Saturday, May 28, 2005

3VL - Fuzzy logic and intuitionism

I can also connect this with fuzzy logic. While Fuzzy logic treats individual truth values in the interval between true and false, Three valued logic distinguishes the endpoints True and False, and the entire interval between as the third value F. It thus acts as a bridge between Classical logic and fuzzy logic.

I have also observed connections with intuitionistic, partly as a result of the comparison I did with other 3-valued systems. When I was at the University of Utah, I wrote a paper with my results and tried, (unsuccessfully) to get one of the professors of Logic there to critique it. However, one person there, John Halleck, took a look at it and included a 3-valued logic evaluator on his web site. Borrowing from his list of axioms of Heyting's Intuitionist PC; I have:

HA1: p=>(p&p) True
HA2: (p&q)=>(q&p) True
HA3: (p=>q)=>((p&r)=>(q&r)) True
HA4: ((p=>q)=>(q=>r))=>(p=>r) True
HA5: q=>(p=>q) contingent
HA6: (p&(p=>q))=>q True
HA7: p=>(p+q) True
HA8: (p+q)=>(q+p) True
HA9: ((p=>r)&(q=>r))=>((p+q)=>r) True
HA10: ~p=>(p=>q) contingent
HA11: ((p=>q)&(p=>~q))=>~p contingent

However, if we use ~<> instead of ~, the last two evaluate as true, and in HA5,
if we use instead, []Q => (p => Q), the expression is true.

As with the case with modal logic, a slight adjustment makes this entirely compatible with intuitionistic logic so that they are not identical, but they are quite similar.

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