## rational to believe x and y?

Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.

### rational to believe x and y?

I think a conjunction of the following is sometimes true:

(1) It is rational for S to believe X.

(2) It is rational for S to believe Y.

(3) It is not rational for S to believe X and Y.

Here is an example:

(1e) It is rational for me to believe that the tomato in front of me is red.

(2e) It is rational for me to believe that the tomato in front of me is orange.

(3e) It is not rational for me to believe that the tomato in front of me is red and that the tomato in front of me is orange.

Is my reasoning sound? Are there other examples?
evaristegalois

### Re: rational to believe x and y?

This is the lottery paradox. There are a hundred million lottery tickets sold, number 1 to 100T.

It's rational to believe that #1 won't win. It's rational to believe #2 won't win. It's rational to believe #3 won't win. Dot dot dot ...

Yet it's NOT rational to believe the conjunction #1 won't win AND #2 won't win AND #3 won't win ..., because SOME ticket must win. [Assuming this is not one of those state lotteries that sometimes has no winner.]
someguy1
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### Re: rational to believe x and y?

Not quite. I think the case for "It is rational to believe that ticket #1 won't win" is much harder to make than what I am looking for. You may have partial beliefs about lottery tickets.
evaristegalois

### Re: rational to believe x and y?

Hi evaristegalois,

Welcome to the Forums.

It would depend on the relationship between X and Y obviously.

1A. I believe the Sky(X) is Blue.
1B. I believe the Grass(Y) is Green.

Since Sky(X) and Grass(Y) are unrelated, both can be rational at the same time.

If X=Y then it would be irrational to believe they are different.

Regards,
Dave :^)

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### Re: rational to believe probability

lottery tickets

rounding errors

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### Re: rational to believe x and y?

Dave,

X and Y stand in for any propositions, which may or may not be related. Naively, one may think that R(X) and R(Y) imply R(X and Y), where R(Z) means "it is rational for S to believe Z." I want to find counterexamples.
evaristegalois

### Re: rational to believe x and y?

I take your meaning to be that

(1p) It is rational for me to believe that pi is 3.14

(2p) It is rational for me to believe that pi is 3.142

(3p) It is not rational for me to believe that pi is 3.14 and that pi is 3.142.
evaristegalois

### Re: rational to believe x and y?

Ok,

R(X): All Humans are Animals.
And
R(Y): All Dolphins are Animals.
Thus
R(Z): All Humans are Dolphins.

That help?

Regards,
Dave :^)

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### Re: rational to believe x and y?

evaristegalois » February 13th, 2015, 2:03 pm wrote:Not quite. I think the case for "It is rational to believe that ticket #1 won't win" is much harder to make than what I am looking for. You may have partial beliefs about lottery tickets.

The state lottery commission appreciates people who think like that. FWIW I didn't make this up, it's a well-known paradox. http://en.wikipedia.org/wiki/Lottery_paradox

I agree with you that it's common to think, "It's incredibly unlikely for me to win but SOMEBODY has to win so here's my dollar," but evidently some philosophers don't regard that as rational.
someguy1
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### Re: rational to believe x and y?

evaristegalois wrote:(3p) It is not rational for me to believe that pi is 3.14 and that pi is 3.142.

Not easy to say without precise definitions of "rational" and the mathematical "is".
Does "is" mean "equals exactly", including the level of precision (i.e. 3.14 implies 3.140 in the context given)?
If "is" may encompass any value that provides useful and reliable results, then it would be reasonable to accept both definitions, each appropriately applied.

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### Re: rational to believe x and y?

Dave_Oblad » February 13th, 2015, 3:03 pm wrote:Ok,

R(X): All Humans are Animals.
And
R(Y): All Dolphins are Animals.
Thus
R(Z): All Humans are Dolphins.

That help?

Regards,
Dave :^)

Dave, you misunderstand. X is a any proposition, and R(X) says "it is rational for S to believe X". I need an X and a Y such that R(X), R(Y) and not-R(X and Y) are true.
evaristegalois

### Re: rational to believe x and y?

Ok,

R(X): I am looking at the Moon. (True if I believe I am looking at the Moon)
R(Y): I am NOT looking at the Moon. (True if I believe I am NOT looking at the Moon)
R(Z): I am both looking at the Moon and Not looking at the Moon.

I think I am having trouble with the word Believe. Is it the case that Proposition (X or Y) must be true at all times? Or do I merely need to believe them to be true?

Regards,
Dave :^)

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### Re: rational to believe x and y?

Here is another one,

I believe I'm looking at a Flying Saucer.
I don't believe in Flying Saucers.
Therefore:
I believe I am looking at a Flying Saucer and I don't believe in Flying Saucers.

or..

I believe I am looking at a Lake in this Desert.
I don't believe there is a Lake in this Desert.
Therefore:
I believe I am looking at a Lake and that I Believe it is not a Lake.

or..

Look only at black X in center of circle.

I Believe I see a missing Red Spot moving around a series of Red Spots.
I Believe I see a Green Spot replacing Red Spots moving around a circle.
I Believe I see only a Green Spot moving around a circle.

In your OP you suggest the Tomato is Red or Orange, but it obviously can't be both.

Anyway, I'm not sure what you are seeking.. Can we omit the word believe? Or, is that critical to your question? Are you trying to find if a person can believe opposing views at the same time?

Take me for example:
X: I honestly Believe Time does not Exist.
Y: I honestly Believe it is Time for me to go home now.
(Both are True)

Back tomorrow...lol.

Best regards,
Dave :^)

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### Re: rational to believe x and y?

evaristegalois » Fri Feb 13, 2015 2:24 pm wrote:I think a conjunction of the following is sometimes true:

(1) It is rational for S to believe X.

(2) It is rational for S to believe Y.

(3) It is not rational for S to believe X and Y.

Here is an example:

(1e) It is rational for me to believe that the tomato in front of me is red.

(2e) It is rational for me to believe that the tomato in front of me is orange.

(3e) It is not rational for me to believe that the tomato in front of me is red and that the tomato in front of me is orange.

Is my reasoning sound? Are there other examples?

1e and 2e are either contradictory, or being observed at two different times or places. Therefore 3e has no relevance to the first two premises. "THE' tomato cannot be observed by a single person as red and orange at the same time and place. Your premises are unsound.

doogles
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### Re: rational to believe x and y?

You all obviously have never grown tomatoes if you think a tomato can't be red and orange at the same time. They can be red, orange, green and yellow and purple all at the same time.

mtbturtle
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### Re: rational to believe x and y?

Heh, heh.

I assume that the OP's premise is that tomatoes are unitary in their color, red tomatoes are entirely red, etc. For purposes of thinking about logic, you can certainly take that approach. In which case, it would not be true that a tomato could be both colors at the same time. But, here's my question: so what? This all seems fairly obvious as a syllogism about objects that have unitary properties.

It is rational for me to believe my dog has fleas, IF my dog has fleas.

It is rational for me to believe my dog has no fleas, IF my dog has no fleas.

It is not rational for me to believe my dog has fleas and also has no fleas, at any particular moment, due to the mutually exclusive nature of the two ascribed conditions at a given moment of time.

The temporal aspect of many attributes is key to the logic.

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### Re: rational to believe x and y?

evaristegalois » Fri Feb 13, 2015 4:34 pm wrote:Dave,

X and Y stand in for any propositions, which may or may not be related. Naively, one may think that R(X) and R(Y) imply R(X and Y), where R(Z) means "it is rational for S to believe Z." I want to find counterexamples.

Ususally, terms like X and Y would not stand for propositions. But as it is there is no problem with holding 2 propositions as true as long as they do not contradict each other which as you've phrased most of them such as the tomato being 2 colors are compatible, don't contradict each other so where is the problem?

mtbturtle
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### Re: rational to believe x and y?

Yes, Dave, I should have specified that I meant the beliefs to be concurrent, so more accurately R(X) means "S is rational to believe X at time t", which means your "looking at the moon" example doesn't go through. The tomato example, however, does work because colour concepts can be understood to be subjective. I guess the following example would work as well:

X: "one should never use a knife to eat spaghetti"

Y: "one may use a knife to eat spaghetti"

It is rational to believe X, rational to believe Y, but the same agent can't rationally hold both beliefs at the same time.
evaristegalois

### Re: rational to believe x and y?

If there is a contradiction it is not rational to believe both at the same time.

mtbturtle
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### Re: rational to believe x and y?

I believe mtb completed the story.

The premises of the whole procedure are incomplete.

- It is rational for S to believe X
- It is rational for S to believe Y

the missing third premise:
option (a): X and Y are compatible
option (b): X and Y are contradictory

a fouth premise (implicit in any logical procedure):
it is rational to simultaneously believe to two compatible items and
it is not rational to simultaneously believe to two contradictory items

conclusion (a): it is rational for S to believe X and Y

conclusion (b): it is not rational for S to believe X and Y

neuro
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### Re: rational to believe x and y?

I rather like the visual trick and intend on using on future forums relating to different topics. Are you able to give me an explanation why the red/pink dots disappear once we fix our attention on the black crosshair. What is the brain chemistry behind it or the retina function process or what not? I would like to know as much as possible.
Percarus

### Re: rational to believe x and y?

Hello Percarus,

Neuro would be the better person to ask.. I would mostly be offering a guess based on what I know of the brain. It is not an optical Illusion effect created by the eyes. Actually, nothing is as far as I know except possibly color blindness. The Brain processes information from the eyes. It fills in the blanks created by the optic nerve on the eyes. It fills in false color information from the non-color sensitive sensors in the eyes. It creates a Mental Model of what it see's through a series of Filters.

One of these filters kills redundant non-changing information.. such as the pinks dots in the periphery portion of the optical input. If everything the eyes see were thrown at the Brains Computer.. it would probably overload or require more resources to process and subtract from more critical thinking. Now if something changes, the filter allows the change through, as it may be important. When the dot turns off, it's a change and is processed. But because it's outside the color sense part of the eye, it is assigned a false color. Best choice (I assume) is the opposite color of what it was. So Red becomes Green and the dots that are not changing are substituted with the background color, so the Red dots disappear.

I like this one in particular because it demonstrates how much of what you see is actually synthesized by the Brain as opposed to what is the real input from the eyes.

Anyway, this is my best guess. Drop a line to Neuro, betting his answer is better.

Regards,
Dave :^)

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### Re: rational to believe x and y?

Dave_Oblad » February 13th, 2015, 8:44 pm wrote:Ok,

R(X): I am looking at the Moon. (True if I believe I am looking at the Moon)
R(Y): I am NOT looking at the Moon. (True if I believe I am NOT looking at the Moon)
R(Z): I am both looking at the Moon and Not looking at the Moon.

I think I am having trouble with the word Believe. Is it the case that Proposition (X or Y) must be true at all times? Or do I merely need to believe them to be true?

Regards,
Dave :^)

What do you mean by "I believe"? If you mean "I think" I am looking at the moon", doesn't the problem lie in not making a positive statement? Either you are or you aren't. Or, do you mean it as you would "believe" a potato is edible - something you absolutely know to be fact.?

It's been years since I've done this but isn't there something wrong with how these are being worded? Something vague? Or something twisted? Perhaps, as you said earlier, two unrelated statements being tied together?. Two contradictory statements? Isn't there something required of each statement to make the syllogism rational? Your syllogism about humans, dolphins and animals has me trying to remember a requirement to make a logical statement.

Several weeks ago, after reading a paradox, I wanted to ask if a paradox involves more than simple logic? If it does, I'll retreat to a quiet corner and listen because I know you know more about this than I do. (Not "I believe you do" but "I know you do".)
vivian maxine
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