Well, that’s quite enough. There is no need to return to our days of enmity. It certainly isn’t worth it to me.
I’m not dodging anything. Why would I attempt to dodge something so painfully obvious as a simple transitive implication? I backed off from S several posts back and said that I meant K. In fact, my first post when our debate started back up refered specifically to K (not S) opposing D.
We’re one step away from my being called disingenous (and it’s implied already). We’re already saying that I “oft” refer to the modal proof thread, when I doubt that I’ve refered to it in one percent of my posts. And we’re already saying that I’m “acting” like this or that.
I just won’t go there. Not with you. I admire you too much, and it makes me too sad. As Willie Mays Hayes said to Rick ‘Wild Thing’ Vaughn in Major League, “It ain’t THAT important.”
I don’t want to come away from debates with you having hurt feelings, but rather having edification.
Again? How many times do I have to say, “I ain’t mad” in the same thread?
If your post opposing necessity and obligation was meant to imply a general retreat to K then I certainly missed that implication. If I had recognized it, I would have asked right then why you think that a modal treatment of truth should be limited to K.
Really, thoug, given the number of times I specifically referred to (M/T) and (B), even to the point of using them to derive the disputed result, I don’t think it’s unreasonable of me to have expected a, “Yes, but I only want to use K” somewhere along the line.
And I didn’t say that you oft referenced the ontological proof of God thread, only that it was oft referenced. In fact, to the best of my knowledge the last two times it has been referenced were both by me.
As to “acting like the context of this discussion has been limited to K,” well, your last post to me included the sentence “But if you will look at the Stanford Encyclopedia of Philosophy papers on modal logic, you’ll see that D is not derivable in M + B.” I found this incongrous with your statement to erl indicating K as the controlling context of this discussion. I still do. Deliberate misrepresentation is not, of course, the only explanation for this, and I have not accused you of lying. It is easy enough to get twisted around in these things debates and lose track of the context(s) in which an argument is framed.
But I still do not think you should act as if this discussion has been consistently limited to K. It has not been, by either party.
Oh, I didn’t think you were mad in my statement, Spiritus. It was a polite way of saying I was getting pissed, lol. My incoherence with startling simple terms should help back me up on that.
Quite. I thought we were talking about M, not K, when in fact we were all the way at S4 (which seems superfluous to the argument about what to do with <> in terms of since the only thing S4 gives us is an ability to shrink strings of like qualifiers, though of course we get the desired result from B+S4).
I’m more interested in backing up to the point where I understand what a recursive entity is, and why it is necessary in order to completely model truth under any theory of truth. Because, let me tell you, I don’t understand.
I’d also be interested in understanding the controversy behind Brouwer’s axiom. As I see it the axiom seems intuitive under the possible world semantics.
Okay, well that’s a fair assessment. And I very much appreciate your clarifications.
The reason I continued talking about D is because you continued talking about D. I think that what I originally said was that truth can’t be mapped deontically. And that’s what led originally to my asking for implications in S. After your post on the prior page convinced me to review my position, it occured to me that the matter is actually trivial in S. (That was what you had convinced me OF.) And then Eris bumped the thread.
By then we got onto the tangent of modeling contingency by necessity. Or at least, I did. By the time Eris spoke up with what you had basically already posted before my regrouping (the transitive implication), it was old news to me. Still, in retrospect, I can’t expect you to have read my mind if I didn’t explicitly state my views.
Anyway, thanks for letting me clear that up. If you would like to still discuss deontics, we can. Because you have not convinced me that it is derived from M + B.
That topic is nearly worth its own thread, and is actually relevant to recursive entities since B is basically T + Brouwer’s Axiom, and T is a reflexive frame.
Lib
Glad we could clear that up. No hurt feelings either direction, I hope.
Well, I’m not saying that deontics is derived from M + B. I’m saying that []A -> <>A can be derived from M + B. In fact, according to the Stanford site you cited it can be derived simply from M, though I haven’t tried to do so.
Deontics is a different matter, because Deontics accepts only the weaker implication (D) and rejects (M), so it cannot really be said to be derivable from (M) + anything.
erl
Well, that Stanford site that Lib linked to does a pretty good job of introducing the issues. Basically, it revolves around an ambiguity in scope in the English “if A then necessarily B” when compared to “necessarily if A then B”. Brouwer’s Theorem is A -> []<>A, which is a different thing from [](A -> <>A).. The second is a provable theorem in M, the first is not. Some folks argue that the theorem should be stated in an equivalent form: <>[]A -> A which eliminates the ambiguity of scope.
I’m a bit leery about applying the necessitation rule like that as I don’t know if it works both ways, but it seems like it would have to be, and in fact I don’t wonder if they didn’t prove by first proving
A-><>A and applying the necessitation rule. I suppose that would be my first step, if I cared that much.
Actually, if you start with the theorem (A -> <>A) then it takes nothing more than (M/T) to get to your conclusion. No need for necessitation:
(A -> <>A)
A -> A…(M)
A -> <>A…(1,2)
A -> <>A…(2,3)
I actually find A -> <>A as a very intuitive theorem for M, since it is a weaker form of the modal axiom and thus should necessarily be derivable from the stronger A -> A. Maybe I’ll mull it over and see if I can’t derive (A -> <>A) from (M).
erl, applying necessitation after step 5 would take you to (A -> <>A), as you surmised.
Step 3 bugs me, but I cannot find any reason to declare it an illegal substitution, and I don’t think that it bugs me any more than “regular” necessitation.
Gah, you’re right! In fact, it is almost directly derivable from K. It never occured to me that deontics could be derived simply by serializing a 4-frame K and adding M. Stanford’s map of relationships made it obvious. I feel like a moron.
Nevertheless, bludgeoned and wounded, I’ll drag on a bit further, and maybe you can put me out of my misery altogether. The problem is that when I add a serial frame to K, I change the meaning of necessity from “true in every possible world” to “true in every world where entities do what they are obligated to do”.
That leads me back to what bothered me before, namely that A -> <>A becomes biconditional, except that, from Stanford’s list of axioms, it happens not in D proper, but when convergence is added to seriality — CD. Because the CD axiom is (gasp) <>A -> A. Hence, A <-> <>A. Necessity and possibility are the same.
Doesn’t that bother you? I mean, if I define God as necessary existence, then He exists axiomatically. No tableau is even required. So, what does necessity mean in CD, “true in every world (period)”? And doesn’t possibility mean the same thing?
Um, Lib, there is no possible world in which entities do not do what they are obligated to do.
If something is truly obligated to act in a certain way, then it acts in that way. If it acts differently, it’s not obligated.
Now, there’s the problem with your ontological proof of God: you’ve failed to take into account the impossible worlds. :rolleyes:
Why do you shun reasoning in the natural languages? Why can’t you see that you’ve shown nothing more than that if you assume something exists, you can show that something exists?
When you add a serial frame you recognize that if a statement is “true in every possible world” then it is also “true in every world where entities do what they are obligated to do”. (D) is weaker form of (M). You do not lose the stronger meaning of (M) by recognizing that the weaker meaning holds, too. It is only when we remove the reflexive frame (M) that the meaning of necessitation has to change.
Yep.
Now is the time for my traditional caveat about not being well-versed in modal logics, but I cannot imagine why I would want to accept (CD) except to study specific degenerate questions. A quick google turned up nothing useful on (CD), but the junction of (D) and (CD) looks very much like The Trivial System to my untrained eye.
Vorlon
In this world divorced parents remain obligated to proide for the care and support of their children. It does not follow that they all necessarily do so.
Well, if you define “obligation” to mean “necessity”, then obviously the two are identical. Neither English nor modal logic, however, defines the term in that manner.
An obligation is what one should do.
A necessity is what one must do.
We can say that a person is obliged to act in a certain way due to their nature. We can say that we’re obliged to reach certain conclusions given certain axioms.
The line between an obligation and a requirement is rather thin.
Does a logical argument determine what should be true, or what must be true?
Here’s the real question: do you see a purpose for having a label that indicates things which ought to be but are not necessarily so?
Many people find this concept useful. In deontic logic, it is indicated by the symbol “O”. In English, it is often indicated by the symbol “obligation”. Now, if you wish to eliminate that usage of teh symbol “obligation” from your vocabulary, you may certainly do so. The rest of us, however, are not obligated to do the same.
