Bertuzzi

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On Logical Contradictions - Removing Misconceptions
« on: February 12, 2014, 09:27:16 am »
So what is a logical contradiction? When does something constitute as such? I've found the term is often used without any logical proof, which is necessary to prove a contradiction. Here I will attempt to do away with the ambiguity. After going through this I think you'll find that logical contradictions require much more than is typically thought. Prepare yourself, it may be a little on the technical side.

There are 3 kinds of contradictions I'll cover, namely explicit, implicit, and formal.

EXPLICIT CONTRADICTIONS

Explicit contradictions are ones of a certain sort and, as such, are rarely ever espoused. And by rarely, pretty much never. An explicit contradiction would be a conjunctive proposition where one conjunct is the negation of the other. For example:

(1) John is a good tennis player, and it is false that John is a good tennis player.

It should be obvious that this conjunctive proposition is explicitly contradictory. Not much more needs to be said here. The other two are much more involved.

IMPLICIT CONTRADICTIONS

Implicit contradictions are those that require additional necessarily true propositions, that when added to the original proposition create a formally contradictory set of propositions. Take the implicitly contradictory proposition "There exists a married bachelor". This is quite clearly contradictory but it is not explicitly contradictory, so defined. It requires an additional necessarily true proposition to create a formal contradiction (which eventually and inevitably explicates an explicit contradiction).

(2) There exists a married bachelor
(3) Bachelors are unmarried

FORMAL CONTRADICTIONS

In the above example, (3) is broadly necessarily true. Here we can see that the conjunct of (2) and (3) create a formal contradiction, but not yet explicit. Formal contradictions are those that, by using the laws of logic, can arrive at a conclusion that when combined with the original set derive an explicit propositional contradiction.

(4) If there exists a married bachelor and bachelors are unmarried, then there exists a married unmarried man.
(5) If there exists a married unmarried man, the law of non-contradiction is false.
(6) But the law of non-contradiction is strictly necessarily true.
(7) Therefore, no married bachelor exists.

The conjunct of (7) and (2) combine to create an explicit contradiction: "There exists a married bachelor, and no married bachelor exists."



I'd be happy to apply this to practical usages such as in the logical PoE, just let me know and we can discuss. Otherwise, feel free to comment!
« Last Edit: February 12, 2014, 10:13:10 am by Bertuzzi »
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Bertuzzi

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #1 on: February 12, 2014, 11:44:55 am »
To add a little controversy, take the following propositions:

(1) God exists
(2) Evil exists

Off the bat, these propositions are not explicitly contradictory. One conjunct is not the negation of the other. (2) would need to read "It is false that God exists". They can only be, potentially, implicitly/formally contradictory. So we must add some necessarily true proposition(s) to derive an explicit contradiction. But which proposition could this be? Any takers?
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Crash Test

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #2 on: February 12, 2014, 12:34:34 pm »
Good thread, +1.
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TheBigOhMan

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #3 on: February 12, 2014, 12:49:26 pm »
¿Can there exists meta-logical contradictions?

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Bertuzzi

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #4 on: February 12, 2014, 01:26:05 pm »
Good thread, +1.

Well thank you sir, any thoughts?
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Bertuzzi

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #5 on: February 12, 2014, 01:26:33 pm »
¿Can there exists meta-logical contradictions?

Not sure what that is?
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Questions11

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #6 on: February 12, 2014, 01:29:44 pm »
BOM, you referring to para consistent logic?

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TheBigOhMan

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #7 on: February 12, 2014, 01:48:10 pm »
¿Can there exists meta-logical contradictions?

Not sure what that is?

A Contradiction that is a contradiction in more than one logical system, not just classical logic.


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Bertuzzi

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #9 on: February 12, 2014, 02:04:59 pm »
¿Can there exists meta-logical contradictions?

Not sure what that is?

A Contradiction that is a contradiction in more than one logical system, not just classical logic.

Prima facie, it would seem so, but I suppose it could be challenged by a dialetheist.

It would depend on which system you espouse. Some theists hold that God can perform the logically impossible, perhaps logical contradictions are only of interest to classical non-dialetheist logicians.
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Re: On Logical Contradictions - Removing Misconceptions
« Reply #10 on: February 12, 2014, 02:06:56 pm »
Currently I think that God can, but won't  perform logical contradictions much in the same way He can, but won't, commit sin.

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TheBigOhMan

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #11 on: February 12, 2014, 02:12:47 pm »
¿Can there exists meta-logical contradictions?

Not sure what that is?

A Contradiction that is a contradiction in more than one logical system, not just classical logic.

Prima facie, it would seem so, but I suppose it could be challenged by a dialetheist.

It would depend on which system you espouse. Some theists hold that God can perform the logically impossible, perhaps logical contradictions are only of interest to classical non-dialetheist logicians.

Is there any way to resolve the issue? If both persons already start with a diferent set of logical axioms, ¿How can they manage to convice the other?

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Bertuzzi

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #12 on: February 12, 2014, 02:19:30 pm »
¿Can there exists meta-logical contradictions?

Not sure what that is?

A Contradiction that is a contradiction in more than one logical system, not just classical logic.

Prima facie, it would seem so, but I suppose it could be challenged by a dialetheist.

It would depend on which system you espouse. Some theists hold that God can perform the logically impossible, perhaps logical contradictions are only of interest to classical non-dialetheist logicians.

Is there any way to resolve the issue? If both persons already start with a diferent set of logical axioms, ¿How can they manage to convice the other?

Far as I understand, dialetheists posit that some contradictions are true, not all. So maybe they take it on a case by case basis.
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Bertuzzi

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #13 on: February 12, 2014, 03:49:12 pm »
My hope for this post was to help show why exactly the logical PoE has failed to be proven. It can sometimes creep up with advocates but rarely, if ever, has the kind of rigorous work needed to explicate an explicit contradiction. Mackie (whom I greatly admire), even at the top of his game was unable to do so.

Does anyone advocate the logical PoE?
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Way

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Re: On Logical Contradictions - Removing Misconceptions
« Reply #14 on: February 12, 2014, 04:07:07 pm »
Creation Problem:

(1) God exists.
(2) God is uncaused.
(a) All existence has a cause.
(a) God caused every existence.
(c) God caused himself.

(2) and (c) are logical contradiction.


Problem of Evil:

(1) God is always Good. (omnibenevolence)
(2) God created and planned everything.
(3) Evil is a lack of Good.
(4) Evil exists.
(5) God created and planned Evil.
(6) God is not always Good.

(5) is a conclusion out of (2) and (4)
(6) is a conclusion out of (3) and (5)
(1) contradicts to (6)

Possible solutions:
(1) is wrong and God is not always Good.
or (2) is wrong and God did not create and plan everything.
or (3) is wrong and evil is not a lack of Good, but in fact a part of Good
or (4)  is wrong and actually Evil does not exist. (3) and (4) in the sense of "all evil is a necessary part of a greater good."
« Last Edit: February 12, 2014, 04:24:22 pm by Wey »
Reason and emotion are what make us human. But reason without emotion is cruelty, emotion without reason is stupidity.