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

  • Context: Graduate 
  • Thread starter Thread starter hatsoff
  • Start date Start date
  • Tags Tags
    Law Logic
Click For Summary

Discussion Overview

The discussion centers around the law of excluded middle in the context of many-valued logic. Participants explore whether it is possible to demonstrate that there exists a proposition P for which the statement ¬(P∨¬P) holds true under certain conditions in many-valued systems.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant questions whether it can be proven that there exists a P such that ¬(P∨¬P) in some many-valued logic.
  • Another participant suggests that there exists a P such that ¬P Λ P, proposing a proof by contradiction, but expresses uncertainty about its validity in non-binary systems.
  • A different participant notes that axiomatic logic with the schema ¬(P∨¬P) for all propositions P could meet the requirement, but implies this may not align with the original intent of the question.
  • Concerns are raised about the implications of rejecting the law of excluded middle, particularly from a constructivist perspective, which challenges the validity of contradiction proofs.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the law of excluded middle in many-valued logic, with some suggesting that it can be rejected while others highlight the complexities involved. The discussion remains unresolved regarding the proof of the initial claim.

Contextual Notes

There are limitations regarding the definitions and assumptions of many-valued logic systems, as well as the implications of constructivist viewpoints on logical proofs.

hatsoff
Messages
16
Reaction score
3
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!
 
Physics news on Phys.org
"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.
 
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.
 
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.
 

Similar threads

Graduate Prove that points which are indistinguishable from 0 exist (using logic)
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad What Are the Axioms of Fuzzy Logic and How Do They Extend Boolean Algebra?
  • · Replies 7 ·
Replies
7
Views
3K
High School Unbounded Logical Trees: Constructing Natural Numbers with Logical Trees
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Many-valued inference possible: extant?
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Classical First-Order Logic, Axiomatic Set Theory, and Undecidable Propositions
  • · Replies 11 ·
Replies
11
Views
10K
Graduate Multiverse theory with impossible universes?
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Clarification regarding argument in EPR paper
  • · Replies 58 ·
2
Replies
58
Views
5K
Undergrad Modern Assesment of Grete Hermann's Philosophy on Quantum Mechanics
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Recursive Ordinals and Creativity
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Help Understanding the Law of the Excluded Middle and Constructivism
  • · Replies 12 ·
Replies
12
Views
4K