I don’t see the difference between <> and just plain <>.
Well, one says that a possibility is necessary and the other doesn’t.
erl
I’m not sure I get the thrust of your quicky (and maybe I’d like to keep it that way ;). Seriously, you said that logic could not address “truth”. I asked why. Now you seem to be saying, “well of course it can, but what does that mean?” I think maybe I should wait for your reformulation before responding, since anything I say would seem to have a high likelihood of becoming moot once that happens. In the spirit of quicky, though, it means whatever we use it for.
An excellent question. The answer is: Yes, if we have done an accurate job in modeling T from “truth” and if the formal language/structure we use is well-chosen for the context in which the “truth” is understood. Our confidence in the first is built, as with all associations, through repeated trial and examination. Our confidence in the second is less simply outlined, but to my mind has aspects of both familiarity with the formal logic in question and a more "qualitative’ element present in our understanding of the “truth” being examined.
Not true. We have gained knowledge/understanding/new perception. You are correct that a deductive inference does not generate new information in the deductive system, but we are not the deductive system. Humanity gained much from the proof of teh Pythagorean Theorem, even though “it was already there”.
We may, if we have built sufficient confidence in our symbolic representation. I say, for instance, that I will have $634.57 in my checking account after the several checks I wrote last night clear (if I put no money in teh account before that time). I assert this by pointing to the mathematical symbols I used to make the inference, not by pointing to any physical collection of currency.
I don’t see your basis for either of these assertions.
I disagree quite strongly with the bolded section and thus with your conclusion. If I open a box and see a cat inside, does it follow that my opening the box added nothing meaningful since the cat was there the whole time?
Lib
My (D) was not derived, it is simply another possible axiom of modal logic. In this case, I took the form from here. Of course, as I noted (D) can easily be derived from (M) and (B), which places it pretty firmly within both B and S5. That derivation does proceed, as you surmise:
A -> A (M)
A -> <>A (B)
<>A -> <>A (M)
A -> <>A
Wait a minute.
That’s conflating K’s necessity with D’s obligation, and K’s possibility with D’s permission. They are merely analogous, not equivalent. OA does not imply A. And PA does not imply [symbol]à[/symbol]A.
System M merely adds (A[symbol]Ù[/symbol]B) [symbol]Þ[/symbol] (A [symbol]Ù[/symbol]B) to K to make S (S4 if necessity is redundant, or S5 if not). Then system B introduces A [symbol]Þ [/symbol][symbol]à[/symbol]A.
When we add D to K, our logic is no longer alethic. Since only non-necessary entities can be deontically judged, introducing an alethic modality into a deontic system is problematic. The Kantian principles of possibility and non-necessity imply directly that the only entities subject to deontic evaluation are contingencies.
Therefore, we can’t use deontic logic to evaluate truth.
Even without formal terms, it is simply intuitive that necessity does not imply possibility. A necessary world can be a metaworld comprised of all possible worlds, even when what might be possible in one world is not possible in another.
For example, it might be possible in one world that [symbol]p[/symbol] is not the ratio of a circle. Call that world W[sub]1[/sub]. But in our world (actual, and therefore possible) it is. Call our world W[sub]2[/sub]. From the reference frame of the metaworld (call it M):
M [symbol]Þ[/symbol] (W[sub]1[/sub][symbol]Ù[/symbol]W[sub]2[/sub])
even though W[sub]1[/sub] and W[sub]2[/sub] are contradictions within their respective reference frames.
Don’t you agree? Or else, where is my flaw?
I meant that you were using logic and reason to prove a point to someone who has denied them. When you say “Logic is meaningless” nothing can have any meaning. To say that is is meaningless is contradictory because it implies that some things have meanings, ie reasons. And if they have a reason, then their is a relationship between them, and we call the study of these relationships logic.
The OP is self-contradictory, and outside of pointing that out, it is pointless to debate with him.
Wow. That is an excellent point, Muad’Dib.
It is very similar to the assertion “You cannot prove a negative”. If the assertion is true, then the assertion is false!
:eek:
That would be an excellent point, Muad’Dib, if in fact I was arguing that informal logic was meaningless. Except I’m not. I am arguing, using informal logic, that formal logic is meaningless.
I HAVE NEVER DENIED INFORMAL LOGIC OR REASON. Indeed, it is the only logic I recognized as having meaning. It seems silly to say at this point, but I feel compelled to note it. I appreciate that I make mistakes in reasoning now and again, a do we all I imagine, but that is a little over the top.
However, I realized last night that I am indeed applying a double standard to formal logic. For it is so that when I write 1, 2, 3, … I mean for you to go on with 4, 5, 6, … and so on. And when I write p->q and then P you are to finish off with q. And so on. And I cannot do this without those things having a meaning, having a use, even if their use is not or cannot be spelled out in their own symbols. That seems to me now to be an unfair criteria for meaning to apply to a language.
This argument, in its entirety, is retracted.
… meaning to apply to constructions of a language (words, sentences, and so on).
And, of course, how could I forget?— :smack:
Tip o’ the hat to Eris. The next time someone claims that debate is futile in this forum, I will point them to this thread. You sir, are indeed a gentleman AND a scholar.
No, though you are correct that that is the analagous form in deontic logic. You asked for an axiom in modal logic which related necessity to possibility. (D) fits that bill. I also provided a link to a presentation including it in a list of possible modal axioms. I think that one usually does not see this in a strictly modal presentation because it is a weaker form of (M). It is most often seen in deontic logic because the analog of (M) clearly does not hold, so (D) is the “most” that can be reasonbly asserted.
The addition of (M) to K results in T. In fact, (M) is sometimes labeled (T) as well, so that might be the source of some confusion. I probably should have used (T), since I think there is no confusion of use for that label.
Actually, requires more than simply system M + K. It requires <>A -> <>A (or an equivalent axiom).
Yep, interestingly enough S4 + B will also produce S5, pretty much by an extension of the derivation I offered above.
It isn’t necessary to think of (D) in terms of obligation-permission. As I showed, the strictly alethic system T + (B) is sufficient to produce the desired result, which is really just a weaker form of (M/T).
Your intuition is very different from mine. If we read necessity as “for every world in W” and possibility as “for some world in W” then it is absolutely intuitive that implies <>.
The flaw is in thinking that A can be satisfied by an A that is not <> in world w[sup]k[/sup]. If w[sup]k[/sup] is part of W, then A cannot be in W if it is not <> in w[sup]k[/sup].
Yes, you can speak of a metaworld comprised of all possible worlds in W. It does not follow, however, that every statement in M is therefore in W.
on preview
Well done, erl. But that doesn’t mean we can’t keep arguing about truth models, does it? ;j
Erm? That’s a different use of possible than I thought modal logic dealt with. I thought possible meant, “The case in at least one world.” Shouldn’t this be so in every world? For wouldn’t something not being possible mean that there exists no world such that A is the case (A is impossible)? Possibility always seems necessary to avoid the contradiction of something being both possible and not possible simultaneously.
Please tell me I’m not way off base here…
Spiritus
I really don’t want to seem nitpicky, and I certainly don’t want to alienate you, but I just can’t accept an analogy for an implication.
And it isn’t just that. Whenever the weak axiom (weak in the formal sense, not weak in the lay sense), OA -> PA, is added to make D, it is the introduction of whole new primitive symbols, “O” and “P”. It’s not merely a new axiom added within the Kripke frame.
Finally, a deontic contingency is not a modal truth any more than is a doxastic belief. So, OA -> PA just doesn’t suffice, at least not for me, as an axiom that A -> <>A either definitively or by reason.
Yeah, I looked at the link. And quite honestly, it took me some time to sort through the errors in spelling and grammar there, along with the non-standard notation, to figure out exactly what he was saying. 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.
Okay, and yeah, I agree with you there. But in this case, the “most” isn’t good enough.
I disagree. And I’ve never seen it asserted that way. In fact, the whole purpose of deriving D is to force obligation into permission, and even then there is significant dissension among deontic logicians whether D itself is sufficient for that. Some say that O(OA -> A) is required as well. Deontic logic is too controversial to say that we can simply paraphrase its base axiom and call it an inference from K (or higher Ks like S).
Yes, but that’s a weak and arbitrarily restrictive reading of the terms, in my opinion.
I already gave you an example of how a modal possibility for some world in W might be a modal impossibility for a different some world in W (where W is a metaworld).
If it were to hold that necessity axiomatically implies possibility, then the implication is that there is only one W (since everything that is necessary has to be possible). Since our W is actual, it has to be possible, and therefore can be the only W.
That, in turn, makes the implication biconditional (!!!), rendering necessity and possibility identical.
Sorry, but I don’t really follow you there. Could you explain?
I don’t think I said that. In fact, if I said that, I’d be disagreeing with myself.
All I’m saying is that if M exists necessarily, then there may be arbitrarily many Ws that are both possible and contradictory among one another, but not contradictory in M.
Eris
I think my example of the two worlds where the ratios of circles are different explains the point. But I’ll try to answer your question with a series of questions in the hopes that it will illustrate the point better.
Let’s call you a metaworld. Now, suppose that your head is in the oven (one world) and your feet are in the refrigerator (another world).
From your perspective, are not both hot and cold possible? But is cold possible from the perspective of the oven, or hot possible from the perspective of the fridge?
Formal logic is how we should think in precisely the same way that formal wear is how we should dress.
Yes. Why wouldn’t one world, where A is the case, be able to recognize that ~A could also be the case, just not here? Why does <>A forbid <>~A? Why should it? In fact, if A is not necessary, and <>A, then can’t we also conclude that <>~A? That, in fact,
<>~A?
I feel like I am really missing something here. It seems to me that the necessity of possibility of something implies the necessity of the impossibility of its opposite. To use your pi example, in w[sub]1[/sub] the infinite series that we can use to represent pi does not correspond to twice the unit circle’s circumference (or whatever). That is, ~A. This world should also be able to postulate that there is a world where A. This means <>A and <>~A. But all this means is ~A (or ~~A, which is one of our axioms!).
So I still don’t see the problem.
Well, let’s start at the beginning. <>A does not preclude <>~A. It precludes (or rather, contradicts) ~<>A. Those should be read like this: <>(~A) and ~(<>A) — note that the latter, not the former, is the negation of <>A.
Second, a mere phenomenological declaration is insufficient for possibility. In our world, an actual and therefore possible world, a square circle is not possible, even though we can say the words “square circle”.
So, now you want to ask, “but is a square circle possible in any world?” Maybe, but maybe isn’t good enough to posit. We may infer the possible from the actual, but not from the necessary. And that’s rather the whole point of my current debate with Spiritus.
The only reason I was able to posit the two pi worlds is because ours is a submetaworld where we have globes, flat planes, and saddles. The radius/circumference ratio is different for each kind of surface — only one of them is pi. Since our world is actual, I could posit the possible.
A (actual A) does imply <>A (possible A). But A (necessary A) does not.
But if A->A and A-><>A then A-><>A, no?
Well, yes, it would. But the M Postulate (A -> A) is not provable in K.
Ah, gotcha.
Yes, because deontic logic is specifically developed to address obligation and permission rather than necessity and possibility. That does not make the modal form of (D) “out of bounds” as an axiom for a modal logic. Deontic systems, in fact, are a subset of modal systems. The change in symbology is arbitrary, but it serves to illustrate that teh formal languages are intended to map different types of “real world” questions. Unless one distinguishes the formal qualities of O from and P from <>, then the change in “how we think of the symbol” is not pertinent as a litmus test for “what qualifies as a modal axiom”. Again, this point is rendered moot since the weaker claim is easily derivable from “standard” modal systems.
I don’t see that at all. In fact, since I have given a simple derivation (others are possible if we accept S4 or S5) I cannot fathom why you would make this claim. Please demonstrate the fallacy in my derivation before continuing.
For the record, what I do see on this site (a couple sentences abov ethe tag for “Deontic Logics”) is the sentence: "[](A -> <>A) is already a theorem of M". (BTW, I see that this site also uses (M) to indicate the Modal Axiom which defines T, so that sentence could be read “theorm of T”.) And, of course, (A -> <>A) -> (A -> <>A) in T, by that same defining axiom.
Really–it is the same reading that you proposed for teh terms in your oft-referenced “ontological proof of God” thread, unless memory fails me. Still, tell me what readings of [] and <> you now prefer to use and I will see if our intuitions suddenly coincide. (I hope not, though, since the relationship in question quite cleary holds in early all modal logics.)
Yes, but that is hardly germane. The question is whether a modal necessity in some world of W might be a modal impossibility in some other world of W.
Not true. This would hold only if possibility implied actuality, which I do not believe either of us is willing to assert.
I don’t see how you close that chain at all. Yes, what is actual in w[sup]k[/sup] must be possible in W. Please show me how you propose to proceed from that to the conclusion that W holds only one w.
Well, consider your hypothetical example. “[symbol]p[/symbol] is the ratio of a circle” is not a necessarily true in W; it is possibly true in W. Your flaw is in equating a possible statement in W to a necessary statement in M. That relationship doesnt hold.
Perhaps you are relying upon the rule of necessitation (A -> A) here, but I would say that the metaworld you construct then palces us quite squarely in the midst of paraconsistent logic. And if we are in a paraconsistent logic, then there is obviously no telling weight to the contradiction you reach.
Okay, but that has no particular consequences for the necessity of statements within W, which is what we are concerned with in deriving (or asserting) that A -> <>A.
erl
Yes.
And on preview
That’s a bit dodgy, Lib, since you started out this whole modal discussion with even with common S4, how am I going to model an infinitely recursive entity?
S4 (and nearly all other useful modal logics) asserts (M/T) axiomatically. Now, if you would like to try and make the case that truth must be modelled exclusively in K as opposed to any of teh more useful modal logics, please feel free. But please don’t act like the context of this discussion has been limited to K when you explicitely began it in S4 and I have explicitely mentioned using both (M) and (B).