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The paradox of material equivalence

Smithereens
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12/12/2015 1:20:23 PM
Posted: 11 months ago
Validity is defined as when an argument entails a conclusion that cannot be false if the premises are true. The following argument is valid:
P1 Smither's is alive if she posts on DDO.
P2 Smither's is posting on DDO.
C: Smither's is alive.

This argument is intuitive sound. However, what if I told you that the following argument is also valid?
P1 DDO exists if DDO can be posted on.
P2 If DDO doesn't exist, DDO can be posted on.
C: Smither's is a troll.

Is this argument valid?
Yes it is.

WTF I hear you ask?

This is the paradox of material equivalence. Or more exactly, the paradox of entailment. The argument is valid because it satisfies the conditions that the premises cannot be true while the conclusion is false. The premises cannot be true because they are mutually negating statements. So I can make any conclusion I like, and because of the impossibility of the premises, the entire argument will always be valid.

This is counter-intuitive, but is actually the case. The confusion arises because of the differences of implication in natural language and formal language. In formal speech,
(~A /\ A) --> B "A and it's negation entail B."
Therefore, we can get statements such as "If I am alive or if I am dead then I will attend the lecture."

To understand why such silly statements are necessarily true, we investigate using a truth table. I will attempt to construct on here in this forum post based on the statement: "If I am alive then I will play mafia."
A__________M__________A-->M
1 True ____True_______ True
2 True ____False ______False
3 False ____True ______True
4 False ____False ______True
1. I am alive, I played mafia. My statement is true.
2. I am alive, I did not play mafia. My statement is false.
3. I am dead, I played mafia though. My statement is true.
4. I am dead, I did not play mafia. My statement is true.

1 is true because I satisfied my conditions. Clearly.
2 is false because I contradicted my conditions. I said if I were alive, I would go to the lecture. But I was alive and not in attendance.
3 Is true because I wasn't alive, so it wasn't really relevant my attendance.
4 is true because 3 is true.

We can conclude that these statements are always true unless the antecedent (first part) is true while the consequent (latter part) is false.

The reason why they sound so contradictory? Implication is essentially "If... then" statements such as "if Socrates is a moral, then socrates can die." Natural language expresses "If... then" statements differently to formal language.
"If Bob is black, then Bob has dark skin." And "If Bob is white, then Bob has light skin." is represented in formal language as:
[B(b)-->B(s)] /\ [W(b)-->W(s)]
However, the following argument can be entailed from this statement:
[B(b)-->B(s)] /\ [W(b)-->W(s)] --> [B(b)-->W(s)] \/ [W(b)-->B(s)]
"If Bob is black, he has black skin, and if he is white he has white skin. Therefore if Bob is black he has white skin or if he is White he has black skin."
Since Bob can't have both black and white skin, the dichotomy is valid.
This further illustrates the point.

In natural language, we use "If... then" for issues regarding everyday epistemology crises. We don't know a thing about Bob. But we know that if he is black, he must have black skin. Formal language doesn't follow these observances, so next time you get challenged to create a valid argument that the earth is flat, use material equivalence.

Also if you have spare time, do read this short article:
http://projecteuclid.org...

Thoughts? Objections? Discuss.
Music composition contest: http://www.debate.org...
skipsaweirdo
Posts: 1,864
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12/13/2015 11:19:24 AM
Posted: 11 months ago
At 12/12/2015 1:20:23 PM, Smithereens wrote:
Validity is defined as when an argument entails a conclusion that cannot be false if the premises are true. The following argument is valid:
P1 Smither's is alive if she posts on DDO.
P2 Smither's is posting on DDO.
C: Smither's is alive.

This argument is intuitive sound. However, what if I told you that the following argument is also valid?
P1 DDO exists if DDO can be posted on.
P2 If DDO doesn't exist, DDO can be posted on.
C: Smither's is a troll.

Is this argument valid?
Yes it is.

WTF I hear you ask?

This is the paradox of material equivalence. Or more exactly, the paradox of entailment. The argument is valid because it satisfies the conditions that the premises cannot be true while the conclusion is false. The premises cannot be true because they are mutually negating statements. So I can make any conclusion I like, and because of the impossibility of the premises, the entire argument will always be valid.

This is counter-intuitive, but is actually the case. The confusion arises because of the differences of implication in natural language and formal language. In formal speech,
(~A /\ A) --> B "A and it's negation entail B."
Therefore, we can get statements such as "If I am alive or if I am dead then I will attend the lecture."

To understand why such silly statements are necessarily true, we investigate using a truth table. I will attempt to construct on here in this forum post based on the statement: "If I am alive then I will play mafia."
A__________M__________A-->M
1 True ____True_______ True
2 True ____False ______False
3 False ____True ______True
4 False ____False ______True
1. I am alive, I played mafia. My statement is true.
2. I am alive, I did not play mafia. My statement is false.
3. I am dead, I played mafia though. My statement is true.
4. I am dead, I did not play mafia. My statement is true.

1 is true because I satisfied my conditions. Clearly.
2 is false because I contradicted my conditions. I said if I were alive, I would go to the lecture. But I was alive and not in attendance.
3 Is true because I wasn't alive, so it wasn't really relevant my attendance.
4 is true because 3 is true.

We can conclude that these statements are always true unless the antecedent (first part) is true while the consequent (latter part) is false.

The reason why they sound so contradictory? Implication is essentially "If... then" statements such as "if Socrates is a moral, then socrates can die." Natural language expresses "If... then" statements differently to formal language.
"If Bob is black, then Bob has dark skin." And "If Bob is white, then Bob has light skin." is represented in formal language as:
[B(b)-->B(s)] /\ [W(b)-->W(s)]
However, the following argument can be entailed from this statement:
[B(b)-->B(s)] /\ [W(b)-->W(s)] --> [B(b)-->W(s)] \/ [W(b)-->B(s)]
"If Bob is black, he has black skin, and if he is white he has white skin. Therefore if Bob is black he has white skin or if he is White he has black skin."
Since Bob can't have both black and white skin, the dichotomy is valid.
Bob could actually both have black and white skin if he has vitiligo.
This further illustrates the point.

In natural language, we use "If... then" for issues regarding everyday epistemology crises. We don't know a thing about Bob. But we know that if he is black, he must have black skin. Formal language doesn't follow these observances, so next time you get challenged to create a valid argument that the earth is flat, use material equivalence.

If Bob is black then he must have black skin is an equivocation fallacy and its conclusion is actually false if by "black" in the first reference you mean race and If in the second reference you are referring to skin color. Vitiligo in African Americans makes their skin white. But if by black you are referring to skin color both times it obviously is a tautology. You could also consider "if Bob is black" to be a fallacy of ambiguity because of the different meanings of the word black when describing a person.
Also if you have spare time, do read this short article:
http://projecteuclid.org...

Thoughts? Objections? Discuss.
Diqiucun_Cunmin
Posts: 2,710
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12/13/2015 6:10:24 PM
Posted: 11 months ago
At 12/12/2015 1:20:23 PM, Smithereens wrote:
Validity is defined as when an argument entails a conclusion that cannot be false if the premises are true. The following argument is valid:
P1 Smither's is alive if she posts on DDO.
P2 Smither's is posting on DDO.
C: Smither's is alive.

This argument is intuitive sound. However, what if I told you that the following argument is also valid?
P1 DDO exists if DDO can be posted on.
P2 If DDO doesn't exist, DDO can be posted on.
C: Smither's is a troll.

Is this argument valid?
Yes it is.

I have an objection :P I don't think it's valid.

http://postimg.org...
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

Don't be a stat cynic:
http://www.debate.org...

Response to conservative views on deforestation:
http://www.debate.org...

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Smithereens
Posts: 5,512
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12/14/2015 2:19:03 AM
Posted: 11 months ago
At 12/13/2015 6:10:24 PM, Diqiucun_Cunmin wrote:
At 12/12/2015 1:20:23 PM, Smithereens wrote:
Validity is defined as when an argument entails a conclusion that cannot be false if the premises are true. The following argument is valid:
P1 Smither's is alive if she posts on DDO.
P2 Smither's is posting on DDO.
C: Smither's is alive.

This argument is intuitive sound. However, what if I told you that the following argument is also valid?
P1 DDO exists if DDO can be posted on.
P2 If DDO doesn't exist, DDO can be posted on.
C: Smither's is a troll.

Is this argument valid?
Yes it is.

I have an objection :P I don't think it's valid.

http://postimg.org...

tisk. That's not the syllogism. It should be E, ~E, T. Hence you can't really draw a table of that info because whenever both are true or both are false you get a contradiction.
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Diqiucun_Cunmin
Posts: 2,710
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12/14/2015 3:22:05 AM
Posted: 11 months ago
At 12/14/2015 2:19:03 AM, Smithereens wrote:
At 12/13/2015 6:10:24 PM, Diqiucun_Cunmin wrote:
At 12/12/2015 1:20:23 PM, Smithereens wrote:
Validity is defined as when an argument entails a conclusion that cannot be false if the premises are true. The following argument is valid:
P1 Smither's is alive if she posts on DDO.
P2 Smither's is posting on DDO.
C: Smither's is alive.

This argument is intuitive sound. However, what if I told you that the following argument is also valid?
P1 DDO exists if DDO can be posted on.
P2 If DDO doesn't exist, DDO can be posted on.
C: Smither's is a troll.

Is this argument valid?
Yes it is.

I have an objection :P I don't think it's valid.

http://postimg.org...

tisk. That's not the syllogism. It should be E, ~E, T. Hence you can't really draw a table of that info because whenever both are true or both are false you get a contradiction.

Argh sorry, it was 2am in the morning when I saw you post and I misread. However, the formalisation should be (P -> E), (~E -> P) |= T based on your post. That is still not valid. The two premises are consistent, and both premises are true and the conclusion is false when P is true, E is true and T is false or when P is false, E is true and T is false).
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

Don't be a stat cynic:
http://www.debate.org...

Response to conservative views on deforestation:
http://www.debate.org...

Topics I'd like to debate (not debating ATM): http://tinyurl.com...
Smithereens
Posts: 5,512
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12/14/2015 4:49:21 AM
Posted: 11 months ago
At 12/14/2015 3:22:05 AM, Diqiucun_Cunmin wrote:
At 12/14/2015 2:19:03 AM, Smithereens wrote:
At 12/13/2015 6:10:24 PM, Diqiucun_Cunmin wrote:
At 12/12/2015 1:20:23 PM, Smithereens wrote:
Validity is defined as when an argument entails a conclusion that cannot be false if the premises are true. The following argument is valid:
P1 Smither's is alive if she posts on DDO.
P2 Smither's is posting on DDO.
C: Smither's is alive.

This argument is intuitive sound. However, what if I told you that the following argument is also valid?
P1 DDO exists if DDO can be posted on.
P2 If DDO doesn't exist, DDO can be posted on.
C: Smither's is a troll.

Is this argument valid?
Yes it is.

I have an objection :P I don't think it's valid.

http://postimg.org...

tisk. That's not the syllogism. It should be E, ~E, T. Hence you can't really draw a table of that info because whenever both are true or both are false you get a contradiction.

Argh sorry, it was 2am in the morning when I saw you post and I misread. However, the formalisation should be (P -> E), (~E -> P) |= T based on your post. That is still not valid. The two premises are consistent, and both premises are true and the conclusion is false when P is true, E is true and T is false or when P is false, E is true and T is false.

Except both premises are not true. Premise 1 is P and premise 2 is exactly ~P. Normally you'd accept that P->E |- T and false antecedent with true consequent also always infers T. But both premises cannot be true. Hence the premises cannot be true with a false conclusion, as the premises negate.
Music composition contest: http://www.debate.org...
Diqiucun_Cunmin
Posts: 2,710
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12/14/2015 5:40:23 AM
Posted: 11 months ago
At 12/14/2015 4:49:21 AM, Smithereens wrote:
At 12/14/2015 3:22:05 AM, Diqiucun_Cunmin wrote:
At 12/14/2015 2:19:03 AM, Smithereens wrote:
At 12/13/2015 6:10:24 PM, Diqiucun_Cunmin wrote:
At 12/12/2015 1:20:23 PM, Smithereens wrote:
Validity is defined as when an argument entails a conclusion that cannot be false if the premises are true. The following argument is valid:
P1 Smither's is alive if she posts on DDO.
P2 Smither's is posting on DDO.
C: Smither's is alive.

This argument is intuitive sound. However, what if I told you that the following argument is also valid?
P1 DDO exists if DDO can be posted on.
P2 If DDO doesn't exist, DDO can be posted on.
C: Smither's is a troll.

Is this argument valid?
Yes it is.

I have an objection :P I don't think it's valid.

http://postimg.org...

tisk. That's not the syllogism. It should be E, ~E, T. Hence you can't really draw a table of that info because whenever both are true or both are false you get a contradiction.

Argh sorry, it was 2am in the morning when I saw you post and I misread. However, the formalisation should be (P -> E), (~E -> P) |= T based on your post. That is still not valid. The two premises are consistent, and both premises are true and the conclusion is false when P is true, E is true and T is false or when P is false, E is true and T is false.

Except both premises are not true. Premise 1 is P and premise 2 is exactly ~P. Normally you'd accept that P->E |- T and false antecedent with true consequent also always infers T. But both premises cannot be true. Hence the premises cannot be true with a false conclusion, as the premises negate.

Not sure what you meant by the bolded part... One certainly cannot derive T from (P -> E)... and your second premise was not the negation of the first. The negation of 'DDO exists if DDO can be posted on.' would be 'It is not the case that if DDO can be posted on, DDO exists' or 'DDO can be posted on and does not exist', not 'if DDO doesn't exist, DDO can be posted on'.
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

Don't be a stat cynic:
http://www.debate.org...

Response to conservative views on deforestation:
http://www.debate.org...

Topics I'd like to debate (not debating ATM): http://tinyurl.com...
Smithereens
Posts: 5,512
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12/14/2015 11:20:12 AM
Posted: 11 months ago
At 12/14/2015 5:40:23 AM, Diqiucun_Cunmin wrote:
Argh sorry, it was 2am in the morning when I saw you post and I misread. However, the formalisation should be (P -> E), (~E -> P) |= T based on your post. That is still not valid. The two premises are consistent, and both premises are true and the conclusion is false when P is true, E is true and T is false or when P is false, E is true and T is false.

Except both premises are not true. Premise 1 is P and premise 2 is exactly ~P. Normally you'd accept that P->E |- T and false antecedent with true consequent also always infers T. But both premises cannot be true. Hence the premises cannot be true with a false conclusion, as the premises negate.

Not sure what you meant by the bolded part... One certainly cannot derive T from (P -> E)... and your second premise was not the negation of the first. The negation of 'DDO exists if DDO can be posted on.' would be 'It is not the case that if DDO can be posted on, DDO exists' or 'DDO can be posted on and does not exist', not 'if DDO doesn't exist, DDO can be posted on'.

T is entailed, not provable excuse that. I'll shorten down the symbols and imply what I mean because I'm typing from a phone atm. I mean for them to be mutually negating. If DDO exists, DDO can be posted on is exactly negative by 'Not' If DDO exists, DDO can be posted on. The point is that a TTF situation does not occur, because both premises are not true simultaneously. The argument is valid however because it isn't TTF, which is the literal definition of validity in standard logic. The formal representation of it should be (P /\ ~P) -> T. P and not P entail T. I'll just say it next time instead of symbolising.

Philosophers (CS Lewis first, and then a bunch of others) create relevance logic systems to solve the dilemma, but it seems you think there is something intrinsically wrong with the formal construction of it?
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Diqiucun_Cunmin
Posts: 2,710
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12/14/2015 1:52:44 PM
Posted: 11 months ago
At 12/14/2015 11:20:12 AM, Smithereens wrote:
At 12/14/2015 5:40:23 AM, Diqiucun_Cunmin wrote:
Argh sorry, it was 2am in the morning when I saw you post and I misread. However, the formalisation should be (P -> E), (~E -> P) |= T based on your post. That is still not valid. The two premises are consistent, and both premises are true and the conclusion is false when P is true, E is true and T is false or when P is false, E is true and T is false.

Except both premises are not true. Premise 1 is P and premise 2 is exactly ~P. Normally you'd accept that P->E |- T and false antecedent with true consequent also always infers T. But both premises cannot be true. Hence the premises cannot be true with a false conclusion, as the premises negate.

Not sure what you meant by the bolded part... One certainly cannot derive T from (P -> E)... and your second premise was not the negation of the first. The negation of 'DDO exists if DDO can be posted on.' would be 'It is not the case that if DDO can be posted on, DDO exists' or 'DDO can be posted on and does not exist', not 'if DDO doesn't exist, DDO can be posted on'.

T is entailed, not provable excuse that. I'll shorten down the symbols and imply what I mean because I'm typing from a phone atm. I mean for them to be mutually negating. If DDO exists, DDO can be posted on is exactly negative by 'Not' If DDO exists, DDO can be posted on. The point is that a TTF situation does not occur, because both premises are not true simultaneously. The argument is valid however because it isn't TTF, which is the literal definition of validity in standard logic. The formal representation of it should be (P /\ ~P) -> T. P and not P entail T. I'll just say it next time instead of symbolising.

Philosophers (CS Lewis first, and then a bunch of others) create relevance logic systems to solve the dilemma, but it seems you think there is something intrinsically wrong with the formal construction of it?

I have no objections to (P & ~P) |= T or that ((P & ~P) -> T) is a tautology, lol... My point was that 'DDO exists if DDO can be posted on.' and 'If DDO doesn't exist, DDO can be posted on.' cannot be formalised as (P & ~P)... Consider that a nitpick...
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

Don't be a stat cynic:
http://www.debate.org...

Response to conservative views on deforestation:
http://www.debate.org...

Topics I'd like to debate (not debating ATM): http://tinyurl.com...
Smithereens
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12/15/2015 6:54:29 AM
Posted: 11 months ago
At 12/14/2015 1:52:44 PM, Diqiucun_Cunmin wrote:
I have no objections to (P & ~P) |= T or that ((P & ~P) -> T) is a tautology, lol... My point was that 'DDO exists if DDO can be posted on.' and 'If DDO doesn't exist, DDO can be posted on.' cannot be formalised as (P & ~P)... Consider that a nitpick...

Ok. Sure. That should be worded differently. Or else a different example entirely..
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skipsaweirdo
Posts: 1,864
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12/19/2015 2:21:03 PM
Posted: 11 months ago
At 12/12/2015 1:20:23 PM, Smithereens wrote:
Validity is defined as when an argument entails a conclusion that cannot be false if the premises are true. The following argument is valid:
P1 Smither's is alive if she posts on DDO.
P2 Smither's is posting on DDO.
C: Smither's is alive.

This argument is intuitive sound. However, what if I told you that the following argument is also valid?
P1 DDO exists if DDO can be posted on.
P2 If DDO doesn't exist, DDO can be posted on.
C: Smither's is a troll.

Is this argument valid?
Yes it is.
Umm, why do you think valid is of any importance in a syllogism! It has nothing to do with the traditional everyday use of the word valid and it is meaningless. If an argument isn't sound then that's actually what's relevant. Valid in form is ridiculous and has no "validity" when it comes to argumentation.
WTF I hear you ask?

This is the paradox of material equivalence. Or more exactly, the paradox of entailment. The argument is valid because it satisfies the conditions that the premises cannot be true while the conclusion is false. The premises cannot be true because they are mutually negating statements. So I can make any conclusion I like, and because of the impossibility of the premises, the entire argument will always be valid.

This is counter-intuitive, but is actually the case. The confusion arises because of the differences of implication in natural language and formal language. In formal speech,
(~A /\ A) --> B "A and it's negation entail B."
Therefore, we can get statements such as "If I am alive or if I am dead then I will attend the lecture."
It's not counterintuitive when valid in logic has no application to arguing soundly. Not even sure why you think saying valid somehow makes an impression of "worth". You simply are using equivocation unintentionally because in everyday language valid is synonymous with being sound. In logic it has no meaning in regards to making an argument that can persuade.
To understand why such silly statements are necessarily true, we investigate using a truth table. I will attempt to construct on here in this forum post based on the statement: "If I am alive then I will play mafia."
A__________M__________A-->M
1 True ____True_______ True
2 True ____False ______False
3 False ____True ______True
4 False ____False ______True
1. I am alive, I played mafia. My statement is true.
2. I am alive, I did not play mafia. My statement is false.
3. I am dead, I played mafia though. My statement is true.
4. I am dead, I did not play mafia. My statement is true.

1 is true because I satisfied my conditions. Clearly.
2 is false because I contradicted my conditions. I said if I were alive, I would go to the lecture. But I was alive and not in attendance.
3 Is true because I wasn't alive, so it wasn't really relevant my attendance.
4 is true because 3 is true.

We can conclude that these statements are always true unless the antecedent (first part) is true while the consequent (latter part) is false.

The reason why they sound so contradictory? Implication is essentially "If... then" statements such as "if Socrates is a moral, then socrates can die." Natural language expresses "If... then" statements differently to formal language.
"If Bob is black, then Bob has dark skin." And "If Bob is white, then Bob has light skin." is represented in formal language as:
[B(b)-->B(s)] /\ [W(b)-->W(s)]
However, the following argument can be entailed from this statement:
[B(b)-->B(s)] /\ [W(b)-->W(s)] --> [B(b)-->W(s)] \/ [W(b)-->B(s)]
"If Bob is black, he has black skin, and if he is white he has white skin. Therefore if Bob is black he has white skin or if he is White he has black skin."
Since Bob can't have both black and white skin, the dichotomy is valid.
This further illustrates the point.

In natural language, we use "If... then" for issues regarding everyday epistemology crises. We don't know a thing about Bob. But we know that if he is black, he must have black skin. Formal language doesn't follow these observances, so next time you get challenged to create a valid argument that the earth is flat, use material equivalence.
Lol, you clearly do not understand why saying an argument is valid in logical form is useless in everyday argumentation. Debate validly but not soundly and see where that ends up....lol
The reason I responded this way is you don't seem to be conveying that valid is not important, at all.
Also if you have spare time, do read this short article:
http://projecteuclid.org...

Thoughts? Objections? Discuss.