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1. Originally Posted by Owl
You guys are, like, forcing me to think and stuff.

Not just a false statement, a contradiction.

If a set of statements includes a contradiction, and the members of that set are taken to be the axioms of a logical language, L, and both modus ponens and universal substitution are taken as primitive transformation rules of L, then L will contain all statements as theorems.

(You could use this language to derive A, where A has n operators, and you could use it to derive A with n+1 operators, i.e. ~A, AvA, A&A etc.)

Wait... You'd need to include certain PC valid theorems as axioms.

sigh...
Yup. Agreed.

Originally Posted by Owl
Which is just to say that if you accept the laws of identity, non-contradiction, and excluded middle as necessary/axiomatic, (i.e. derivable from the empty set), then there is only one contradictory hypothesis.
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?

2. Originally Posted by Owl
Which is just to say that if you accept the laws of identity, non-contradiction, and excluded middle as necessary/axiomatic, (i.e. derivable from the empty set), then there is only one contradictory hypothesis.
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.

3. Originally Posted by ygolo
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?
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).

4. Originally Posted by nocturne
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.
This is my understanding too. Cool, I think I interpreted you rightly.

5. AHHHH!!

I just realized that nocturne is an ESFJ!

6. Originally Posted by ygolo
But when you are imagining "complete" sets of hypothesis, you take things too far. Remember Godel's incompleteness theorem?
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.

7. 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.

8. Originally Posted by ygolo
2) However, it is not clear to me that C(A)=C(B) implies A=B.
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.

9. 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.

10. Originally Posted by ygolo
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 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.

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