Modal Logic
in [Logic]
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From: Confutus on 5 Oct 2006 08:07 Frederick Williams wrote: > Confutus wrote: > > > the huge gulf between multi-valued logics and modal logics > > The gulf may not be that huge. See my reference to Lukasiewicz. Mind > you, his logic is a bit odd, iirc. > Lewis and Langford discussed Lukasiewicz 3-valued logic but rejected it, and later workers proved that Lewis's modal logics could not be reduced to 3 values. Ever since then, almost the entire community of logicians has followed along, and multi-valued logics and modal logics have grown up as separate kingdoms that speak different languages and rarely discuss one another. It's hard to find anyone who's bilingual. Lukasiewicz logic *is* a bit odd, but if one wants deal with the problems associated with the forbidden "third" squarely instead of tiptoeing around them, it's the best tool for the job. It just hasn't been used properly.
From: Jan Burse on 5 Oct 2006 13:02 Hi The gulf between (finite) multivalued logic and modal logic, can be compared to the gulf between basic algebra in undergraduate school and analysis in graduate school. In basic algebra you might have +, -, *, /, and in analysis you add differentiation and integral. So the gulf is rather one level above another. Propositional modal logic is richer than finite propositional multivalued logic. At least this my impression. The consideration collapses when one does not use porpositional logic, but first order logic. Because most of the modal logics can be modelled in first order logic. Bye Confutus wrote: > Yes, I have studied the literature I could find, and I'm familiar with > the huge gulf between multi-valued logics and modal logics and the > reasons, both logical and historical for it. I also know that modal > logics and multivalued logics *as they have been developed* are > different and incompatible. What I have is a development of 3-valued > logic that looks and acts very like one of the Lewis systems and their > many, many, descendants and kindred, until you look at it closely and > see just how unrelated it is. > > Jan Burse wrote: > >>There are two names around. Some logics >>are called modal, and some logics are >>called multi valued. >> >>It can be shown that modal logics correspond >>with certain infinitely valued logics. >>But on the other hand it can also be shown >>that finitly valued logics do not correspond >>with certain modal logics. >> >>So that is why some logics are called modal >>and other are called 3-valued, and they >>do not correspond to each other. >> >>If you want a reference on my claims, >>please let me know. There have also already >>been posts in this news group that support >>this claim. >> >>Bye >> > >
From: "Dmitry Sustretov dmitry.sustretov@gmail.com>" on 5 Oct 2006 14:41 This is true, but it is really not an interesting result, because modal logics have a real (relational Kripke) semantics and a rich model thery (not as rich as FO logic, but suitable for lots of applications in philosophy and computer science). A good book on ML from amsterdam school: Blackburn,de Rijke,Venema "Modal Logic". Jan Burse wrote: > It can be shown that modal logics correspond > with certain infinitely valued logics.
From: Confutus on 5 Oct 2006 15:06 Thanks for the references. Frederick Williams wrote: > > J Lukasiewicz "A System of Modal Logic" Journal of Computing System" > vol i, no 3 July 1953 > > but I'm now having doubts about my memory--I haven't looked at this > stuff in 25 years. > > What I am sure about is that connections between n-valued logic (for > small n) and modal logic are discussed in > > A N Prior "Formal Logic" OUP > A N Prior "Time and Modality" OUP (1979 Greenwood reprint) > > and indeed a 3-valued modal logic is discussed. >
From: Confutus on 5 Oct 2006 16:42
Jan Burse wrote: > Hi > > The gulf between (finite) multivalued logic and > modal logic, can be compared to the gulf > between basic algebra in undergraduate school > and analysis in graduate school. I've seen it as more like the difference between abstract algebra in undergraduate school and analysis in graduate school. Both are in some sense based on high-school algebra, but they go in entirely different directions. Similarly, multivalued logic and modal logic start with classical, two-valued logic as a basis, but they take it in entirely different directions. For instance, modal logic typically takes the theorems and formulas of classical two valued logic for granted, includes all of them, and adds the modal features with additional axioms. Multivalued logic does not take these theorems for granted, but is concerned more with which remain valid with an n-th logical value added. Unfortunately, the track record of multi-valued logic in successfully extending classical logic is dismal. I don't think anyone actually does deductive reasoning with any version of it. The issue is not whether a modal logic based on 3VL is identical with any of the various existing modal systems (it isn't). It's whether anything remotely comparable even exists. This kind of thing is supposed to be well beyond the power of multi-valued logic, a curiosity reminiscent of the quote from Samuel Johnson: >A woman's preaching is like a dog's walking on his hinder legs. It is not done well; but you >are surprised to find it done at all. |