Classical Theism, Philosophy, and Religion Forum / Does Russell's paradox undermine logic?

**Posted by RomanJoe** 5/12/2017 12:24 pm

Does Russell's paradox reveal that we are not operating within a formal system of logic? In a determinate system of logic when we take proposition A, only A or not-A can be true. But the paradox that 'S is the set of all sets that don't contain themselves' shows that we don't have such a determinate system of logic.

Last edited by RomanJoe (5/12/2017 3:41 pm)

**Posted by John West** 5/12/2017 4:25 pm

RomanJoe wrote:
Does Russell's paradox reveal that we are not operating within a formal system of logic?

Nah. It doesn't even do in every form of set theory—i.e. both ZF and ZFC are built on standard first-order logic, and the law of the excluded middle is part of that.

**Posted by RomanJoe** 5/12/2017 5:53 pm

John West wrote:
RomanJoe wrote:
Does Russell's paradox reveal that we are not operating within a formal system of logic?
Nah. It doesn't even do in every form of set theory—i.e. both ZF and ZFC are built on standard first-order logic, and the law of the excluded middle is part of that.

How would one go about showing formally that it excludes the middle term?

**Posted by John West** 5/12/2017 6:35 pm

RomanJoe wrote:
John West wrote:
RomanJoe wrote:
Does Russell's paradox reveal that we are not operating within a formal system of logic?
Nah. It doesn't even do in every form of set theory—i.e. both ZF and ZFC are built on standard first-order logic, and the law of the excluded middle is part of that.
How would one go about showing formally that it excludes the middle term?

The law of the excluded middle is an axiom of the predicate calculus (usually, anyway).
Curious: how did the Russell's paradox thing come up?

**Posted by RomanJoe** 5/12/2017 8:57 pm

John West wrote:
RomanJoe wrote:
John West wrote:
Nah. It doesn't even do in every form of set theory—i.e. both ZF and ZFC are built on standard first-order logic, and the law of the excluded middle is part of that.
How would one go about showing formally that it excludes the middle term?
The law of the excluded middle is an axiom of the predicate calculus (usually, anyway).
Curious: how did the Russell's paradox thing come up?

I'm in an argument with someone who denies that our use of formal propositional logic is determinate. He believes our minds are reducible to the indeterminate physical processes of hormone secretion and neural firing. So, consequently, this indeterminate process cannot conform to a determinate set of inflexible logical laws. He claimed that an admissible proposition A in a determinate system of logic only can yield A or not-A. The paradox thus shows that our logical system is indeterminate because it is a proposition that yields neither A or not-A.

**Posted by RomanJoe** 5/13/2017 1:50 am

The law of the excluded middle is an axiom of the predicate calculus (usually, anyway).

Oh yeah, I was thinking of the fallacy of the undistributed middle.

**Posted by RomanJoe** 5/15/2017 1:32 pm

What do you think about the liar's paradox?

(A) This statement is false.

If (A) is true, then "This statement is false" is true. Therefore, (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.If (A) is false, then "This statement is false" is false. Therefore, (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.

The law of the excluded middle states that a proposition is either A or not-A. So is this paradox an exception?

Last edited by RomanJoe (5/15/2017 1:32 pm)

**Posted by John West** 5/15/2017 5:57 pm

RomanJoe wrote:
What do you think about the liar's paradox?

The short answer is: I don't think the Liar expresses a proposition.

The law of the excluded middle states that a proposition is either A or not-A. So is this paradox an exception?

So no.

**Posted by RomanJoe** 5/15/2017 6:40 pm

Why do you think it's not a true proposition?

**Posted by John West** 5/16/2017 10:17 am

I think that, on closer analysis, the liar sentence comes out meaningless.

But the liar paradox is irrelevant to the argument from reason, anyway. Suppose we agree that it refutes the law of the excluded middle, and cope by adopting a three-valued logic. How do we account for rational inference in the three-valued logic in terms of “indeterminate physical processes of hormone secretion and neural firing”?