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Is there a diff. between the propositions 'P' and 'P is true'?

Is there a diff. between the propositions 'P' and 'P is true'?
derrick.farnell
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Posted Mar 26, 2012 - 3:45 AM:
Subject: Is there a diff. between the propositions 'P' and 'P is true'?

Hello

Please could you help me with the above question.

For example, is there a difference between the propositions:

  1. Edinburgh is the capital of Scotland.

and

  1. The proposition 'Edinburgh is the capital of Scotland' is true.

?

Given that the second proposition is a proposition about the first proposition, it seems to me that they can't be the same proposition.

Instead, each merely directly implies the other.

That is, if Edinburgh is the capital of Scotland, then the proposition ‘Edinburgh is the capital of Scotland’ is true - and vice versa.

Is there a term for a pair of propositions that directly imply each other?

Derrick

fdrake
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Posted Mar 26, 2012 - 4:01 AM:

Propositions in the same language which entail eachother are logically equivalent. One thing to watch is that P and " "P" is true " aren't necessarily propositions in the same language, as P's being true isn't P. Even if P, then "P is true" is true, and if "P is true" then P, the entailments might be different. As in "P => "P is true"", as far as I'm aware, isn't a well-formed-formula of propositional logic, so the entailment we use there isn't the material conditional. P => P is true says more than P=>P.
derrick.farnell
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Posted Mar 26, 2012 - 6:16 AM:

Thanks very much for your reply fdrake.

You say 'Propositions in the same language which entail each other are logically equivalent.', but perhaps I was wrong to say that these particular propositions directly imply each other. That is, perhaps, instead of saying that P => P is true, I should have said:

  1. P
  1. If P, then P is true.

Therefore,

  1. P is true.


If so, then they don't entail each other, but require another premise.

Given this, would you therefore agree that 'P' and 'P is true' are distinct propositions?

As I said, given that 'P is true' is a proposition about the proposition P, it seems to me that they can't be the same proposition.

derrick.farnell
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Posted Mar 26, 2012 - 6:21 AM:

Sorry, upon re-reading your post, I see that you seem to indeed be agreeing that they're distinct propositions: 'P => P is true says more than P=>P'
Willemien
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Posted Mar 26, 2012 - 7:10 AM:

derrick.farnell wrote:
Hello

Please could you help me with the above question.

For example, is there a difference between the propositions:

Edinburgh is the capital of Scotland.

and

The proposition 'Edinburgh is the capital of Scotland' is true.
?

Given that the second proposition is a proposition about the first proposition, it seems to me that they can't be the same proposition.

Instead, each merely directly implies the other.

That is, if Edinburgh is the capital of Scotland, then the proposition ‘Edinburgh is the capital of Scotland’ is true - and vice versa.

Is there a term for a pair of propositions that directly imply each other?

Derrick


There are books written about this.
And all those writers disagee with eachother.

some opinions:

- Tarski : "is true" is not a predicate so "X is true" is not a logical statement.

- ? : they have the same logical content

- ? : the X is true is a stronger statement than X
derrick.farnell
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Posted Mar 26, 2012 - 7:52 AM:

Thanks Willemien.

Re your selection of other's opinions, I think that my opinion is not just different from the first two, but also from the third - that is, it's not that I think that 'P is true' is 'stronger' than P, but that it's just different, because it's a proposition about P, and so can't be the same proposition as P.
ughaibu
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Posted Mar 26, 2012 - 8:07 AM:

On the face of it, it seems to be obvious that they're not the same. Let P be the proposition X is false. X is false is not the same as X is false is true.
Owen
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Posted Mar 26, 2012 - 1:23 PM:

ughaibu wrote:
On the face of it, it seems to be obvious that they're not the same. Let P be the proposition X is false. X is false is not the same as X is false is true.


I don't agree.

p =df (p is true). ~p =df (p is false)

(p is false) is true <-> (~p) is true <-> ~p.
(p is false) is false <-> (~p) is false <-> ~(~p) <-> p.

Edited by Owen on Mar 26, 2012 - 1:38 PM
Pneumenon
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Posted Mar 26, 2012 - 5:45 PM:

Is the sentence, "Everything John says is true," equivalent to "Everything John says"?
Owen
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Posted Mar 26, 2012 - 6:52 PM:

Pneumenon wrote:
Is the sentence, "Everything John says is true," equivalent to "Everything John says"?


Of course..If all propositions that John says include p1, p2, p3, etc. then what John asserts is that
p1, p2, p3, etc., are true.
That is to say, (p1, p2, p3, ..) <-> (p1, p2, p3, ..) is true.

'Is true' is a redundent predicate of propositions.
(((p is true) is true) is true) <-> p.
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