Re: Basic Fuzzy Logic

thomaz wrote: 
I'm quite new in the field of Fuzzy Logic

and I heard about the basic predicate fuzzy logic tautology's
one of them I don't get:

forall x (A->B) -> (A -> forall x B)

Can some one explain this one?
Please...

Dear Tomaz,

The sentence you refer to is - according to Hajek - the third axiom of Basic Predicate Logic. Hajek refers to this one as (forall 2); see page 111 of Hajek's seminal book (reference enclosed below). Note that your formulation is not fully complete. You have to require that the variable x is not free in the formula A.

This axiom means, provided that the quantified variable is not free in the first argument of an implication, that you can draw the quantifier into the implication. Perhaps it's more understandable if you consider this axiom as a generalization of the well-known correspondence from Boolean logic:

   ((A -> B) & (A -> C)) -> (A -> (B & C))

[Remark: this formula is not provable in BL if you consider & as the strong conjunction, but it holds for the weak conjunction /\. In Goedel logic, the two coincide and the above sentence is provable.]

On the semantic level, if we consider the unit interval [0,1] as the domain of truth values and if we model the implication by the residual implication of a continuous t-norm and the universal quantifier by the infimum, then the axiom above breaks down to the right-continuity of residual implications. Since this is always the case, the above axiom is a tautology in every t-algebra.

I don't know if this answers your question, I hope so.

Best regards,
Ulrich


--

@book{Hajek98,
  author =	 {P. H\'ajek},
  title =	 {Metamathematics of Fuzzy Logic},
  publisher =	 {Kluwer Academic Publishers},
  volume =	 {4},
  series =	 {Trends in Logic},
  address =	 {Dordrecht},
  year =	 {1998}
}
.

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