Deeper than primes

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HatRack said:
I asked for you to point out the logical flaw in the statement.
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You force Something (the statement "Every member X of {}") as if it is Nothing ("").
 
doronshadmi said:
HatRack please explain what is exactly Every member X of {}?
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Don't even worry about the informal version. Just focus on the formal version:

[latex]$$$ \forall x, \, x \in \emptyset \Rightarrow S(x) $$$[/latex]​

Where is the logical flaw in this?
 
HatRack said:
Don't even worry about the informal version. Just focus on the formal version:

[latex]$$$ \forall x, \, x \in \emptyset \Rightarrow S(x) $$$[/latex]​

Where is the logical flaw in this?
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What is x in the case of {} ?
 
[latex]$$$ \forall x, \, x \in \emptyset \Rightarrow S(x) $$$[/latex]​

Translation:

"For all Nothing if Nothing is a member of {} then Nothing is a set".
 
doronshadmi said:
"For all Nothing if Nothing is a member of {} then Nothing is a set".
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Well Doron, you have exactly one of two choices here:

1) If Nothing is an existing object, then Nothing is not a member of {}, because {} has no members. Hence, the proposition is vacuously true.

2) If Nothing is not an existing object, then you cannot substitute it into that formula.
 
doronshadmi said:
Because your formula can't deal with concept like Nothing.
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This all depends on what you mean by Nothing. I suspect you are committing the same fallacy as seen in the conclusion of the following statement.

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness
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doronshadmi said:
EDIT: It depends exactly on Nothing (in the total sense), which you can't comprehend by your relative existent-only reasoning.
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Perhaps you can explain to us the flaw in the following line of reasoning:

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness.
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Since you get Nothing as an existent thing then what you wrote is:

"There is no other thing that is better than eternal happiness; a ham sandwich is better than anything else (here you do not follow after your first statement); therefore, a ham sandwich is better than eternal happiness."
 
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doronshadmi said:
You get Nothing as an existent object.
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No, I'm afraid that's not the flaw. Let's look at the statement carefully.

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness.
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This section of the statement is readily interpreted. Taking > to mean "better than" and H to mean "ham sandwich", this reads formally as:

[latex]$H > \emptyset$[/latex]​

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness.
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Now, here is where the fallacy occurs. Upon first glance, and letting E mean "eternal happiness", one may interpret the above as:

[latex]\emptyset > E[/latex]​

Of course, under the reasonable assumption that the "better than" relation is transitive, this leads to:

[latex]$H > E$[/latex]​

Which corresponds with:

Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness.
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The flaw in this reasoning is that the author of this quotation incorrectly assumed that "nothing" meant the same thing in both phrases. In fact, nothing actually has a different meaning in "Nothing is better than eternal happiness" than it does in "a ham sandwich is better than nothing".

The latter we interpreted correctly, but the former we did not. In the former, nothing is correctly interpreted to mean the contents of the empty set, not the empty set itself. Hence, the correct formal rendering of this statement is:

[latex]\forall x, x > E \rightarrow x \in \emptyset[/latex]​

Since the empty set has no members, this formula correctly expresses the fact that that there is no existing thing better than eternal happiness. And, we can no longer come to the conclusion:

[latex]$H > E$[/latex]​

This is precisely the reason why we use formal language as opposed to informal language in mathematics. There is no room for misinterpretation in formal language as there is in natural language.

You are committing precisely the same fallacy in your posts. You are equivocating the different ways of interpreting the word "nothing", and using this to draw incorrect conclusions. This type of fallacy is well known and well dealt with in the study of formal logic, as I have shown above. And this is exactly the reason you can't/won't put your reasoning into a formal system.

Sorry Doron, but the only mathematical discovery you've made is the discovery of logical fallacy. Which, unfortunately, is already well known, and has been for some time.

ETA:

[latex]\forall x, x > E \rightarrow x \in \emptyset[/latex]​

An equivalent form of this statement is:

[latex]\lnot \exists x, x > E \land x \notin \emptyset[/latex]​
 
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HatRack said:
You are equivocating the different ways of interpreting the word "nothing",
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Again you show you inability the understand that Nothing is not an existent thing (where one of the existent things is the word "nothing")

There is no failure in my reasoning, which (as you claim) is equivalent to different ways of interpreting the word "nothing", because by my reasoning Nothing is a total concept, which can't be changed by different interpretations of a given word.

Again you demonstrate your inability to grasp the totality of Nothing.
 
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doronshadmi said:
Again you show you inability the understand that Nothing is not an existent thing (where one of the existent things is the word "nothing")
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1) If your idea of "nothing" is not existent in the logical sense, then you can't talk about it. That's like talking about invisible pink unicorns. You can draw whatever conclusions you want, but they're all baseless.

doronshadmi said:
There is no failure in my reasoning, which is equivalent to different ways of interpreting the word "nothing", because by my reasoning Nothing is a total concept, which can't be changed by different interpretations.
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2) See 1 above.

doronshadmi said:
Again you demonstrate your inability to grasp the totality of Nothing.
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The only one who has demonstrated their inability to grasp "nothing" is you.
 
HatRack said:
1) If your idea of "nothing" is not existent in the logical sense, then you can't talk about it.
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It is logical, and it is not equivalent to some word that is used to talk about it.

Again you demonstrate your weak reasoning that can't distinguish between "Silence" and Silence, because your reasoning works only if you make sounds like "Silence".
 
HatRack said:
For something so logical, it's strangely resistant to being expressible in formal language. :rolleyes:
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You simply can't get the notion that some expression of X is not necessarily X (and by analogy the expression "Silence" is not Silence itself, because Silence itself is not expressible).
 
doronshadmi said:
You simply can't get the notion that some expression of X is not necessarily X (and by analogy the expression "Silence" is not Silence itself, because Silence itself is not expressible).
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Doron, we use words, phrases, and notation to describe things. I, along with every other human being in the world, am well aware the word silence is not the same physical thing as the phenomena it describes.
 
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