Classic reasoning problems

[goobafish]

I will post mine without explanation and with only reading the first post. I am actually taking a course on Critical Thinking right now studying syllogisms, but it's not done yet , so these may be wrong.

1. You can infer that there is an Ace in the hand,
2. These two premises do not have a logical conclusion.
3. The Happy Cube is GOKE.

[Smmenen]

@ Anusien

I know nothing about programming so I don't quite follow what you are saying.   My truth tables are based in propositional logic.

Perhaps if I elaborate it will make a bit more sense:

What can you infer about the hand in question from the following if-then statement?

If there is a king in the hand, then there is an ace, or else if there isn't a king in the hand, then there is an ace

Now, keep in mind that I have both my initial answer (in my first post) and my revised answer (second post), based upon a different interpretations of the sentence, which I think is semantically ambiguous (think Time Vault! yay!)

My first reason for the second interpretation has to do with Diceman's prefatory remark.

In logic a statement is a sentence that is either true or false.  It is the basic building block of logic.  Sentences sometimes contain multiple statements.  For example:

"I went to the shore and I ate lunch."   That's two separate statements:
1) I went to the shore
2) I ate lunch

Diceman specifically states that the "following" is an "if-then" "statement".   That's why I said to Rich that if the "Or" was a disjunctive wedge that effectively created TWO "If...then" statements, then he was either sloppy or wrong in describing the following as an "If...then statement."    He should have said:
The following "sentence" or the following "statements" (note the plural).

The second key piece is the phrasing and the use of the comma.

If there is a king in the hand, then there is an ace, or else if there isn't a king in the hand, then there is an ace

If this sentence was intended to contain two "If...then" statements, then it should read:
"If there is a King in hand, then there is an Ace; if there isn't a king in hand, then there is an ace."  Or something along those lines.   

A third reason for interpreting the sentence that way is that if it were merely two Conditional statements, then the answer would be VERY easy.   My first answer in the thread shows this.   

But i think the trick is ignoring the meaning of the sentence and focus on the structure/logic of the sentence.

Let's focus on the sentence again.

If there is a king in the hand, then there is an ace, or else if there isn't a king in the hand, then there is an ace

So, we can symbolize the following:
K = King in hand
A = Ace in hand
From there, propositional logic can fill in the rest of the meanings

So it says:

If K, then A, or else not K, then A.

So the two possible interpretations are:

(K-->A) v (-K--> A)
 
Or

K--> (A v (-K-->A)) 

As I told Rich, I thought that this is a case semantic ambiguity, which means that the sentence can be interpreted in more than one way. 

[dicemanx]

Hi everyone,

It looks like these logic questions generated some very interesting discussion, including the possible interpretations of first question. Note that I did not write any of these questions - the questions came directly from a research study designed to generate insight into cognitive processes (references are at the bottom for those interested). I'll give only the answers first in case you got any incorrect and would like to see if you can determine the source of the error as a further challenge.

(1) Nothing can be inferred

(2) Nothing can be inferred

(3) Happy Cube


---------------------------------------------------------------------------------------------------------------------------------------------
Rationales:

(1)

Question number 1 is definitely the hardest of the three in my opinion. It is even purposefully ambiguous, because regardless of how it is interpreted you will never be able to infer anything about the presence or absence of the Ace.

The sentence is a disjunction (featuring the word OR) that can qualify as either:

1) inclusive (either statement A, or B, or BOTH are true) or
2) exclusive (statement A or B are true, but not both)

The statements A and B can also be interpreted as:

1) "one way" conditionals (K ->A and ~K->A, which translate into "If King present, then Ace present" and "if king not present then ace present") or
2) biconditionals (K<-->A and ~K<-->A, which translate into "if king present then ace present, and if ace present then king present" and "if king absent then ace present, and if ace present then king absent").

Nevertheless, let us go with the most likely interpretation that people will use: an exclusive disjunction with one way conditionals. You can try the exercise with other interpretations and you will reach the same result.

Given that either the first conditional is true OR the second conditional is true, it is possible that we have a hand with a K but no A (in this case, conditional 1 is false, and conditional 2 is true). It is likewise possible to have a hand with no K and no A (conditional 1 is true, conditional 2 is false). Therefore, we could have a case with no A in hand and the disjunction is still true. You can repeat the exact same rationale for A in hand. Given that it is possible that there is no A in hand and that A could be in hand (the disjunction is true in both cases), we cannot infer anything about the presence or absence of the A.

Congratulations to Demonic Attorney (who PMed me) for answering this question correctly . I got this question incorrect the first time I tried it.


(2)

While all Frenchmen are wine drinkers, it doesn't automatically follow that all wine drinkers are Frenchmen. Therefore, we cannot infer that any Frenchmen are gourmets, because the wine drinkers that are gourmets might not be Frenchmen.

(3)

If the unhappy dodecahedron is a GOKE, I must have written down either unhappy for attitude or dodecahedron for shape, but not both. In other words, I have either written down "unhappy cube" or "happy dodecahedron". This means that according to my rule for GOKE, "happy cube" must be a GOKE.



For more discussion of problem #1, check out the following publication:

http://webscript.princeton.edu/~mentmod/publications/1999/1999illusory.pdf

A discussion of all three questions can be found here:

http://www.rpi.edu/~brings/SELPAP/LMINDS/lminds.cogsci/node2.html

There is a fourth question posed there that was used by Wason in his investigations of the psychology of reasoning.

[Anusien]

(1)

Question number 1 is definitely the hardest of the three in my opinion. It is even purposefully ambiguous, because regardless of how it is interpreted you will never be able to infer anything about the presence or absence of the Ace.

The sentence is a disjunction (featuring the word OR) that can qualify as either:

1) inclusive (either statement A, or B, or BOTH are true) or
2) exclusive (statement A or B are true, but not both)

The statements A and B can also be interpreted as:

1) "one way" conditionals (K ->A and ~K->A, which translate into "If King present, then Ace present" and "if king not present then ace present") or
2) biconditionals (K<-->A and ~K<-->A, which translate into "if king present then ace present, and if ace present then king present" and "if king absent then ace present, and if ace present then king absent").

Nevertheless, let us go with the most likely interpretation that people will use: an exclusive disjunction with one way conditionals. You can try the exercise with other interpretations and you will reach the same result.

Given that either the first conditional is true OR the second conditional is true, it is possible that we have a hand with a K but no A (in this case, conditional 1 is false, and conditional 2 is true). It is likewise possible to have a hand with no K and no A (conditional 1 is true, conditional 2 is false). Therefore, we could have a case with no A in hand and the disjunction is still true. You can repeat the exact same rationale for A in hand. Given that it is possible that there is no A in hand and that A could be in hand (the disjunction is true in both cases), we cannot infer anything about the presence or absence of the A.

Congratulations to Demonic Attorney (who PMed me) for answering this question correctly . I got this question incorrect the first time I tried it.

One of us (Noel) notes that this particular problem may be confusing to people with some background in computer programming. (Do you see why?) We are planning an experiment running in parallel to J-L's in which the phrase `or else if' is replaced with `or' and the phrase `, but not both' is added at the end.

[dicemanx]

To be honest at first I did not differentiate between the word "OR" versus the phrase "OR ELSE". However, I found some info here:

http://www.thelogician.net/2_future_logic/2_chapter_26.htm

The word "OR" encompasses both inclusive and exclusive disjunctions.

The phrase "OR ELSE" specifically indicates exclusive disjunction.

Therefore my analysis that the statement contains an ambiguity in terms of the type of disjunction isn't correct, but this doesn't impact the solution.

[MadManiac21]

Ah, didn't know the or applied to the whole statements. Guess that was the trick to think of the entire if/then.

Yeah, don't know jack then.

I'm still curious about the rest of you; anyone else here take logic classes before?

[LordHomerCat]

I came up with Ace for the first one (lots of computer programming background and I guess I didn't think it out carefully enough), and then the correct conclusions for the other two.  I've had multiple classes with logic, including some basic logic in HS, digital logic in classes like Computer Architecture, then more propositional logic (and some more advanced forms) in Artifical Intelligence. 

The first one, most of the difficulty was definitely in deciphering the sentence, as I was definitely not confident in how I interpreted it at first ('or else' is weird looking).  The second was just a matter of careful reading, and I reasoned out the third one very quickly, although I did get a little confused reading all the different answers and started to doubt my own.

Neat problems Peter, got any more?

[Apollyon]

The if then else interpretation changes the or from a disjunctive or else to a conjunctive or, meaning that the or loses any logical meaning, since it's merely a construct of English.

So that interpretation then becomes "(if (king in hand) then (ace in hand)) and (if king not in hand) then (if king not in hand) then (ace in hand))". Since the king is either in your hand or not in your hand, the condition must be true for one of those two. Therefore, since either the ace is in hand or the ace is in hand, the ace must be in hand, using the if ... then ... else ... interpretation.

[Anusien]

To be honest at first I did not differentiate between the word "OR" versus the phrase "OR ELSE". However, I found some info here:

http://www.thelogician.net/2_future_logic/2_chapter_26.htm

The word "OR" encompasses both inclusive and exclusive disjunctions.

The phrase "OR ELSE" specifically indicates exclusive disjunction.

Therefore my analysis that the statement contains an ambiguity in terms of the type of disjunction isn't correct, but this doesn't impact the solution.

Computer scientists will find this difficult in two ways.  First, the IF-ELSE will be interpreted the way I demonstrated earlier:
If ( ) THEN ( ) ELSE IF ( ) THEN ( )
The "or" is syntactically meaningless there

First the rules for precedent and how closely certain operations bind together are different in CS and logic I suppose.  You parse the statements differently.

[dicemanx]

CS and logic aside, the comparison

(ace in hand...OR...if)

doesn't follow proper parallel structure and is grammatically incorrect, so should be ruled out as a possible interpretation. Proper parallel structure would occur in the case of the intended interpretation:

(if...OR...if)

[Smmenen]

CS and logic aside, the comparison

(ace in hand...OR...if)

doesn't follow proper parallel structure and is grammatically incorrect, so should be ruled out as a possible interpretation. Proper parallel structure would occur in the case of the intended interpretation:

(if...OR...if)

I don't think that's "logic aside" - that was exactly my basis for my concluding that way.   If you translate the sentence structure into propositional logic, that's really the way the sentence turns out.  The problem is that the sentence is unusual and people get caught up in the nonsensical meaning rather than the actual syntax.  IMO, people SHOULD be trying to reconstruct the sentence into an IF or IF structure because as human beings we have to make sense of what we read.  As a logic question, however, a more careful parsing is called for, which is why I changed from initial conclusion from my first to my second post. 

[dicemanx]

I don't think that's "logic aside" - that was exactly my basis for my concluding that way.   If you translate the sentence structure into propositional logic, that's really the way the sentence turns out.  The problem is that the sentence is unusual and people get caught up in the nonsensical meaning rather than the actual syntax.  IMO, people SHOULD be trying to reconstruct the sentence into an IF or IF structure because as human beings we have to make sense of what we read.  As a logic question, however, a more careful parsing is called for, which is why I changed from initial conclusion from my first to my second post. 

I'm not sure I understand completely - ignoring grammar, using propositional logic either interpretation is valid because parallelism isn't an issue. However, grammatically only one interpretation is valid (i.e. there is only one proper way to parse) by following proper parallel structure. I'm assuming that the original experimenters weren't concerned with any ambiguity issues because you must start with the grammatically correct sentence before beginning the translation even for a "logic question". The question didn't start out using propositional calculus, it started with an English sentence that had proper syntax with no ambiguities regarding meaning so long as grammatical rules were followed.

The question is essentially - can you parse ignoring rules of grammar when translating into propositional logic.

[runeblayde]

As far as question number 2 goes, while it is true that you can state that (A is B), it is hardly sound logic if there is an absence of A to provide a basis for this statement.  Therefore, while nothing can be inferred about anyone from both statements, it can be reasonably assumed that there are both frenchmen and gourmets in the room in question, thus my answer.  However, with pure logic in question, it is true that nothing at all can be inferred.