jiml@alberta.UUCP (05/28/87)
I hope this is the correct group in which to post an article on a curiosity in epistemic logic. It seems that there are a class of sentences which, while true, may not be believed. I'll give first an example of an entirely reasonable statement, followed by a close cousin which is unbelievable but true. let p = "it is raining" 1. Bel(Smith, p & ~Bel(Jones,p)) Smith believes that it is raining but that Jones doesn't believe it. 2. ~Bel(Jones, p & ~Bel(Jones,p)) Note that in this case, p & ~Bel(Jones,p) may well be true, but that Jones is incapable of believing it. It's been a while since I've seen the derivation, but I recall that the following is a theorem of epistemic logic: 3. (x)(p)~Bel(x, p & ~Bel(x,p)) provided that 4. Bel(x,p) > Bel(Bel(x,p)) is an axiom. Good grief, my memory is getting rusty. Is it S4 that is typically used to model epistemic logics? Anyway, can anyone out there add to this curious class of sentences?
kube@cogsci.berkeley.edu (Paul Kube) (05/29/87)
In article <1150@cavell.UUCP> jiml@cavell.UUCP (Jim Laycock) writes: > >Good grief, my memory is getting rusty. Is it S4 that is typically >used to model epistemic logics? Which set of modal axioms you use for your epistemic logic of course depends on what intuitions you want to capture, but also on whether you're after a logic of belief or of knowledge. I've seen T, S4, and S5 used as axiom sets for logics of belief, but the axiom schema LP -> P in each of them makes better sense interpreting L as `knows' than as `believes'. >It's been a while since I've seen the derivation, but I recall that >the following is a theorem of epistemic logic: > > 3. (x)(p)~Bel(x, p & ~Bel(x,p)) > >provided that > > 4. Bel(x,p) > Bel(Bel(x,p)) > >is an axiom. -L(P & -LP) (i.e., (x)(p)~Bel(x, p & ~Bel(x,p)) ) is a theorem in each of these systems. You don't need LP -> LLP (which is missing from T), only LP -> P : 1. L(P & -LP) (assume for contradiction) 2. P & -LP (from 1. by LP -> P) 3. P (from 2. by conjunction elimination) 4. LP (from 3. by necessitation) 5. -LP (from 2. by conjunction elimination) 6. LP & -LP (from 4., 5.) For anyone who wants to get into this stuff, Hintikka's book _Knowledge and Belief_ is the place to start; and there's been a bunch of recent stuff in the AI knowledge representation literature (e.g. see D. McDermott, "Nonmonotonic Logic II: Nonmonotonic Modal Theories", JACM, January 1982). --Paul kube@berkeley.edu, ...!ucbvax!kube