!=b, c=d (eq C). So c=d and c!=d.... another contradiction.

Correct. But not mutually satisfiable just means they're not all true at the same time, right?

[araver]

First, statement C is not true.

Assume C is true then:

- if B is true then B=/=C but they're both true - contradiction

- if B is false then B=C which means B is actually true - contradiction.

So C is false.

C actually means (A=B AND C=A) OR (A=/=B AND C=D). C is false so nonC is true.

Applying DeMorgan laws:

nonC = non((A=B AND C=A) OR (A=/=B AND C=D))

=non(A=B AND C=A) AND non(A=/=B AND C=D)

=(non(A=B)OR non(C=A)) AND (non(A=/=B)AND non(C=D))

=(A=/=B OR C=/=A) AND (A=B OR C=/=D)

Assume A =/= B.

........Then C=/=D (because the second part of non-C must be true).

........If A is true, then from A we get A = C (else statement) which means A is false contradiction.

........Therefore A must be false.

........Since A = (C=D AND A=B) OR (C=/=D AND A=C) is false

........non A = (C=/=D OR A=/=B) AND (C=D OR A=/=C) must be true.

........But since C=/=D and A=C (they're both false) the second part (C=D OR A=/=C) cannot be true. Another contradiction.

Therefore A=B.

........Then C=/=A is true (first part of nonC must be true) which means A and B are both true.

........Looking at A (which we know is true) we see that if C=/=D then A=C impossible.

........Therefore C=D. Which means D is False (the answer is not riff-raff it's FALSE ).

Answers:

A - True

B - True

C - False

D - False

Or if you prefer more "standard" answers:

A: Yes. B. Yes. C. No D. No.

[Guest]

No one has put down the right answer...

It appears to be a contradiction, there is still a logically valid way to evaluate and answer these questions as they demand. I strongly suggest those here consult the truth tables of If-Then-Else (ternary) clauses.

[araver]

No one has put down the right answer...

Sorry,

that the answers Yes, Yes, No, No violate in any logically way the setting of the problem.

And unless you as OP or anyone else can contradict that, it is a valid answer to the problem.

Maybe not the right answer from your point of view, but a valid one nevertheless.

[Guest]

that the answers Yes, Yes, No, No violate in any logically way the setting of the problem.

And unless you as OP or anyone else can contradict that, it is a valid answer to the problem.

Maybe not the right answer from your point of view, but a valid one nevertheless.

Read the questions again, instead of the formalisms. You're close, I'll give you that much. Remember: You must answer the questions.

[bonanova]

There are no questions. tartolb: want to restate the puzzle?

Hint: There are also no statements.

So, there are also no truth values.

The OP comprises four instructions. Instructions are imperatives.

Imperatives don't have answers, and they don't have truth values.

[Guest]

You must treat them as if they were questions. I know, it is somewhat deceptive, but that is how the problem is stated.

[Guest]

Moreover, even an imperative can be treated as a statement of a logical sort; however, there is such a thing as "imperative logic", so the point is moot.

I guarantee: that there is a solution to all four questions; that the solutions are logically valid all the time.

What are they?

[bonanova]

If what you want is answers, it's fairest to the problem solvers to ask questions. A good puzzle defines the requested response. One thing you might do is change the instructions to statements [conditionals] and ask for a consistent set of truth values. If that's in fact what you desire, why not explicitly ask for it?

Otherwise you're left with something like this:

If 2+2=5 then New York is a small city.

If I were to call that a question, what answer would you give?

Imperatives do not have truth values: Go sit in that chair. True? False?

[Guest]

It is fair when the problems state that they are questions, even if they aren't stated as questions. There is nothing unclear about the problem! This has been shown by araver getting very close the the actual solution.

[bonanova]

Moreover, even an imperative can be treated as a statement of a logical sort; however, there is such a thing as "imperative logic", so the point is moot.

I guarantee: that there is a solution to all four questions; that the solutions are logically valid all the time.

What are they?

All of A, B, C, and D, treating them as statements, not imperatives, are FALSE.

Every statement that says THIS MUST BE THAT is false. Anything can be anything. That way nothing is specified. That eliminates all contradiction.

[Guest]

That isn't the right answer.

these are ternary expressions. This isn't a question of merely "true" and "false". The statement can be false in one part while being true in the other. My formalism helps clarify the problem, but doesn't provide the avenue to the solution, which is why I don't consider it a hint. But in that formalism, you have a clarified depiction of the problem.

Edited October 28, 2010 by tartolb

[superprismatic]

The only way I can see to make any

sense of this mess is to take araver's

analysis and just restate it as answers

to questions. Here, I'll assume that

riff-raff has its usual meaning in

English: Rubbish, Trash, False.

So, I suppose the answers are:

A is not riff-raff

B is not riff-raff

C is riff-raff

D is riff-raff

[bonanova]

The only way I can see to make any

sense of this mess is to take araver's

analysis and just restate it as answers

to questions. Here, I'll assume that

riff-raff has its usual meaning in

English: Rubbish, Trash, False.

So, I suppose the answers are:

A is not riff-raff

B is not riff-raff

C is riff-raff

D is riff-raff

If "riff-raff" means false, then statement D becomes:

D: This statement must be false, whether you like it or not.

The unanswered question lies in whether "This MUST BE That" is an imperative, implying a statement whose truth is not open to question.

If so, then A-D all become statements that are true statements because they must be true statements.

This interpretation leads to contradictions - your

If not, the must's may all be taken to be false, and the OP contains no guidance.

If the must's translate to necessarily true statements, i.e. the answer to D must be 'riff-raff',

then 'riff-raff' cannot carry the connotation of falsehood, to avoid self contradiction.

The OP does not specify the nature of an acceptable answer: 'riff-raff'? or True/False?

He implies the latter, by referring to truth tables.

Even there, where there is a compound statement [A and C] is the truth at issue the highest level inference [only]?

[Guest]

The only way I can see to make any

sense of this mess is to take araver's

analysis and just restate it as answers

to questions. Here, I'll assume that

riff-raff has its usual meaning in

English: Rubbish, Trash, False.

So, I suppose the answers are:

A is not riff-raff

B is not riff-raff

C is riff-raff

D is riff-raff

That is a valid answer, but the precise reasoning is different from araver's original answer.

[Guest]

If "riff-raff" means false, then statement D becomes:

D: This statement must be false, whether you like it or not.

The unanswered question lies in whether "This MUST BE That" is an imperative, implying a statement whose truth is not open to question.

If so, then A-D all become statements that are true statements because they must be true statements.

This interpretation leads to contradictions.

If not, the must's may all be taken to be false, and the OP contains no guidance.

If the must's translate to necessarily true statements, i.e. the answer to D must be 'riff-raff',

then 'riff-raff' cannot carry the connotation of falsehood, to avoid self contradiction.

No. There is no need to assume that "riff-raff" means "false". I will explain the reasoning behind the valid answer in my next post, since a valid answer has been obtained, but through invalid reasoning (guessing).

[Guest]

In this post, I will describe the steps that lead to a valid solution to this problem, as simple as it is:

Let's consult the formalism:

A) If C = D, then A = B, else A = C.

B) B =/= C.

C) If A = B, then C = A, else C = D.

D) D = riff-raff.

From here, all we need to note is that B and D must be true; therefore, it is clear that the first half of premise C is false, since it contradicts premise B by claiming indirectly that C = B. Since the first half of C is false, then the second ("C = D") must be true. Therefore, observing A, it is clear that the first half is true, which in turn means A = B, not A = C. In conclusion, any answer for A and B is acceptable insofar as the answers to both are truly equivalent in all respects, whereas C and D must both be "riff-raff".

[superprismatic]

If "riff-raff" means false, then statement D becomes:

D: This statement must be false, whether you like it or not.

The unanswered question lies in whether "This MUST BE That" is an imperative, implying a statement whose truth is not open to question.

If so, then A-D all become statements that are true statements because they must be true statements.

This interpretation leads to contradictions - your

If not, the must's may all be taken to be false, and the OP contains no guidance.

If the must's translate to necessarily true statements, i.e. the answer to D must be 'riff-raff',

then 'riff-raff' cannot carry the connotation of falsehood, to avoid self contradiction.

The OP does not specify the nature of an acceptable answer: 'riff-raff'? or True/False?

He implies the latter, by referring to truth tables.

Even there, where there is a compound statement [A and C] is the truth at issue the highest level inference [only]?

Bonanova, I know from whence you come.

But tartolb's insistance on answers to

questions, together with his cryptic

remarks about ternary expressions, as well

as remarks like 'This isn't a question of

merely "true" and "false". The statement

can be false in one part while being true

in the other', leads me to believe he is

fishing for something like what I gave.

I grew up in a world where sentential

connectives pulled in all the false and

true bits under one umbrella. The

reference to "imperative logic", with its

lack of rigor, further egged me on. So,

I don't disagree with you, I just threw

something out as one would do while engaging

in real fishing.

[bonanova]

In this post, I will describe the steps that lead to a valid solution to this problem, as simple as it is:

Let's consult the formalism:

A) If C = D, then A = B, else A = C.

B) B =/= C.

C) If A = B, then C = A, else C = D.

D) D = riff-raff.

From here, all we need to note is that B and D must be true; therefore, it is clear that the first half of premise C is false, since it contradicts premise B by claiming indirectly that C = B. Since the first half of C is false, then the second ("C = D") must be true. Therefore, observing A, it is clear that the first half is true, which in turn means A = B, not A = C. In conclusion, any answer for A and B is acceptable insofar as the answers to both are truly equivalent in all respects, whereas C and D must both be "riff-raff".

First:

.

1. D must be true
2. D must be "riff-raff"
   .

Second:

.

1. If some of A-D might be false, what is the logical import of "must be" instead of the more usual "is"?
2. To what question is either statement an answer?
   .

It may have improved the puzzle to add:

Determine the truth values of these four statements.

Then give [with statement D as an example] an appropriate response to each.

I don't mean to prolong a discussion about form, just guidance to help solvers get at what the puzzle is asking.

SP: Understand.

Intractable was meant to point to OP not to a deficiency in the answer.

The candidate answer pointed to uncertainty of the OP.

And if truth values for A-D were the object to be determined, that does

open the can of peas as to truth or falsity on what level, when the

statements are compound. Usually the highest level.