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1. Originally Posted by nocturne
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.

I thought we understood each other on the original point, but it seems like there is still some confusion left.

What does it mean to "express the same hypothesis?"

If we are defining it to mean that:
"C(A) = C(B) iff A and B express the same hypothesis,"
then the statement,
"if C(A) = C(B) then A and B express the same hypothesis"
simply follows from the definition, and is devoid of real content.

2. Originally Posted by ygolo
Simply follows from the definition, and is devoid of real content.
Well, doesn't everything in logic and mathematics simply follow from definitions? In any case, that is why I couldn't understand your objections before on what seemed to me to be an uncontroversial issue. I defined a hypothesis by its logical content, not by any particular set of premises which can be used to express that hypothesis. That seems to accord nicely, I think, with what people generally mean by 'hypothesis'.

3. Originally Posted by nocturne
Well, doesn't everything in logic and mathematics simply follow from definitions?
Well, the deeper results usually have more steps. Generally we try to keep the # of defnitions used to a minimum.

In a way, we are trying to see the full implications of a small set of definitions.

That definition seemed non-standard.

Again, just an exercise in semantics.

Originally Posted by nocturne
In any case, that is why I couldn't understand your objections before on what seemed to me to be an uncontroversial issue. I defined a hypothesis by its logical content, not by any particular set of premises which can be used to express that hypothesis. That seems to accord nicely, I think, with what people generally mean by 'hypothesis'.
I suppose here we can agree to disagree, since that is a matter of preference.

4. ygolo,

I am still confused as to why we should 'agree to disagree'.

The statement 'every raven is black' can be said to express the hypothesis that every raven is black. The statement 'there does not exist a nonblack raven' can also be said to express the hypothesis that every raven is black.The same sentence can be written in English, French, German, Spanish, or whatever, and can still be said to express the hypothesis that every raven is black, it can be written in big or small letters, in one or many colours, by a string of 0s and 1s, or written in any font.

It is a common occurence that two scientists or mathematicians create the same hypothesis, but this is not recognised for sometime as they expressed the same hypothesis in different words. If, or when, this equivalence is discovered, nobody persists in identifying them as two different hypotheses, but rightly recognise them as two different methods of saying the same thing. If we investigate, analyse, or explore some hypothesis, we are studying its logical form and logical content, or its internal relations and external relations when conjoined with other hypotheses i.e. the logical content of conjuncting the hypothesis with another hypothesis.

This all seems quite uncontroversial.

5. Originally Posted by nocturne
This all seems quite uncontroversial.
I don't think there is a controversy just multiplicity.

Even though differential equations, and system-level modeling express the same information, they are thought off as different things and learned seperately. Many don't even realize the equivelence.

In short, we do persist in identifying and working with the two systems indpendently despite the equivalence.

Every mechanichal system has an equivelent electrcal analog (that is the electrical network analysis entails mechanical analysis) but constantly making the transformation is unnatural (even for those trained in the electrical domain first).

I just think it is a preference.

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