Modal Logic
in [Logic]
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From: Confutus on 6 Oct 2006 12:45 Frederick Williams wrote: > > I don't get it. Intuitively speaking, if something is not logically > true it may be logically false or merely factually true. (Mind you I'm > not sure that the difference between matters of fact and matters of > logic is as clear-cut as is often made out. Didn't Quine have something > to say about this?) But if this is the interpretation, this is implicit in the truth tables. P []P ~[]P T T F t F T f F T F F T *Not logically true" does include the categories "logically false", "(merely) factually false" and "(merely) factually true".
From: Owen on 6 Oct 2006 15:42 I posted a very detailed response to your questions, but there was a Google problem?! And my detailed post did not show. I will try again, with less rigour. Confutus wrote: > Some time ago I developed a four-valued extension of the three valued > logic I have been working with. This comes very close to the same > thing. On the basis of my experience with it, I'd like to make some > suggestions. > > 1) in your table for implies, third line first entry, why f -> t = t? Because, (f -> t) <-> (~f v t) <-> (t v t) <-> t. > > Try [](p & (p -> q)) ->q and watch what happens when p=f & q = t. > []P should evaluate as T for any tautology P. > I would suggest f -> t = T instead. ([](f & (f -> t)) -> t) <-> ([](f & t) -> t) <-> (F -> t) <-> (T v t) <-> T. Where is your problem. Note, that this 4-valued logic does not represent S5 As I have stated, modal logic (S5) of two propositional variables require a system of 16 truth values. > > Try this definition: > p => q :: [](p -> q) and see what happens. > > 2) In the table for equivalence, there are three problems. > a) first line, second entry, you have t <-> f = F. This doesn't match > the entry for implies; I would sugges instead, t t <-> f = f i > b) second line, first entry you have f <-> t = T. Surely this must be > an error. I would suggest > f <-> t = f in accord with 2a) > c) third line, second entry is a duplicate of third line, first entry > and there is thus no entry for F <-> f. I would suggest F <-> f = t, > for symmetry with the second line, third entry. > Yes, you have cought me in two typos. (t <-> f) <-> F <-> (f <-> t). > These changes would allow you to define the "equivalence" > p <-> q :: (p->q) & (q -> p) > > I'd also suggest that the "equivalence" you mention should be renamed > the "biconditional" and that equivalence be reserved for a new function > ==, defined thus: > p == q :: [](p <-> q), or (equivalently) > P == q :: (p => q) & (q => p) > > It isn't S5 or any of the other traditional systems of modal logic, but > it is functional. Wrong. All of the axioms of classical propositional logic and Lewis' (S5) are tautologies in My sense! Clearly, within Lewis' logic, (p => q) =df [](p -> q) And, (p <=>q) =df [](p <-> q), ...and much more,
From: Confutus on 6 Oct 2006 18:22 Owen wrote: > I posted a very detailed response to your questions, but there was a > Google problem?! > And my detailed post did not show. On closer examination, your four-valued system and mine are not as nearly identical as I had thought on first glance. There are differences in your definition of & and v as well as in -> and <-> which eliminate the inconsistencies I thought I saw. I said > > Try [](p & (p -> q)) ->q and watch what happens when p=f & q = t. I should have had []((p & (p -> q)) ->q), but never mind. I wasn't using your definition of &, so my objection isn't valid. Likewise, I'll withdraw my other suggestions. Your definitions look a little strange to me, but except for the typos (Your table for & is also missing an entry) they do appear to be consistent. What I don't quite understand is: > Note, that this 4-valued logic does not represent S5 > As I have stated, modal logic (S5) of two propositional variables > require a system of 16 truth values. > > It isn't S5 or any of the other traditional systems of modal logic, but > > it is functional. > > Wrong. > All of the axioms of classical propositional logic and Lewis' (S5) are > tautologies in My sense! So, what's wrong with what I said?
From: Chris Menzel on 7 Oct 2006 17:09 On 6 Oct 2006 04:24:08 -0700, Owen <owenholden(a)rogers.com> said: > ...the axioms and theorems of modal propositional logic (eg.S5) are > truth functional tautologies. Do you have a proof of this? (Would seem to be an easy induction on length of proof.) > This logic greatly expands classical logic. In what way does it greatly expand classical logic (by which I take it you mean classical propositional modal logic)? Are some of its tautologies not theorems of S5? (If so, is that a good thing?) If not, and if your claim above is true, your system is semantically equivalent to S5 and simply provides an alternative decision procedure -- on the face of it, one that is exponential and hence computationally harder than the usual S5 decision procedures.
From: Owen on 8 Oct 2006 08:38
Confutus wrote: > Owen wrote: > > I posted a very detailed response to your questions, but there was a > > Google problem?! > > And my detailed post did not show. > > On closer examination, your four-valued system and mine are not as > nearly identical as I had thought on first glance. There are > differences in your definition of & and v as well as in > -> and <-> which eliminate the inconsistencies I thought I saw. > > I said > > > Try [](p & (p -> q)) ->q and watch what happens when p=f & q = t. > > I should have had []((p & (p -> q)) ->q), but never mind. I wasn't > using your definition of &, so my objection isn't valid. Likewise, > I'll withdraw my other suggestions. Your definitions look a little > strange to me, but except for the typos (Your table for & is also > missing an entry) they do appear to be consistent. > > What I don't quite understand is: > > > Note, that this 4-valued logic does not represent S5 > > As I have stated, modal logic (S5) of two propositional variables > > require a system of 16 truth values. > > > > It isn't S5 or any of the other traditional systems of modal logic, but > > > it is functional. > > > > Wrong. > > All of the axioms of classical propositional logic and Lewis' (S5) are > > tautologies in My sense! > > So, what's wrong with what I said? > I said > > > Try [](p & (p -> q)) ->q and watch what happens when p=f & q = t. > I should have had []((p & (p -> q)) ->q), Both of these propositions are tautologous, but, the complete truth table for two propositional variables has not yet been given. > > > It isn't S5 or any of the other traditional systems of modal logic, but > > > it is functional. My claim is that this system, when expanded to include functions of two variable, includes all of the axioms of S5. |