Prove that the law of excluded middle does not hold in some many-valued logic

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In summary: What you need instead are "constructive proofs," which are proofs that show that a certain property follows from certain axioms. In other words, they're not about contradiction, they're about showing that the property is logically implied by the axioms.
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Hi, all.

Wikipedia says:

In logic, the law of the excluded middle states that the propositional calculus formula "P ∨ ¬P" ("P or not-P") can be deduced from the calculus under investigation. It is one of the defining properties of classical systems of logic. However, some systems of logic have different but analogous laws, while others reject the law of excluded middle entirely.​

(emphasis added)

My question is, can we prove that the bolded claim is true? For some many-valued logic, can we show that for some condition, there is P with ¬(P∨¬P) ?

Thanks!
 
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  • #2
"can we show that for some condition, there is P with ¬(P∨¬P) ?"

There exists a P such that ¬(P∨¬P)
There exists a P such that ¬P Λ P

Lets prove this by contradiction. It seems really easy, although I'm not sure if it is valid since the law of excluded middle isn't meant to be applied to non-binary systems.

For every P, (P V ¬P)
In a multivalued system, this isn't true, because P can be something besides true or false.

Therefore the opposite, ¬P Λ P, is true.

Okay, that's my poor attempt at it. I would give it more of a mathematical go if I had more time. Actually, I'm procrastinating right now.
 
  • #3
hatsoff said:
My question is, can we prove that the bolded claim is true? For some many-valued logic, can we show that for some condition, there is P with ¬(P∨¬P) ?

Surely the axiomatic logic with schema
¬(P∨¬P)
for all propositions P would meet your requirement, but I imagine that's not what you intend.

Also, be careful: there are logics which reject the law of the excluded middle (for all propositions P, P∨¬P) but accept the law of noncontradiction (for all propositions P, ¬(P∨¬P)). Intuitionistic logic would be an example.
 
  • #4
When they say that some systems reject the law of the excluded middle, I believe what they're talking about is the constructivist viewpoint. Basically, if you are a constructivist, contradiction proofs go out the window.
 

1. What is the law of excluded middle?

The law of excluded middle is a fundamental principle in classical logic that states that for any proposition, it must either be true or false, there is no middle ground or third option.

2. What is many-valued logic?

Many-valued logic is a type of logic that allows for more than two truth values, in contrast to classical logic which is limited to true or false. This allows for a more nuanced and complex understanding of propositions.

3. How is the law of excluded middle violated in many-valued logic?

In many-valued logic, there are propositions that do not have a definite truth value, but instead have a range of possible truth values. This means that the law of excluded middle does not hold as there is a middle ground between true and false.

4. What are some examples of many-valued logic systems?

There are several different many-valued logic systems, such as fuzzy logic, intuitionistic logic, and paraconsistent logic. These systems differ in their approach to handling propositions with multiple truth values.

5. How does the violation of the law of excluded middle impact logical reasoning?

Many-valued logic allows for a more nuanced understanding of propositions and can better capture the complexity of real-world situations. However, it also means that traditional logical reasoning methods may not be applicable and new methods must be developed to accommodate the multiple truth values.

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