Law of excluded middle

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Cirion Spellbinder
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Law of excluded middle

Post by Cirion Spellbinder »

I really want classical logic to be a better model of reality than constructive logic because it makes its objectivity a lot smoother, but I’m not really sure how to justify its use anymore. I recognize that classical logic is broader than constructive logic, and that this could mean that it is more capable of describing the scope of reality, but I don’t find that particularly persuasive anymore for some reason. Furthermore, Occam’s razor would have to favor constructive logic, since the axioms which lead to the law of excluded middle are removed.

What are your thoughts? I suspect I’m completely misunderstanding the issue as I tend to do with logic, which is why it’s important to me that I get input. Thanks!

P.S. Another unrelated question: is there a way to deduce if a system will have any contradictions without just stumbling upon them? And if not, could it be that some, all, or all but one of the competing systems of logic are invalid, and that we simply just don’t know yet?
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Re: Law of excluded middle

Post by carnap »

Whether or not a logic has any "contradictions" is a question of whether the logic is consistent or not and that is something you can readily prove. Classical logic, etc are consistent.

Occam's razor wouldn't really apply here, its a preference for the more simple explanation/theory. If the two logics had the same outcome then you'd certainly want to go with the one with fewer rules but that isn't the case. They have very different outcomes. Also using this logic you'd actually have to prefer so called "minimal logic" which is weaker than constructive logic (you remove the rule I mention below).

The issue with classical logic is conceptual in nature, classical logic allows for the proof of entities without knowing how to construct those entities. That is, you can prove the existence of some object or property without ever knowing how to build the number or property. This property limits the usefulness of any system built on top of classical logic as you'll have a bunch of statements like "there exists....." but no way to construct it which limits any potential application. In contrast everything provable in intuitionistic logic you can construct which is why its often also called "constructive logic".

Also the law of the excluded middle is a consequence of classical logic, the real issue is the reductio rule. That is the rule that allows you to conclude P if you can show a contradiction when assuming not P. Constructive logic replaces this rule with another that says that you can prove anything from a contradiction (this rule is a basic consequence of the reductio rule so classical logic is "stronger").
I'm here to exploit you schmucks into demonstrating the blatant anti-intellectualism in the vegan community and the reality of veganism. But I can do that with any user name.
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Re: Law of excluded middle

Post by mkm »

The Classical logic applied to the reality sucks anyway. For example try to evaluate the following sentence:

If Oswald hadn't killed Kennedy, someone else would.

:)
Cirion Spellbinder
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Re: Law of excluded middle

Post by Cirion Spellbinder »

Thank you for suffering my stupidity.
carnap wrote: Wed Apr 11, 2018 11:51 am Whether or not a logic has any "contradictions" is a question of whether the logic is consistent or not and that is something you can readily prove. Classical logic, etc are consistent.
How do we know classical logic is consistent? What I was wondering is if we need to consider all possible statements in a system to know that it is consistent or if there is a way to tell without doing that. Like, for example, in my bad understanding of math history, Russell discovered a paradox in set theory, so the foundations of mathematics needed to be replaced.
carnap wrote:Occam's razor wouldn't really apply here, its a preference for the more simple explanation/theory.
Is it unreasonable to consider systems that equally describe something hypotheses, which can be scrutinized with Occam’s razor?
carnap wrote:If the two logics had the same outcome then you'd certainly want to go with the one with fewer rules but that isn't the case. They have very different outcomes.
I think that’s true of all applications of Occam’s razor. Competing hypotheses compete because they are different.
carnap wrote:The issue with classical logic is conceptual in nature, classical logic allows for the proof of entities without knowing how to construct those entities. That is, you can prove the existence of some object or property without ever knowing how to build the number or property. This property limits the usefulness of any system built on top of classical logic as you'll have a bunch of statements like "there exists....." but no way to construct it which limits any potential application. In contrast everything provable in intuitionistic logic you can construct which is why its often also called "constructive logic".
How do we know this?
carnap wrote:Constructive logic replaces this rule with another that says that you can prove anything from a contradiction (this rule is a basic consequence of the reductio rule so classical logic is "stronger").
Are you referring to the principle of explosion? I was taught that it was part of classical logic as well?

Also, could you elaborate on what you mean by stronger?
Cirion Spellbinder
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Re: Law of excluded middle

Post by Cirion Spellbinder »

mkm wrote: Thu Apr 12, 2018 5:15 am The Classical logic applied to the reality sucks anyway.
How else can we justify an objective good?
mkm wrote:If Oswald hadn't killed Kennedy, someone else would.
If it is true, then this means what it says. If it is false, it means if Oswald had killed Kennedy, someone else wouldn’t have. What is wrong with that?
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Re: Law of excluded middle

Post by carnap »

Cirion Spellbinder wrote: Sat Apr 14, 2018 5:59 am How do we know classical logic is consistent? What I was wondering is if we need to consider all possible statements in a system to know that it is consistent or if there is a way to tell without doing that. Like, for example, in my bad understanding of math history, Russell discovered a paradox in set theory, so the foundations of mathematics needed to be replaced.
You don't need to evaluate "every possible statement" but prove that there is no formula P such that both P and ~P is provable. There are a few techniques you can use to proof but they are complicated and rely on a variety of other propositions.

In terms of Russell's paradox, that doesn't apply to set theory as known today but instead of "naive set theory" that was created by Cantor. The paradox comes about when you allow any well defined set to exist. If you assume that, then take the following set:

A = { x | x is not in x} . But in that case if A is in A then that means A is not in A which is a contradiction and vice verse.

This paradox is resolved by only allowing sets that be defined from previously established sets.
Cirion Spellbinder wrote: Sat Apr 14, 2018 5:59 am I think that’s true of all applications of Occam’s razor. Competing hypotheses compete because they are different.
The hypothesis are different but they attempt to describe the same phenomena. Occam's razor doesn't make sense if you're comparing hypotheses for different subject matters.
Cirion Spellbinder wrote: Sat Apr 14, 2018 5:59 am How do we know this?
Not sure what you're asking about here.

Cirion Spellbinder wrote: Sat Apr 14, 2018 5:59 am Also, could you elaborate on what you mean by stronger?
What I mean by "stronger" is that a logic A is "stronger" than logic B if anything that is provable in A is also provable in B and there are some statements that are provable in B that aren't provable in A. In this sense minimal logic < constructive logic < classical logic.
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Re: Law of excluded middle

Post by mkm »

Cirion Spellbinder wrote: Sat Apr 14, 2018 6:02 am How else can we justify an objective good?
Can we? It's not the best practice to hold to some framework just because it implies things you like.
Cirion Spellbinder wrote: Sat Apr 14, 2018 6:02 am If it is true, then this means what it says. If it is false, it means if Oswald had killed Kennedy, someone else wouldn’t have. What is wrong with that?
Try to formulate it in the classical logic then and show me a truth table for that sentence.
And your negation is not correct, but it's more clear with the following: If I had gone to school I would have learnt new things. A negation of it doesn't mean that if I hadn't gone to school I wouldn't have learnt new things. These statements gives you sufficient conditions for something to occur, not necesseary.
Cirion Spellbinder
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Re: Law of excluded middle

Post by Cirion Spellbinder »

mkm wrote: Mon Apr 16, 2018 5:19 amCan we? It's not the best practice to hold to some framework just because it implies things you like.
That is true, but I would still be happy if I knew you could in another framework.
mkm wrote:Try to formulate it in the classical logic then and show me a truth table for that sentence.
And your negation is not correct, but it's more clear with the following: If I had gone to school I would have learnt new things. A negation of it doesn't mean that if I hadn't gone to school I wouldn't have learnt new things. These statements gives you sufficient conditions for something to occur, not necesseary.
A : Oswald hadn't killed Kennedy
B: Someone else would

A→B

We wan't to consider ¬(A→B) which is logically equivalent to A∧¬B
A B.........¬(A→B) || A∧¬B
T T.........F
T F.........T
F T.........F
F F.........F

I think that means if it is true that Oswald hadn't killed Kennedy and if it is not true that someone else would, then Oswald hadn't killed Kennedy and someone else wouldn't kill Kennedy
Last edited by Cirion Spellbinder on Mon Apr 16, 2018 1:28 pm, edited 1 time in total.
Cirion Spellbinder
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Re: Law of excluded middle

Post by Cirion Spellbinder »

carnap wrote: Sat Apr 14, 2018 8:21 pmYou don't need to evaluate "every possible statement" but prove that there is no formula P such that both P and ~P is provable. There are a few techniques you can use to proof but they are complicated and rely on a variety of other propositions.
Ah, okay. Does it have a name so I can look into it?
carnap wrote:The hypothesis are different but they attempt to describe the same phenomena. Occam's razor doesn't make sense if you're comparing hypotheses for different subject matters.
Couldn't it be said that logic describes the phenomena of valid argumentation?
carnap wrote:Not sure what you're asking about here.
How do we know that all things can be constructed in intuitionistic logic? Also, why is it useful to construct things as opposed to simply have proof of their existence?
carnap wrote:What I mean by "stronger" is that a logic A is "stronger" than logic B if anything that is provable in A is also provable in B and there are some statements that are provable in B that aren't provable in A. In this sense minimal logic < constructive logic < classical logic.
Perfect.
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Re: Law of excluded middle

Post by brimstoneSalad »

Cirion Spellbinder wrote: Mon Apr 16, 2018 1:26 pm
carnap wrote:The hypothesis are different but they attempt to describe the same phenomena. Occam's razor doesn't make sense if you're comparing hypotheses for different subject matters.
Couldn't it be said that logic describes the phenomena of valid argumentation?
Carnap is right here.

It's hard to think of a good analogy on the spot, but I'll try.

Occam's razor applies well to to explaining phenomena as long as all of the (measurable) outcomes are the same, not to describing it.
A simpler explanation is no good if it predicts the wrong results.
Descriptions benefit from precision(here the scope of consideration) and accuracy, and Occam's razor is one dimensional (thus all analogies are probably going to be poor).

You could define all apples as red, and that might be simpler than saying apples come in red and green, but it leaves out all of the green apples that also exist. Such as it is with more limited forms of logic. It might be pretty accurate, but the scope is bad.

The red apples described ARE indeed apples, but that doesn't mean there aren't other apples it failed to identify.
The equivalent with more conservative logical frameworks is that the arguments it identifies as valid ARE valid, but there are many more arguments that are likely valid that it is unable to assess.

Hopefully that helps. Carnap might have a better analogy.
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