Fuzzy Logic - T-norms

clytaemnestra

New member
Joined
Mar 7, 2021
Messages
7
Hi, I've got the following tasks to solve:

1. calculate the following statement by using the Godel's t norm:
(ϕ ∧ ψ) ⇒ (ϕ ⇒ ψ)
Godel's t norm: tG (a, b) = min(a, b)

2. calculate the following statement by using the product t norm:
¬(ϕ ∨ ψ) ⇒ (¬ϕ ∧ ¬ψ)
product's t norm: tΠ(a, b) = a · b

3. calculate the following statement by using the Lukasziewisz's norm:
(ϕ ⇒ ψ) ⇒ (¬ϕ ∨ ψ),
Lukasziewisz's norm: tL(a, b) = max(0, a + b − 1)

Now, I checked the coursebook, Wiki and googled out some videos and materials - all of them describe it in some way, but none of them has specific instructions on how to plug it into a statement and solve it with it (for a dummy like me). Now, I don't want to come off az lazy - I'm of course willing to try to solve it on my own, before asking for help, but I couldn't find any materials on it. Thus I'd like to ask, if anyone has materials or any clue how to solve it? I'm not sure even how to start.

Thank you in advance
 

lex

Full Member
Joined
Mar 3, 2021
Messages
661
Unfortunately, I know nothing about this.
(I notice all 3 are tautologies under 'normal' predicate logic).

I see your problem, when you do searches on the internet.
Here are a few possibles:
http://math.uchicago.edu/~may/REU2014/REUPapers/Mendelssohn.pdf

(Table 2 may be of use in the following):
https://res.mdpi.com/d_attachment/philosophies/philosophies-06-00021/article_deploy/philosophies-06-00021.pdf

I think the book to get your hands on if you could is Hájek’s "Metamathematics of Fuzzy Logic". Chapter 2 may be helpful.
(Reviewed here: https://www.sfu.ca/~jeffpell/papers/ReviewHajek.pdf)

https://en.wikipedia.org/wiki/T-norm_fuzzy_logics

Sorry. Let me know if you get a good text.
 
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