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For The Uber Nerds
| Good [+1]Toggle ReplyLink» basdini replied on Fri Apr 9, 2010 @ 9:39am |
 Coolness: 145295 | if you don't know what the fuck you re talking about shut your word hole
however, if you do know what you re talking about i welcome any comments or suggestions you may have the hardest thing i ever wrote... *************************************************************** *************************************************************** *************************************************************** Kripkian semantics is a formal semantic for a language with (*) for necessity and (#) for possibility. It is sometimes called ‘possible world semantics’ or ‘relational semantics’. A Kripke frame F (or modal frame) for a sentential language is a pair (W, R), where W is an non empty set (the possible worlds), and R is a binary relation on W. The relation is usually interpreted either by means of ‘relative possibility’ or ‘accessibility’. The model M on the frame (W, R), is the triple (W, R, V), where V is the valuation function that will assign the truth values to the sentence letters at worlds. Given this, we can establish truth conditions for (*). If w € W then a sentence like (*)A, is true at world w in the model (W, R, V), if A is true at all worlds v € W. So, we may say informally, (*)A is true at world w, if A is true at all worlds that would be possible if w were actual. When speaking of the notion of validity, it can be applied to both the model M, and to the frame F. In the case of a model A is valid in the model (W, R, V), if it is true at all worlds w € W, in that model. In the case of the frame, it is valid in the frame (W, R) if it is valid in all frames. The accessibility relation is often explained in terms of the modal operators, which seems circular because it uses the very modalities it interprets to try to give an explicit analysis. Furthermore, issues regarding quantification in modal systems (specifically over different types of sets) are tied up in deeply controversial questions with regards to the nature of set theory and the status of natural kinds. A modal system typically uses (*) and (#) to mean something akin to (1)‘it is necessary that’ and (2) ‘it is possible that’. (1) and (2) can be defined in terms of each other, this is what is known as the ‘dual form’ or ‘modal inter-definability’ such that: (*)p = ¬(#)¬p (#)p = ¬(*)¬p Modal operators are one place connectives like negation. Modal operators don’t really express truth in the sense that we can’t give truth values (┬ or ┴) like other connectives, rather we must regard (*) and (#) as quantifiers over possible worlds. A possible world is to be understood as a complete way in which things might have been or might be. In set theoretic terms, (*)p means ‘is true in all possible worlds, (#)p is true in at least one possible world. We might conceive of a special world, call it w^, this world represents all the statements that hold true in the actual world, that is, the world we have all come to know and love. A statement like (*)(B→(#)(C٨D) is true at world w^ if (B→(#)(C٨D) is true at every world w. Either, B is false or else that at some other world u, C and D are both true. We can calculate the truth value of (*)(B→(#)(C٨D) provided we know the truth values of sentences B, C, D in every world. W is the set of possible worlds (including w^) and a specification for each world w, of the truth values of all sentences at each w. All this to say that what is necessary or possible may in fact depend on how the world actually is. Given the way things are, certain things may be possible that may not have been possible, had the world been a different way. An example may illustrate this some what better than clumsy natural language. I come from a particular sperm and egg. That is to say I come from S(1) and E(1). In a number of possible worlds I could have come from either a different S or a different E, but not a different S and a different E (S(1) and E(n) or S(n) and E(1) where n is any natural number greater than or equal to 2). This yields the following interpretations for: S(1)E(2) and S(2)E(1). Where S(1)E(2) = same sperm different egg, call this world u. And Where S(1)E(2) = different sperm same egg, call this world v. Thus it would seem that u and v are possible relative to the actual world, but strangely enough, not possible relative to each other. The reason for this, perhaps in this case, is that if I had both a different sperm and a different egg, I would cease to be myself, identity would not hold. The question arises then which worlds are possible relative to each other? To determine relative possibility in a model, we identify R with a collection of pairs of the form (u , v) where each u and each v is in W. If a pair (u , v) is in R, v is possible relative to u. If (u, v) is not in R, v is impossible relative to u. A simpler yet much more lurid example involves a cat that a friend has you take care of for a weekend, if the cat dies, taking it to the vet will not help. The cat has passed from one state to another (living to dead) and presumably the process is asymmetrical, that is as far as we can possibly know, for ordinary living things the passage from life to death is a one way street. This is to say that given certain states of affairs, certain other states of affairs are not accessible. Put differently certain facts about states of affairs prevent or impede the creation of other states of affairs. |
| I'm feeling surly right now.. | |
| Good [+1]Toggle ReplyLink» flo replied on Fri Apr 9, 2010 @ 11:20am |
 Coolness: 146420 | At first I thought it was for a paper (I mean, scientific publication) so I made some corrections about phrasings and explanations; but then I stopped when I got at the examples in the second part :)
I also inserted some comments between <<<chevrons>>>. It's interesting and I didn't know about that stuff; however I'm familiar with "Kripke structures", which seem to be almost the same thing as "Kripke frames/models", the former being a set-theoretical model and the latter appearing to me as a philosophical view of this same model. ========================= Kripkian semantics is a formal semantic for a language with (*) for necessity and (#) for possibility. It is sometimes called ‘possible world semantics’ or ‘relational semantics’. A Kripke frame F (or modal frame) for a sentential language is a pair (W, R), where W is an non empty set (the possible worlds), and R is a binary relation on W. The relation is usually interpreted either by means of ‘relative possibility’ or ‘accessibility’. A model M on the frame (W, R) is the triple (W, R, V), where V is the valuation function that will assign truth values to the sentence letters at worlds. Given this, we can establish truth conditions for (*). If w € W, then a sentence like (*)A is true at world w in the model (W, R, V) if A is true at all worlds w' € W. So, we may say informally, (*)A is true at world w, if A is true at all worlds that would be possible if w were actual. When speaking of the notion of validity, it can be applied to both the model M, and to the frame F. In the case of a model, A is valid in the model (W, R, V) if it is true at all worlds w € W, in that model. In the case of the frame, it is valid in the frame (W, R) if it is valid for any valuation function V. The accessibility relation is often explained in terms of the modal operators, which seems circular because it uses the very modalities it interprets to try to give an explicit analysis<<<I don't know how it's usually explained, but I think you can see it as a transition relation between worlds: R is a subset of the cartesian product WxW, and w R w' means that you can go from w to w', i.e. (V(w,(*)A) ٨ wRw') implies V(w',(*)A)>>>. Furthermore, issues regarding quantification in modal systems (specifically over different types of sets) are tied up in deeply controversial questions with regards to the nature of set theory and the status of natural kinds<<<I'm not sure what your point is in this sentence, since it's more like a philosophical discussion than a theoretical statement/scientific fact. By reading the end, I think you may want to do an analogy between (*) and Ɐ as well as between (#) and ⱻ>>>. A modal system typically uses (*) and (#) to mean something akin to (1)‘it is necessary that’ and (2) ‘it is possible that’. (1) and (2) can be defined in terms of each other, which is what is known as the ‘dual form’ or ‘modal inter-definability’ such that: (*)p = ¬(#)¬p (#)p = ¬(*)¬p Modal operators are one-place connectives, like negation. Modal operators don’t really express truth in the sense that we can’t give truth values (┬ or ┴) like other connectives, but rather that we must regard (*) and (#) as quantifiers over possible worlds. A possible world is to be understood as a complete way in which things might have been or might be. In set theoretic terms, (*)p means p is true in all possible worlds, and (#)p means p is true in at least one possible world. We might conceive of a special world, call it w^; this world represents all the statements that hold true in the actual world, that is, the world we have all come to know and love. A statement like (*)(B→(#)(C٨D) is true at world w^ if (B→(#)(C٨D) is true at every world w. Either B is false or, at some other world u, C and D are both true. We can calculate the truth value of (*)(B→(#)(C٨D) provided we know the truth values of sentences B, C, D in every world. Hence, we need a model (W,R,V) in which W is the set of possible worlds (including w^) and V is a specification of the truth values of all sentences at each w. All this to say that what is necessary or possible may in fact depend on how the world actually is. Given the way things are, certain things may be possible that may not have been possible, had the world been a different way. An example may illustrate this somewhat better than clumsy natural language. I come from a particular sperm and egg. That is to say I come from S(1) and E(1). In a number of possible worlds I could have come from either a different S or a different E, but not a different S and a different E (S(1) and E(n) or S(n) and E(1) where n is any natural number greater than or equal to 2). This yields the following interpretations for: S(1)E(2) and S(2)E(1), where S(1)E(2) = same sperm different egg, call this world u, and where S(1)E(2) = different sperm same egg, call this world v. Thus it would seem that u and v are possible relatively to the actual world, but strangely enough, not possible relatively to each other. The reason for this is perhaps that if I had both a different sperm and a different egg, I would cease to be myself; identity would not hold. A question then arises: which worlds are possible relatively to each other? To determine relative possibility in a model (W,R,V), we identify R with a collection of pairs of the form (u, v) where each u and each v are in W. If a pair (u, v) is in R, then v is possible relatively to u. If (u, v) is not in R, v is impossible relatively to u. A simpler yet much more lurid example involves a cat that a friend has you take care of for a weekend. If the cat dies, taking it to the vet will not help. The cat has passed from one state to another (living to dead) and presumably the process is asymmetrical, that is, as far as we can possibly know, for ordinary living things, the passage from life to death is a one-way street<<<You might as well cite Schrödinger!>>>. This is to say that given certain states of affairs, certain other states of affairs are not accessible. Put differently certain facts about states of affairs prevent or impede the creation of other states of affairs. |
| I'm feeling at home! right now.. | |
| Good [+1]Toggle ReplyLink» Kire replied on Fri Apr 9, 2010 @ 1:02pm |
 Coolness: 66795 | what ? lol |
| I'm feeling stringin right now.. | |
| Good [+1]Toggle ReplyLink» AlienZeD replied on Fri Apr 9, 2010 @ 1:18pm |
 Coolness: 509655 | someone's on crack. ;) |
| I'm feeling good in the hood right now.. | |
| Good [+1]Toggle ReplyLink» flo replied on Fri Apr 9, 2010 @ 2:21pm |
| Coolness: 146420 | oh so good crack :)
Originally Posted By FLO
I think you may want to do an analogy between (*) and Ɐ as well as between (#) and ⱻ>>>. The symbols might not appear correctly... I meant (*) is somehow like "for all", i.e. universal quantification, and (#) is like "there exists", i.e. existential quantification. I also forgot to say that I found the text rather well-written :) |
| I'm feeling at home! right now.. | |
| Good [+1]Toggle ReplyLink» greatjob replied on Fri Apr 9, 2010 @ 2:44pm |
 Coolness: 282550 | - \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.
The general solutions are: \psi(x) = A e^{ikx} + B e ^{-ikx} \qquad\qquad E = \frac{\hbar^2 k^2}{2m} or, from Euler's formula, \psi(x) = C \sin kx + D \cos kx.\! The presence of the walls of the box determines the values of C, D, and k. At each wall (x = 0 and x = L), ψ = 0. Thus when x = 0, \psi(0) = 0 = C\sin 0 + D\cos 0 = D\! and so D = 0. When x = L, \psi(L) = 0 = C\sin kL.\! C cannot be zero, since this would conflict with the Born interpretation. Therefore sin kL = 0, and so it must be that kL is an integer multiple of π. Therefore, k = \frac{n\pi}{L}\qquad\qquad n=1,2,3,\ldots. The quantization of energy levels follows from this constraint on k, since E = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2h^2}{8mL^2}. |
| Good [+1]Toggle ReplyLink» nothingnopenope replied on Fri Apr 9, 2010 @ 4:10pm |
 Coolness: 201320 | [ www.articlesbase.com ]
loved how trigonometry came roaring back when I was doing complex sound work. |
| I'm feeling meow right now.. | |
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