# introduction to formal logic + truth tables with an annoying card question

#### higs

##### Omg wow imo
If there is a queen in the hand then there is an ace in the hand,

or else if there isn't a queen in the hand then there is an ace in the hand.

There is a queen in the hand. What follows?

---------------------------

Stop reading here if you don't want the breakdown of my answer to this (which is the correct one ofc )

As many have pointed out, the statement is phrased sort of ambiguously.

The crucial ambiguity lies with the interpretation of "or else" which is the logical operator. As the logical operator, it describes the relation between two propositional units.

A) (if there is a queen in the hand then there is an ace in the hand)

B) (if there isn’t a queen in the hand then there is an ace in the hand)

But these propositions themselves are complex units that can be further broken down into simpler units with their own operators.

The breakdown of A) is :

(pq) or “if p) (queen) then there is q (ace)

The breakdown of B) is

(~pq) or “if not p (queen) then there is q (ace)

I am going to try and explain why I believe very strongly the answer is "no ace."

There are three relevant logical operators that “or else” seems to be commonly interpreted as.

1) [⊕] Exclusive disjunction (exclusive “or”) : Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if one but not both of its operands is true.

2) [V] Logical disjunction (inclusive “or”) : Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if at least one of its operands is true.

3) [^] Logical conjunction (“and”) :
Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if both of its operands are true.

1) I am only going to talk about the exclusive disjunction in my OP. It is in my opinion the best interpretation of “or else.” which typically suggests two alternatives one of which will occur and the other will not. If it were just “or” then 2) would perhaps be a stronger interpretation, but “else” seems to me to be added commonly to denote exclusivity. For example :

(Give me that donut) or else (I will kill you.)” This threat only makes sense because you understand that if you give me the donut, then I will not kill you, but if you don’t give me the donut then you will die. If this operator is in play, you know that if you make one operand true, then the other operand is necessarily false. The overall statement gives a value of TRUE if one but NOT BOTH of the operands are true.

(You love it) or else (you hate it)
(the sun is out)
or else (it is raining)
(My unicorn is pink)
or else (Macron has a massive head)

((If there is a queen in the hand then there is an ace in the hand,)

or else (if there isn't a queen in the hand then there is an ace in the hand.))

There is a queen in the hand
. What follows?

Now I am going to talk about truth tables and specifically the exclusive disjunction operator. I will detail the steps for how one fills in a truth table during my explanation.

1) this is my prepared truth table. What we first need to do is give the truth value for the operators and operands within the complex propositions A) and B)

First step is to fill in all the combinatory possibilities for the operands.

Second step is to determine the truth value of the overall statement In A) and in B) respectively In this case: () your basic implication which produces a value of false if the first operand is true and the second operand is false, and a value of true otherwise.

Feel free to check the work according to the rule I just stated.

Now comes the REALLY EXCITING BIT !!!!!

We will fill in the truth value for the overall statement, that is (A) B)) . The important truth values to compare are in columns 2 and 6 (circled), because that is the truth values of individual propositions A) and B)

As we mentioned earlier, The rule for [⊕] is that produces a value of true if
one but not both of its operands is true.

We now have all the possible truth values for the overall proposition A) or else B).

The table is now complete and irrefutable according to our own rules, all we have to do is look for the result.

Starting condition is that there is a queen in the hand. Or in other words, there is p)

Here I have circled in the table in which cases p) is true.

And then we can see that there is only one case in the table where the overall statement (A B) is true, and that p) is also true, which is the second line.

So....
....

....What follows? if we look at "q" on this line ?

Conclusion : In the case when p is true, and the overall statement is true and the main operator is an exclusive disjunction, q is false.

It's counter intuitive, but there it is.

The answer is ~q or “no ace.

------------------

I have shown my work, I can discuss it if you like. I actually think I can see how the reasoning works now that I've done the demonstration, and can even describe it in non formal logic terms, but it still sounds fucked and counter intuitive when I try.

PS : here is an article, specific problem is around page 204
file:///tmp/mozilla_kora0/10.1.1.333.6706_1.pdf

yep

#### higs

##### Omg wow imo

That's fine if you think that who am I to judge, but could you flesh out your reasoning

#### Firehazard159

##### ¯\_(ツ)_/¯
There are several cases where I can get ace exists, some cases where I can get "unknown/not enough information" and zero cases where I can get "no ace."

I followed you up until the conclusion of your truth table, which is the opposite of what I would expect.

My take using more normal language:

If it's a wedding, I wear a formal outfit.
If it's not a wedding, I wear a formal outfit.

The assumption should be in this situation, that regardless of the wedding or not wedding, you wear a formal outfit. The assumption should not be, "I wear non-formal outfits"- this is the "lying presenter" interpretation because it by default means that "If it's not a wedding, I wear a formal outfit" is not always a true statement, whereas "if it's a wedding, I do" is always true.

So the only way you can get around the logical interpretation is to assume the presenter of the information is lying to you. There are situations beyond the two where they do not act as they describe in the premise.

So we end up in these possible solutions:

Option A:
It's a wedding. I wear a formal outfit. If it's not a wedding, see B:

Option B:
It's not a wedding. I do wear a formal outfit.

There is no "If it's not option A nor option B, then we go with option C" Where:
Option A is in place, and it is not a wedding, skip B to reach C:
There is no wedding, I do not wear a formal outfit.

If the options are to be disconnected (exclusive or), they should be written separately:
1: Interpretation one: you've ignored the second half of the overall question and taken it out of context
Rule: It's a wedding, I wear a formal outfit.
it's a wedding. do I wear a formal outfit? <yes>
it's not a wedding. do I wear a formal outfit <unclear>

2: Interpretation two: you've ignored the first half of the overall question and taken it out of context
Rule: It's not a wedding, I wear a formal outfit.
it's a wedding. do I wear a formal outfit? <unclear>
it's not a wedding. do I wear a formal outfit <yes>

3: Programmer interpretation
RuleA: It's a wedding, I wear a formal outfit.
RuleB: It's not a wedding, I wear a formal outfit.
Either statement A. or B. is true. Do I wear a formal outfit? <yes>

So again, there are situations where it is clear, situations where it is unclear, and situations where it is unambiguous. But I can't see the logic where we have an affirmation that there is no formal outfit worn (no ace).

#### washti

##### pablo
For me OR ELSE sounds more like 'in another case' rather than exclusion. So I bet on the logical(inclusive) disjunction.

#### Animekitty

##### baby marshmallow born today
queen and ace
not queen and ace
queen and not ace
not queen and not ace

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#### ZenRaiden

##### One atom of me
That's fine if you think that who am I to judge, but could you flesh out your reasoning

I think its not logical to say it rains outside and it is wet or else it does not rain outside it is wet
and then say it rains outside and it is dry. Something is missing.
or else cannot negate previous statement.

#### higs

##### Omg wow imo
@Firehazard159

I know this is really annoying and I really am sorry, but no one on this thread can really reject the conclusion of the truth table, unless you reject the principles of formal logic. I am certain that my demonstration is correct because I made a logician friend do it on their own end and they were like "fck off of course it's not "no ace" until they did the truth table and arrived at the same conclusion as me, and one of the source articles says that my conclusion is valid. I have an irrefutable demonstration according to formal reasoning rules that I have executed correctly. Science and math sometimes give counter-intuitive conclusions, there is no reason why logic should not sometimes give counter-intuitive conclusions also, it's the only reason that this enigma is somewhat famous, because the valid answer is extremely strange. (this is why I said it was a great question )

I don't understand how it could be that programmers would follow a different logic, philosophers and mathematicians and computers use the same rules as far as I know. Maybe your programming language is different from other languages and the "or else" is a different logical operator, but you definitely have the possibility of implementing an "or else" that IS an exclusive disjunction in some form or other. In fact I have read that this problem was originally stumbled upon because a computer kept giving the "no ace" answer, and the dude inputting it thought there was a bug in the program, but when he checked it by hand, he found the computer was correct.

That being said, as you pointed out in an earlier discussion, the language in this version of the problem is hazy and the answer depends on the interpretation of "or else." We have seen in the article that accompanied it, that the explicit and clear version of the problem is preceded by "one of the two following statements is false", which means that actually, my interpretation of "or else" is the one that was meant even if it was admittedly badly communicated. But as latte and cheese pointed out, EVEN when the clear version is given, people can't get to the corect conclusion. This is a case where, for several reasons, you need to do something equivalent to a truth table to get the answer, and when you do it it's still extremely surprising, it is a logical aberration. What's also extremely surprising is that this is a very simple statement to give a counter-intuitive answer. I have ignored no parts of the question, I have translated all the parts and their relations to each other into a complete logical proposition. Which is [(p->q) (~p ->q)] The statements are not as you say, "disconnected" from each other by the exclusive disjunction, on the contrary they are put into relation.

You're not giving me the logical operators between your wedding/formal outfit statements, you are disconnecting them in fact. ALL statements have a truth value, they can always be either TRUE or FALSE. If they cannot be true or false, then they are not valid statements. So you have to first admit that both your statements have a truth value, that is that they could be, as you say, a "lie".

RuleA: It's a wedding, I wear a formal outfit.
RuleB: It's not a wedding, I wear a formal outfit.
Either statement A. or B. is true. Do I wear a formal outfit? <yes>

KEY WORD and logical operator between statement A and B : EITHER

Is this an inclusive "or" or an exclusive "or" or something else ?

Finally you have also not given me your starting condition so that I can give my answer. Is it a wedding (p) or not a wedding (~p) ?

#### Firehazard159

##### ¯\_(ツ)_/¯
Well, I would say I did give it, because I painted all the scenarios, so:

RuleA: It's a wedding, I wear a formal outfit.
it's a wedding. do I wear a formal outfit? <yes>
it's not a wedding. do I wear a formal outfit <unclear>

RuleB: It's not a wedding, I wear a formal outfit.
it's a wedding. do I wear a formal outfit? <unclear>
it's not a wedding. do I wear a formal outfit <yes>

Would then have to reconjoin and lead to:
It's a wedding: (rule a) do I wear a formal outfit? <yes> && (rule b) do I wear a formal outfit? <unclear> (therefore no? in formal logic)

It's not a wedding: (rule b) do I wear a formal outfit <yes> && (rule a) do I wear a formal outfit <unclear> (therefore no? in formal logic)

Which fits Animekitties format:
Wedding and Formal
not Wedding and Formal
Wedding and not Formal*
not Wedding and not Formal*

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*I would contest here too, in that we never know if there are other conditions that give rise to formal attire (or an ace in hand.) We only know the condition of the wedding's impact on formality, and that whether it is or isn't, one wears formal attire, yet one of those statements can be wrong (and therefore be wearing formal attire.)

However, because we *KNOW* one of those statements (rule A or B) to be false, then all we end up with is:
It's a wedding: (rule a) do I wear a formal outfit? <yes> *OR* (rule b) do I wear a formal outfit? <unclear>

It's not a wedding: (rule b) do I wear a formal outfit <yes> *OR* (rule a) do I wear a formal outfit <unclear>

Inevitably, the only valid answer seems to be "not enough information/unclear" OR yes (dependent on which is false, but again, still not enough information, because we have to know which is true to answer), but never an explicit "No," unless, with additional information, we can confirm that the "unclear" route can indeed lead to "no formal outfit."

This is why the example of deck-type (additional information) would prove your "no aces" logic wrong. If we had a deck of 51 aces and 1 queen and you drew a hand of 2, no matter what, a queen would always lead to an ace, but no queen would also lead to an ace.

If you're using a standard deck (which again, is additional information) - then we know rationally that the odds are slim that drawing a queen and an ace isn't 100%, and of course, we know that drawing queens do not force the draw of aces through some mystical force. But because they *can* be drawn means that you would say "no aces" and potentially be wrong, because the premise itself was faulty.

If you're using a deck with no aces, then that is the only way you can confirm no aces to exist.

#### higs

##### Omg wow imo
@washti

Here is the truth table with the inclusive "or", another reasonable interpretation of the question.

Here is your truth table washti. Your conclusion is still somewhat counter-intuitive however, because all you can say is "cannot confirm if ace or no ace."

You can see that with starting condition p there are two possibilities for q, which means that you cannot confirm if q or ~q. You would be able to confirm "ace" if you were told "p" and also "both of the statements are true."

#### higs

##### Omg wow imo
My conclusion to all this, is that people have a bias in which they do not tend to consider the possibilities of falsehood and do not include it so naturally into their reasoning.

#### higs

##### Omg wow imo
If you're using a deck with no aces, then that is the only way you can confirm no aces to exist.

No no no no no no

There is no deck, there are no actual cards, this is NOT empirical. All the information is in the text. And my "no ace" answer (given an exclusive disjunction) is irrefutable.

The enigma could just as well be :

If there is a flippyflop then there is a cthulhu,

or else if there isn't a flippyflop then there is a cthulhu.

There is a flippyflop. What follows?

(No Cthulhu)

##### think again losers
That breakdown was excellent Higs, the truth table makes sense. Unfortunately you're still wrong ¯\_(ツ)_/¯

It can be interpreted as an exclusive disjunction, but it can also be interpreted as an inclusive one. You've given reasoning (the threat on your life thing) but that's soft at best.

When you (a well known psychopathic killer) say "give me the donut or I'll kill you", are you really saying that once I have gifted you the treat I am immune to your threats of death? What if I then take out my phone and start calling the cops? Walk up and take your wallet? Try and kill you?

There are worlds where the donut buys me enough goodwill for you to restrain your killer tendencies while I shamelessly slash your tires, but there are a whole bunch of worlds where it doesn't.

Giving me the donut and then you killing me is definitely still on the table.

You assume that it's an exclusive disjunction, you basically introduce the premise to facilitate your preferred interpretation. I think your reasoning outside of that assumption is cool and clever, and if I were to grant you that assumption we'd be on the same page.

#### higs

##### Omg wow imo
That breakdown was excellent Higs, the truth table makes sense. Unfortunately you're still wrong ¯\_(ツ)_/¯

It can be interpreted as an exclusive disjunction, but it can also be interpreted as an inclusive one. You've given reasoning (the threat on your life thing) but that's soft at best.

When you (a well known psychopathic killer) say "give me the donut or I'll kill you", are you really saying that once I have gifted you the treat I am immune to your threats of death? What if I then take out my phone and start calling the cops? Walk up and take your wallet? Try and kill you?

There are worlds where the donut buys me enough goodwill for you to restrain your killer tendencies while I shamelessly slash your tires, but there are a whole bunch of worlds where it doesn't.

Giving me the donut and then me killing you is definitely still on the table.

You assume that it's an exclusive disjunction, you basically introduce the premise to facilitate your preferred interpretation. I think your reasoning outside of that assumption is cool and clever, and if I were to grant you that assumption we'd be on the same page.

omg you are so super wrong hado.

The fact that the killer could kill me anyway does NOT change the meaning communicated in the threat when she says it , that is, as she wants me to believe it. If she told me "Give me the donut and I will kill you anyway" I have no reason to give her the donut anymore do I ? Threatening someone as a power play makes no sense without an exclusive disjunction.

It is not my preferred interpretation, or at least, no more than yours is.

Not only is the exclusive disjunction in several ways the most reasonable interpretation of "or else" following classical usage of the term as can be seen here :

And here :

https://dictionary.cambridge.org/dictionary/english/or-else

used to say what will happen if another thing does not happen:
We must be there by six, or else we'll miss the beginning.

used to compare two different things or situations:
She's either really talkative and you can't shut her up or else she's silent.

used as a threat, sometimes humorously:
He'd better find it quickly, or else (= or I will punish him in some way)!

But it turns out it was also the original intent as can be seen from the subsequently clarified version of the problem in one of the source articles, whereby it is stated beforehand "one of the following statements is false" which definitively sets the "or" as being an exclusive.

Also my reasoning is not that clever, truth tables are dumb, you just fill in squares in the right order (my post was nicely edited though wasn't it ?) . Fire is probably being more creative and interesting in his reasoning, even if he super wrong !!!!

(like the rest of you.)

Anyway, I admit that the unclear text can legitimately be interpreted as an inclusive "or" like washti has done. But then, your answer is "cannot confirm." (which I think you accept ?)

#### Firehazard159

##### ¯\_(ツ)_/¯
If you're using a deck with no aces, then that is the only way you can confirm no aces to exist.

No no no no no no

There is no deck, there are no actual cards, this is NOT empirical. All the information is in the text. And my "no ace" answer (given an exclusive disjunction) is irrefutable.

The enigma could just as well be :

If there is a flippyflop then there is a cthulhu,

or else if there isn't a flippyflop then there is a cthulhu.

There is a flippyflop. What follows?

(No Cthulhu)

I understand that it's not empirical; hence why I gave multiple examples, because the point I was making was in line with @hado's:

> Giving me the donut and then me killing you is definitely still on the table.

Similarly, I can have a deck where there are 51 aces and 1 queen, or a deck with no aces. There could be a world where flippyflops and cthulus exist together, and one where they don't. You're right in that the example itself doesn't matter. There simply isn't enough information without assuming more information than is presented.

When I say they are lying in the premise, I'm referring to this:
" "Or else" means used to prove that something must be true, by saying that the situation would be different if it was not true. E.g: It’s obviously not urgent or else they would have called us straight away. "

In other words, by tying them together with an "or else"
RuleA MUST be true, and if it is not (and we know ONE of them is false, yeah?)
then: RuleB Must be true.

The alternative is that the premise itself, not the rules, is not true.

So then we end up with
Queen in hand <--- ruleA is true, or it's a lie. Or, ruleB is true, or it's a lie, but rule B is not in play because "Queen is in hand." The answer is either True (Yes, we have an ace) or (Uncertain, there may or may not be an Ace, depending on whether ruleA or ruleB was false, which we cannot know, therefore, uncertain is the only true answer. If we can confirm the answer by looking at the hand, we then introduce the information necessary to know, which is how the computer program works. The programmer getting an unexpected result is due to not realizing they made a mistake in logic - they made a rule they *expected* to be true/false, but was actually false/true respectively, which is very, very, very common.)

I think this answer sums up the usage of or/or else nicely:
"You really don’t need else after or, because or means that the upcoming phrase or clause is an additional alternative to what has come before: "

You said it yourself here:
" ALL statements have a truth value, they can always be either TRUE or FALSE. If they cannot be true or false, then they are not valid statements. So you have to first admit that both your statements have a truth value, that is that they could be, as you say, a "lie". "

We have a queen:
True--------True
Queen --> Ace <----statement is true, premise was true; statement is valid
If A then B - A:B

True--------False
Queen --> !Ace <----Premise was a lie, statement must be false/invalid. This DOES mean that there is no ACE, but it also means the entire premise is a falsehood, which is probably why people get confused and answer unintuitively.
If A then B - A:!B

You're essentially saying one of these is true (or false):
If Green, then Go.
If Red, then Go.
(If Not-Green, then Go.)

The light is green. Do you go?
"Which one is true?" is the only valid answer. You cannot know without first determining the truth or lie of each statement.

The light is red, or at least, Not-Green. Do you go?
"Which one is true?" is the only valid answer. You cannot know without first determining the truth or lie of each statement.

#### higs

##### Omg wow imo
Saying "maybe the killer will murder you anyway" does not invalidate the statement she made. If I give the donut and she does kill me, it simply means that the statement did not describe the world accurately. The statement was still "valid" as in it could have been true or false, it had a truth value. The fact that she kills me anyway just means that on the truth table we would select a column where "or else" is false. This statement is ALWAYS valid as a logical proposition but is false in two cases : if I give the donut and she kills me, and, if I do not give the donut and she doesn't kill me. i think it would be pretty consensual to say that in that case, the threat that she communicated was "a lie". Wouldn't you agree ?

With this puzzle we have no real world confirmation or refutation,in other words, we do not have a description of reality. we have theoretical items in relation with each other.

You can reject the whole puzzle as false, but in that case your two options left (refer to the truth table and look at the two horizontal lines where ⊕ comes up "F" ) are "you do not have p" (this means that the starting condition in the narrative was untrue) or "both statements A and B are true" and in that case it was NOT an exclusive disjunction. So, you are effectively rejecting the WHOLE narrative , the way I am arguing it is to be interpreted, and the way in which it is MEANT to be interpreted according to the more explicit version given later on :

Only one of the following statements is true :

If there is a queen in the hand then there is an ace in the hand,

or else if there isn't a queen in the hand then there is an ace in the hand.

There is a queen in the hand. What follows?

In this version, there is no longer any controversy on whether or not the disjunction is exclusive, and the answer is definitively "no ace" unless you reject the whole narrative as false, which you can, but then, that really is kind of pointless...

##### think again losers
Higs you have cancer. Take chemotherapy or else you'll die.

If you then die from chemo, you wouldn't think your doctor was lying. You wouldn't have misunderstood what they meant. It wouldn't be some clever exploit of language. The doctor wants you to believe it what they say, it's in your best interest to believe them, and your understanding does not differ from theirs. All the premises are true and yet this story concluding with your survival is not guaranteed.

This is the definition of invalid reasoning Higs.

You say the killer isn't implying they'll kill you anyway, and I agree. But the equivalent of your proof where you can then deduce unlikely events (no ace) from the premises, is me deducing unlikely events like killer Higs not being able to defend themselves from me after I give them the donut.

You are treating your own analogy differently to the original problem by explaining counter-examples away with the killers intentions and interpretation of their meaning. I think the doctor example works better because there is less subtext than in a death-threat.

I also think you might be presenting evidence for the author's intentions selectively, but don't have time to looks at it right now. Maybe after work.

#### washti

##### pablo
Here is your truth table washti. Your conclusion is still somewhat counter-intuitive however, because all you can say is "cannot confirm if ace or no ace."

You can see that with starting condition p there are two possibilities for q, which means that you cannot confirm if q or ~q. You would be able to confirm "ace" if you were told "p" and also "both of the statements are true."

OR confirms the truth of statement if one of the propositions (here in the form of implications) is true. And that's the case in the truth table.

OR operating on these two implications means that the occurrence of ace isn't dependent on the occurrence of the queen.
It sounds rather intuitive.
It confirms the presence of ace in both cases - when the queen is there and when she is not.

#### higs

##### Omg wow imo
Higs you have cancer. Take chemotherapy or else you'll die.

If you then die from chemo, you wouldn't think your doctor was lying. You wouldn't have misunderstood what they meant. It wouldn't be some clever exploit of language. The doctor wants you to believe it what they say, it's in your best interest to believe them, and your understanding does not differ from theirs. All the premises are true and yet this story concluding with your survival is not guaranteed.

This is the definition of invalid reasoning Higs.

You say the killer isn't implying they'll kill you anyway, and I agree. But the equivalent of your proof where you can then deduce unlikely events (no ace) from the premises, is me deducing unlikely events like killer Higs not being able to defend themselves from me after I give them the donut.

You are treating your own analogy differently to the original problem by explaining counter-examples away with the killers intentions and interpretation of their meaning. I think the doctor example works better because there is less subtext than in a death-threat.

I also think you might be presenting evidence for the author's intentions selectively, but don't have time to looks at it right now. Maybe after work.

First of all I would point out the obvious implicit atomic proposition in the doctor example. The doctor is telling me that I will not die because of the cancer, which is a different propositional item than dying of chemo.

But let me ignore this and do the truth table for chemo/exclusive disjunction /general death anyway

chemo or else death
T
F T
T
T F
F
T T
F
F F

Your scenario is top line, aaaaand....oh shit GODDAMNIT HADO YOU JUST BROKE LOGIC FFS DO YOU HAVE ANY IDEA HOW LONG IT TOOK US TO COME UP WITH THIS CRAP AND THEN YOU JUST COME ALONG LIKE THAT AND BREAK IT OH GOD WTF

Well so? The statement is just false, obviously. It failed to describe the world correctly in this instance. It was still a logically valid statement though. If it hadn't been, then we wouldn't have been able to say anything about it's truth or falsity at all.

None of this is relevant, because here we are dealing with a purely theoretical scenario and a series of propositions that are articulated with each other into one larger statement. By process of elimination and with some conditions the narrative asks us which part of the statement is true when X condition is met and given Y. The truth table allows you to look at all possible truth values for all the items. In the case of "or else" (exclusive) There is only one fit.

In the case of an inclusive "or" given the conditions, it is true in two cases, if you look at the truth table that I did for Washti, she has two possibilities for q that oppose each other, this means that given what she knows, she "cannot confirm."

Here, you should enjoy the paper that this is from, it's psychology. Why don't you check it out ? See for yourself whether I am being selective @washti this is what was in the broken link if you're still interested.

EDIT : be careful if you read the paper, There are a couple of similar scenarios in it that are not the one we are talking about here. Make sure you use the right one if you want to evaluate my claim.

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#### washti

##### pablo
In the case of an inclusive "or" given the conditions, it is true in two cases, if you look at the truth table that I did for Washti, she has two possibilities that oppose each other, this means that she "cannot confirm."
you misrepresent the meaning of the operator OR.

@washti this is what was in the broken link if you're still interested.
errr are you sure it's right pdf? On which page is the puzzle?
I searched for 'ace' and 'queen' there and ...nothing?

Is this a real answer? have I just killed suspense?

#### higs

##### Omg wow imo
In the case of an inclusive "or" given the conditions, it is true in two cases, if you look at the truth table that I did for Washti, she has two possibilities that oppose each other, this means that she "cannot confirm."
you misrepresent the meaning of the operator OR.

@washti this is what was in the broken link if you're still interested.
errr are you sure it's right pdf? On which page is the puzzle?
I searched for 'ace' and 'queen' there and ...nothing?

Is this a real answer? have I just killed suspense?

It's king and ace in this paper.

Uhhh go around pages 210, 212, 204-205 there is relevant stuff.

I do not misrepresent OR ... how

#### washti

##### pablo
The operator OR doesn't have option - 'you cannot confirm'.
Inclusive disjunction (also called or) is a logic operation. It normally takes two inputs. It is false when both inputs are false. Otherwise, it is true.
A V B is true if A is true, or if B is true or if both A and B are true.
source

When using OR two values/propositions in the statement can oppose each other and if one is true then whole statement is true.

If there is a queen in the hand then there is an ace in the hand,

or else if there isn't a queen in the hand then there is an ace in the hand.

You would have to have both propositions to be false to OR giving False.
But you cannot have - 'cannot confirm'. That's not how OR works.

-----------
there is no king as well. In the Attached PDF
Sticks or Carrots? The Emergence of Self-Ownership - that's the paper title. and it has 18 pages. So wrong pdf? I mean it's late, right?

#### higs

##### Omg wow imo
Oh shit wrong pdf hahaha sorry, good thing it was that and not something weirder. I will give you the right pdf in a few hours

For the logic question I am still good though. «Cannot confirm» is not the truth value i assign to OR, but my answer to the question «what follows?» It is in reference to the fact that you have 2 rows where OR is true and these two rows have different truth values for q in statement A/(p—>q). This means that if someone asks «is there q/an ace» you don't have sufficient info to eliminate one of the two instances where «or» and p turn up True ! You are left with two possibilities concerning the state of q/ace, both possibilities fulfill the premises, but they contradict each other. Am I clear ? It is very late.

#### washti

##### pablo
«Cannot confirm» is not the truth value i assign to OR, but my answer to the question «what follows?»

The answer to this question depends on the result provided by using the operator. In this case OR.

You cannot confirm ace only if the 3rd sentence There is a queen in the hand. What follows? is a new premise independent of the previous sentences. That would just mean - there is a queen. That's not the case.

It is in reference to the fact that you have 2 rows where OR is true and these two rows have different truth values for q in statement A/(p—>q).

You claim that for OR turn True for Ace in What follows? you must have the same values of q from separate rows. That would be doing conjunction (ANDing) q values from separate rows! And if they are both the same (both T or both F) that's mean that OR operator proves Ace. That's nonsense. You are still misrepresenting OR.

Using logical operators we consider several separate situations for rationale by prescribing different value configurations for premises.

In the first row, the Truth of p implies the Truth of q OR the False of ¬p implies True of q. Both implications are True so OR is True.

In the second row, the Truth of p doesn't imply that q is False OR the False of ¬p implies that q is False.
One implication is False and another is True. So OR by definition is True.
(And so on for the following rows)

If you wanna AND rows you need to create more complex theorem connecting them.

puzzle: pq OR ELSE ¬p q

rows: pq OR ¬(¬q)⇒q AND p⇏¬q OR ¬(¬p)¬q

Both OR are True so AND is True.

Btw, the drawn table seems incomplete (no OR for two false implication results)

if someone asks «is there q/an ace» you don't have sufficient info to eliminate one of the two instances where «or» and p turn up True ! You are left with two possibilities concerning the state of q/ace, both possibilities fulfill the premises, but they contradict each other.

No. This means that the occurrence of the second condition does not depend on the occurrence of the first condition. And not that the presence of the second cannot be confirmed because the first condition is True once and the other time is False.

#### Firehazard159

##### ¯\_(ツ)_/¯
Essentially you have 3 views:
A: Exclusive Disjunction - Philosophically Trained No Ace
B: Inclusive Disjunction - Layman Unknown (Not using a non-colloquial definition of "or else" which to them, is distinction without difference)
C: With Constraint - Programmers View - Only one of these statements has truth value (rather than both) - leading to a contradiction/unknown.

So how this works in "post-processing":

A: Builds truth tables assuming all arguments are valid, but not sound. They cannot stop at RuleA because a full truth table must be built, leading to RuleB, which identifies a contradiction that necessitates that "what follows" must be true (RuleB) which flips a "false" to be"true", clarifying the unknown to be now known, affirming that there is no ace under a completed truth table. (This likely also works if you start with RuleB and lead to RuleA as well.)

B. Builds a truth table assuming all arguments are valid, but not sound, but does not exclude based on Else, therefore, neither side can be invalidated, leading to the selection of two possible outcomes, thus, outcome unknown.

C: Builds a truth table assuming one argument is valid and sound, and one argument is valid but not sound (which is the false, and thus, discardable statement), which is implied by the "or else" && the "only one of these statements is <true> meaning "valid AND sound." This leads to only one rule being processed, ever, because only one can be true. So to them, it looks like:
Q->A or else !Q->A; Alternatively,
Q->A or else ~!Q->A
based on what is actually in the hand. Ultimately, the same as B in the outcome, but with a different rationale due to computer science training.

So really this is a question of:
"what is the meaning of true"

Because philosophy apparently sees:
"Valid but not sound" as "True"
Where as programmers see
"Valid and sound" as "True"

AND
"what is the meaning of or/or else"
(exclusive/inclusive)

#### higs

##### Omg wow imo
@washti

Maybe I haven't been clear.

It's not a question of whether or not OR has a "cannot confirm" option. You're right, it doesn't. Or is simply an operator that gives a value of "true" or "false" according to the set of rules that govern it.

Premises of the puzzle (in case of inclusive OR) are this :

1) P [there is queen]
2) (p->q) V (~p->q) [(queen implies ace) OR (not queen implies ace)

Question : given this, what is state of q ?

The truth table in photo is finished, all the combinatory possibilities for (p->q) V (~p->q) are in it. There is no possible scenario outside of the truth table left for this statement.

The puzzle tells you premises 1 and 2 are true.

Central column indicates when premise 2) is true or false (as you can see, for this statement it is always true.)

Premise 1) being true is ONLY the first two rows, first column. These are the only instances where 2) is true whilst 1) is true.

In row 1, q [ace] is true.
In row 2, q [ace] is false.

So you have two possibilities where premise 1) and 2) are true, and you have no further information to discriminate between them.

You have two possible states of q given the fulfilled premises, and no further info to be able to say it's one option and not the other.

The rows are separate possibilities yes, given the premises they are two equal logically possible outcomes. We are no longer applying any operators here, just looking for the answer ?

______________

Sorry if I have repeated myself, I don't really understand your objection so my intention is to fully clarify. If you still do not agree with the reasoning please be patient with me and point out what you think is flawed again.

EDIT : Oh this is interesting when you say :
Btw, the drawn table seems incomplete (no OR for two false implication results)

I think I understand what you are saying. It is not incomplete, because the statement uses the same operators several times. "p" cannot be "not p" at the same time in the overall statement see ? So every time that "p" is true on the right, then "not p" is false on the left, do you see ?

This is the complete truth table for "or" working on this statement.

If you want to be really annoying you can start saying that you think that the operators are all different cards, different aces and different queens, and then yes, the truth table will be different. (I think that would be kind of a dishonest interpretation of the puzzle though, because then they would have just used different card names.)

#### washti

##### pablo
Premises of the puzzle (in case of inclusive OR) are this :

1) P [there is queen]
2) (p->q) V (~p->q) [(queen implies ace) OR (not queen implies ace)

Question : given this, what is state of q ?

It seems you have now constructed the argument in a different way than in OP by reading the puzzle backward.
You take the resulting sentence as the first premise and the first two sentences as the second compound premise.

3 st sentence - 1st premise) p
1st &2nd sentence - 2nd premise) p-> q V ~ p-> q

The rows are separate possibilities yes, given the premises they are two equal logically possible outcomes. We are no longer applying any operators here, just looking for the answer?
How would you come to answer without using the operator? you unconsciously apply a connection between premises in the argument to form a conclusion. In your case 'being equal' means AND.
for 1st premise - p(True) AND 2nd premise p-> q V ~ p-> q] (True for V - whole premise) - ∧ AND is True.
for 1st premise - p(True) AND 2nd premise p-> q V ~ p-> q] (False for V - whole premise) - ∧ AND is False.

And yeah it doesn't make any sense because:
a) you constructed argument wrong way
b) you don't believe that you need to apply operators between premises while you apply one
c) your choice of the applied operator is wrong.

--
When building an argument according to the direction of the puzzle's content, we have two premises in the form of implications, separated by OR ELSE (for me in the sense of OR)
The third sentence is the conclusive implication of both premises: "What follows" from p if I have those 2 above?

1st sentence - 1st premise) p-> q
OR ELSE
2nd sentence - 2nd premise )~ p-> q
3rd sentences - implication from the above sentences. if I have p -> q OR ~ p -> q ---> THEN WHAT imply p?

the conclusion is q.
I think I understand what you are saying. It is not incomplete, because the statement uses the same operators several times. "p" cannot be "not p" at the same time in the overall statement see ? So every time that "p" is true on the right, then "not p" is false on the left, do you see ?
Whaa?
In the table, you list every possibility to see every correct outcome for the operator.
Your table states that OR is always True and that's not the case.
It's False if both of the values are False. So it should include

p ----> q OR ~p ----> q

T - F - F - F - T - F - F

And yeah it's the same meaning for both implications. The same case in your first row and you didn't cut it out.
If you want to be really annoying you can start saying that you think that the operators are all different cards, different aces and different queens, and then yes, the truth table will be different. (I think that would be kind of a dishonest interpretation of the puzzle though, because then they would have just used different card names.)
I don't wanna be annoying and I didn't say anything like that. So ... ?

#### higs

##### Omg wow imo
The order in which the premises are stated changes absolutely nothing, they both apply just the same to the reasoning.

If I say
1) (p->q) V (~p->q) [(queen implies ace) OR (not queen implies ace)
2) p

It's exactly the same result. The premises constrain the interpretation of the table.

The truth table is not incomplete. the value is always "true" in the results under "or" because this is not the truth table for "or", but the truth table for THE WHOLE PROPOSITION with OR as one of the operators.

Idk what to say if you don't agree with the method, that's just how it is. Ask someone who knows truth tables who isn't me. I've already had it vetted by two people who have done formal logic, to about a master's level, and this is extremely basic level stuff. They both agree it is completely correct.

And I don't even know how to answer the assertion that I am using "and", that's just weird. I'm just interpreting the results of the table. I am not using "and", I am simply stating the results concluded in the table. I am saying there are two results that could apply given the premises.

I don't wanna be annoying and I didn't say anything like that. So ... ?

Sorry, I didn't mean to sound like I was saying you were being annoying specifically or that you said anything like that, I was just talking about possibilities ! I had no intention of being rude I promise !

#### washti

##### pablo
higs you put initially your OP musings in the frame of classical formal logic ( using truth tables, two values), and then started evading its constraints.

By changing the order of the premises arbitrarily, combining the meanings of different operators, making arbitrary comparisons within the elements of different equation lines, and saying that apart from T and F you have the option 'cannot be confirmed' - you stopped moving within the framework of classical formal logic.

There are others like intuitionistic or multivalue logic, fuzzy logic, etc. Maybe they'll better suit your needs. I will gladly educate myself from you if you are fluent in them. I'm not.

And I will quit now, cause we definitely don't share platform for conversation anymore, likely operating on different brain waves.

Your constant appeal to authority (of the unknown people without displaying their reasoning about the issue) isn't convincing at all.
You also didn't provide the original source for the puzzle.

I wish you luck. I'm sure your logical prowess will manifest itself one day even for such skeptics like me.

#### higs

##### Omg wow imo
I wish you luck. I hope your logical prowess will manifest itself one day even for such skeptics like me.

The dripping sarcasm in this kind of upsets me ngl, I don't feel like I really deserve it. And I don't know what all that not sharing brainwaves/platforms stuff is about either, we are unable to communicate because I took a break from the forum and you stopped using discord ?

I know my appeals to authority can seem annoying, I'm only saying that because this thread is already exhausting (I am trying to answer everyone in it) and explaining all the norms and rules would be exhausting. Simultaneously I am trying to figure out exactly why the answer given is this. Fine by me if you don't think it's correct, I highly encourage you to look into it yourself if you care !

And yes I forgot the article again, one second.

#### Attachments

• illusoryinferences.pdf
183.9 KB · Views: 101

#### higs

##### Omg wow imo
@Firehazard159

Well, like I said in the previous post, I am kind of exhausted, so I hope you get something out of this.

A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound.

This being a self contained purely theoretical puzzle, the soundness of the argument is circularly given by the fact that you accept the first two premises, (and decide to play the game, if you will.)

These two premises are :
1) p
2) (p->q) or else (~p->q)

These you have to accept as having the value "true" if you want to answer the question ((what is state of q ?) and play the game.

All of the statements contained within the main statement are valid, that is to say that they are all structured correctly and can be assigned a value of "true" or "false". A valid statement is still valid even if it produces the value of "false."

If you accept the overall statement, (premise 2) as true (thereby making the puzzle "sound", you know that one of the two statements composing it must be false.

--------------------

Let's try this another way and see if anyone finds this more comprehensible. (Boy you ppl have worked me hard.)

PREMISES (ALL ARE NECESSARILY TRUE)
1) p
2) (p->q) or else (~p->q)

Let us suppose all outcomes lead to "q" (there is an ace), like people intuitively seem to do.

In this case (p->q) is true !

and (~p->q) is ALSO true !

But If that is the case, then 2) (our untouchable premise which must always be true) is FALSE, because or else says ONE OF THEM MUST BE FALSE. And either one of the two statements being false has a result of ~q or "no ace".

Because of my limited knowledge, I still don't really understand what natural language reasoning you could use to arrive at the conclusion that A is the false one and B the true one like the truth table shows. So I am going to stop here for today

#### Firehazard159

##### ¯\_(ツ)_/¯
These two premises are :
1) p
2) (p->q) or else (~p->q) <---Main argument

These you have to accept as having the value "true" if you want to answer the question ((what is state of q ?) and play the game.

I was not contesting this. (Added the bolded part for reference later)

The part where we are communicating past each other seems to be:
The arguments *within* and their validity/soundness (which is what we are trying to determine).

So while I agree with the premise for the encompassing puzzle, what you're missing is that there are additional sub-arguments:
P->Q <---Sub-argument 1
OR ELSE <---Main argument (see above)
~P->Q <---Sub-argument 2

Which is where I now have to reiterate:
Because philosophy apparently sees:
"Valid but not sound" as "True"
Where as programmers see
"Valid and sound" as "True"

I am now arguing that:
Programmers see both Subargument1 (SA1) and subargument2 (SA2) as VALID, but one must not be SOUND to satisfy the OR ELSE. Whichever one is not sound, invalidates that *entire* subargument - not the main argument. Their truth table is effectively 1 part - though 2 truth tables would exist, as they are both valid, one truth table gets a big X through it when proven unsound, this is all the exclusive or does - exclusive or does not have a truth table component, like in your philosopher version. This leads to the unknown conclusion until we reveal whether the ace is in hand or not, and tells us which exclusive or was SOUND.

Philosopher see both SA1 and SA2 as VALID regardless of SOUNDNESS- it is assumed to be TRUE but not SOUND (While true in the programming sense requires SOUNDNESS) - so you have to fully complete a 3 part truth table - one for SA1, one for SA2, and one for the "OR ELSE" that sandwiches them together - like a sort of Schrodinger's puzzle - both are valid *and* sound until one is proven not to be, and one is proven *not* to be once you observe the full table.

So a programmer process it like:
Is there a queen? Yes
Is there an ace? Unknown, cannot continue. If Exists, SA1 is the truth - all conditions met, SA2 is false (because a queen exists). If Ace does not exist, SA1 is false, and SA2 is true because all conditions are met. Exclusive Or is validated, only One Truth is established, no conflict.

You're processing it like:
Is there a queen? Yes
Is there an ace? No, because the existence of both arguments being valid meaning "true" (again, programmer requires soundness, but you do not) necessitates a contradiction that cannot exist, so when we take a look at all 3 as valid and sound arguments, we find the one that is actually not sound.

It's almost a timing problem. In the programming case, it literally cannot continue until it knows the variable (Is q and ace, or not an ace) which will validate the true statement against the false statement.

In philosophy, we are imagining a scenario that leads to the inevitable decision that there mustn't be an ace because we forced both to be SOUND - which cannot be, thus, the ace is eliminated by the fact that there is a Queen and the results of SA2 that I struggle to put into words exactly because I keep getting lost in rationalizing it.

Edit: I had to change some verbiage because I'm struggling to communicate the concepts of Sound/Valid/True as seen between programmers and philosophers.

#### Tomten

##### Member
Here are two inconsistent premises:
Premise 1: p
Premise 2: not p
Does q follow? Does "not q" follow?

#### EndogenousRebel

##### We're all trying our best. Aren't we?
I don't see why 'or else' is being seen as two concepts being combined to make one new concept.

I've been familiar with two coding languages, both of them simply used else. I'm assuming 'or' is dropped is because it slows coding down? I've seen 'else if' which I'm sure generally works to narrow down possible options from most to least desirable, hence 'if' to denote that its checking parameters.

The thing I'm stuck is that "or" in logical notation (∨), means a very specific thing.
Code:
If A or B are divisible by 2 then T = T
Else T = F

A = 3
B = 2
T = T
It's mostly used to say either of these choices will satisfy X but not both. Replace or with and, and the opposite is true, a set of things must be true.
Code:
If A and B are divisible by 2 then T = T
Else T = F

A = 3
B = 2
T = F

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

[CONDITIONAL]+[DEPENDENT CLAUSE]+[CONDITIONAL CONSEQUENCE]+[INDEPENDENT CLAUSE] [comma] [COORDINATING CONJUNCTION]+[CONDITIONAL CONSEQUENCE]?+[DEPENDENT CLAUSE]+[CONDITIONAL CONSEQUENCE]+[INDEPENDENT CLAUSE]

This is how I see it. Funnily enough, the only parts of this sentence that can stand on their own are: there is an ace in the hand, there is an ace in the hand.

I'm pretty sure the logic behind if else if, is: if this then X, if anything else then Y.
The 'or' throws me off, but it shouldn't because, well doesn't matter what you pick the answer is still the same. It doesn't matter how or else if is cut up, it all can be reduced to otherwise.

Is this not the way I should be looking at it?

#### Haim

##### Worlds creator
Your mistake is that you assume a state can be when that state is not possible , the state which is there is no ace.
It similar to the function x/x=1
Then you go x=0
0/0=1
Nope 0 is invalid input for your function, you just can not have that input while having the function make any sense.You see a function have a range of valid inputs which "no ace" is outside said range.
Now that we have proof that 0 is invalid input we can say x!=0
Code:
function isAceInHand(isQueenInHand:boolean):boolean
{
var aceInHand:boolean;
if(queenInHand)
{
aceInHand=true;
}
else if(!queenInHand)
{
aceInHand=true;
}
return aceInHand;
}

var wasAceInHand:boolean;
wasAceInHand=isAceInHand(true);
print(wasAceInHand);// print true
wasAceInHand=isAceInHand(false);
print(wasAceInHand);// also print true

As you see in the code there is not state where function isAceInHand result is false, it would only be possible if it was other function.

#### higs

##### Omg wow imo
Asking a program to solve it in that way is useless, you're just telling a computer what to print in advance and cutting out the relevant possibilities. These are statements with a truth value so the factors are queen/no queen and ace/no ace. Your program is not adapted to solving the problem as it was intended to be interpreted. Programming languages are not constructed to do this. I mean, you could build a program that could do what I did on paper ofc, but that would demand understanding formal logic rules and honestly who cares except for dusty analytical philosophers.

One of the statements is false, which means that either statement one is in reality:

"If there is a queen then there is no ace in the hand." (p->!q)

or else statement two is

"If there is no queen in the hand then there is no ace in the hand." (!p->!q)

The fully explicit version of the problem is this :

One of the following statements is true and the other is false:

If there is a queen in the hand then there is an ace in the hand, (p->q)

or else if there isn't a queen in the hand then there is an ace in the hand. (!p->q)

There is a queen in the hand. What follows?

Kuu pointed out to me in another discussion that you could use Wolframalpha to give you the relevant truth table :

https://www.wolframalpha.com/input/?i=(p+=>+q)+xor+(not+p+=>+q)

SEEING AS YOU ALL TRUST COMPUTERS MORE THAN MY LOVELY HAND WRITTEN WORK

Can't be assed to explain it much more though, everything I have to say is in the thread already, though feel free to discuss it among yourselves for as long as you all like )

In retrospect this was a really rubbish problem to introduce formal logic with, because the answer is so counter-intuitive

#### Haim

##### Worlds creator
Well of course I will be wrong if you do not give the all problem(of course I am not actually wrong as I covered myself with " only be possible if it was other function ")

#### higs

##### Omg wow imo
Well of course I will be wrong if you do not give the all problem(of course I am not actually wrong as I covered myself with " only be possible if it was other function ")

When you listen to netanyahu do you believe everything he says or do you consider that some of the statements may be false and could be other functions.

I'm only half teasing, i am aware that its an annoying question (hell its in the title) thread

#### EndogenousRebel

##### We're all trying our best. Aren't we?
We weren't trying to figure out the validity of the statements relative to x conditions set doe. " One of the following statements is true and the other is false:" changes the entire framing of the question. We were trying to solve for x assuming that the conditional statements were fine. One should've figured the inclusion of 'or' was weird (I kinda did,) but the OP was a misrepresentation.

#### higs

##### Omg wow imo
We weren't trying to figure out the validity of the statements relative to x conditions set doe. " One of the following statements is true and the other is false:" changes the entire framing of the question. We were trying to solve for x assuming that the conditional statements were fine. One should've figured the inclusion of 'or' was weird (I kinda did,) but the OP was a misrepresentation.

Well I did say the interpretation of "or else" could be subject to controversy, I only came across the updated version later on. I will agree yet again that the puzzle is poorly worded. You can find the intended interpretation in it, but it's also misleading. So you know, fair enough !