Yup. Agreed.
I still don't agree with this. Yes. All contradictory hypothesis have the same logical content.
But when you are imagining "complete" sets of hypothesis, you take things too far. Remember Godel's incompleteness theorem?
Yup. Agreed.
I still don't agree with this. Yes. All contradictory hypothesis have the same logical content.
But when you are imagining "complete" sets of hypothesis, you take things too far. Remember Godel's incompleteness theorem?
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield
If you want more than one contradictory hypothesis then simply delete the law of excluded middle i.e. where P v ~P is not implied by the empty set. That is why some have suggested that intuitionist logic be used for analysing and investigating different contradictory sets of premises, because with the law of excluded middle they all end up as the same thing. That said, I haven't looked into it much myself, so cannot elaborate further.
A criticism that can be brought against everything ought not to be brought against anything.
I may have missed some nuance in nocturne's argument, but his use of natural deduction seems to assume PC.
I interpreted nocturne's argument thusly: given a language in which all (and only) PC valid theorems are axioms, (enabling the use of natural deduction), then, if a contradiction is added as an axiom, then the resulting language is formally inconsistent.
Of course, if a language is stronger than PC (say, a mathematical language) then a contradiction can be taken a axiomatic without producing a formally inconsistent language.
I could be wrong. I'll think more about it.
edit (#2): needed to add (and only).
Last edited by Owl; 04-10-2008 at 03:02 PM. Reason: editing jargon
AHHHH!!
I just realized that nocturne is an ESFJ!
My head is about to explode!
I think that I understand your objection. There is an ambiguity in my definition of 'logical content', but I do not think it is anything other than a minor quibble.
If C(A) is the set of provable statements from A, then there will be statements about C(A) which are true, but are not provable from A. Therefore, C(A) will not capture every logical consequence of A. Is that correct?
I admit that my definition has been somewhat ambiguous in this regard, but I do not think that it has any serious consequences for my general point. In short, if C(A) = C(B), then the set of statements which are true but not provable from A are the same for B, which means that if C(A) = C(B) then they have the same logical consequences. This also holds for the true statements about the set of true statements which are not provable from A which are not provable, and so on and so on. There will be no logical consequence at any stage where which C(A) has and C(B) does not or vice versa--otherwise we run up against the law of identity.
Perhaps not. I am no mathematician.
A criticism that can be brought against everything ought not to be brought against anything.
I actually agree with you on the decidable parts of the last post, Nocturne, but there is still some things that stick out in my mind.
We can cicumvent the undecidable points by sticking to 0-th order logic.
1) Initiially, I was thinking axiaomatically, that is not using natural deduction. So I was thinking there could be implicit propositions in the derivation. That was the source of my objection.
But now that I know you are using propositional logic (0-th order logic), then it is much clearer what you meant. --thanks Owl
2) However, it is not clear to me that C(A)=C(B) implies A=B. This may be ture for non-contradictory hypothesis (infact my intuition indicates it is likely). But I hanven't thought about the proof.
IMO, what you presented is actual a counter example of "C(A)=C(B) implies A=B." That is the fact that two different cotradictory hypotheses can have the same logical content--the set of all propositions.
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield
If I wrote this then that was a mistake. In any case, I did not mean to imply this.
Obviously, A and B can be different and yet have equal logical content. For example.
P & Q = ~(P → ~Q)
If 'P & Q' is A and '~(P → ~Q)' is B, then clearly A and B are not the same, yet C(A) and C(A) are the same.
I do not identify 'hypotheses' as the particular statements which express it, but rather by its implications and content. If we say that we are going to investigate or analyse a hypothesis, we do not mean that we are going to analyse the statement which expresses the hypothesis, but rather its logical consequences and meaning. It could be said that we analyse the proposition, not the statement itself, and we recognise that the same proposition can be expressed by many different statements, not just in logical form (see the above example), but also language (English or French?), and by medium (pictures or writing?), etc.
A criticism that can be brought against everything ought not to be brought against anything.
Yet another case of working through semantics. I must learn to do this more proficiently.
Still, for 0-th order logic, it seems like one could (dis)prove:
C(A)=C(B) implies A=B. Of course, the (dis)proof-itself would likely need predicate calculus (1st order logic).
I have been running on very little sleep due to pressure at work, and still need to get some stuff done tonight. So I need to preserve my "technical thinking" till the weekend.
Accept the past. Live for the present. Look forward to the future.
Robot Fusion
"As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
"[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
"[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield
I need to clarify. I am not trying to suggest the following:
If C(A) = C(B) then A = B
That is not any part of my argument. This is:
If C(A) = C(B) then A and B express the same hypothesis
The sets A and B may differ in some way. For example, if A = {P} and B = {~~P} then A = B is false. However, they express the same proposition and have the same logical content.
If I said 'A is equal to B' I meant only with respect to the expressing the same proposition or hypothesis, and not that both sets of statements were identical.
A criticism that can be brought against everything ought not to be brought against anything.