# Re: Fitch's paradox and OWA

From: vldm10 <vldm10_at_yahoo.com>
Date: Sun, 3 Jan 2010 07:00:41 -0800 (PST)

On Jan 2, 10:57 pm, stevendaryl3..._at_yahoo.com (Daryl McCullough) wrote:
> vldm10 says...
>
>
>
> >On Jan 2, 4:14=A0pm, stevendaryl3..._at_yahoo.com (Daryl McCullough) wrote:
>
> >> W = the type of possible worlds
> >> A = the type of atomic propositions
> >> P = the type of all propositions
>
> >I am not sure that propositions are types???
> >Let me give you the following example:
>
> >This sentence is false.
>

I guess that some objects can be treated as types if they have some characteristics in common. We expect the propositions which will have same properties of concern to logic, i.e. propositions are types. This is not the case with Liar paradox.
The liar paradox contains a sort of self-reference and the predicate ‘- is true’ and it is applied to name its own sentences. This paradox is important, for example “in proving the first incompleteness theorem, Gödel used a slightly modified version of the liar's paradox”

> In the higher-order type theories that I know of, the liar is not
> expressible (which is good, since it would lead to a contradiction).

There are self-references without predicates ‘- is true’. Let me give you two examples which are related to other kind of the selfreference:

Example1.
Here we have two sentence:

Tom is a mathematician. Tom is a mathematician.

They have the same truth value in any model.

Example 2.
I have two sheets of paper. One is marked with P1 and another with P2. I will put the following sentence into every of the paper:

The sentence which is on paper P1 have red letters.

So on each of the mentioned paper there is the same sentences and nothing else.
However semantically these sentences are not the same.

(These examples are inspired from the following two books: John Burdian on Self-Reference by Hughes, G.E; Classical Mathematical Logic by R.L. Epstein)

It is interesting to find models for the proposition that contains self-referencing.
Regarding abstract objects it is also interesting the following question: Can the propositions come to an existence and cease to exist?

>
> --
> Daryl McCullough
> Ithaca, NY