# Rewriting a set of propositions that includes a circular proposition

"For any proposition P, if I believe that P then this paragraph (everything that is written between the quotes) entails that I believe that P.
I believe that I exist.
For any proposition P, if I believe that P then I believe that I believe that P.
For any proposition P, if I don't believe that P then I believe that I don't believe that P."

How can I write a set of propositions that doesn't include circular propositions and is logically equivalent to the set of propositions that are in the quotes?

• @Logikal Isn't there an entire modal logic built to reason deductively about beliefs? Commented Sep 12, 2019 at 18:10
• @Logikal "the symbolism in Math can be vague because the standards for what counts as a proposition has been lost ... Almost any nonsense would qualify as a proposition today" Huh? That's a strong claim, can you back it up at all? Commented Sep 12, 2019 at 18:18
• @Logikal Well, some modal logics are non classical, but when you say "model logic" you think of classical modal logic, which absolutely is bivalent. And non bivalent logics are still deductive Commented Sep 12, 2019 at 19:32
• @DanielPrendergast You might want to read some of the other answers by Logikal on this site - if their last couple comments didn't make it clear, they're not exactly arguing in good faith (and I'm bowing out at this point.) Commented Sep 12, 2019 at 19:47
• "Math was not invented to argue in the same sense lawyers would argue or activist for a cause would argue. That is closer to Rhetoric than math" if that'ds true, then Maths influence on logic can only be a good thing for philosophers. We want to be more precise and rigorous than activists in our argumentation Commented Sep 12, 2019 at 20:23

Here is the original set of propositions:

"For any proposition P:
If I believe that P then this paragraph (everything that is written between the quotes) entails that I believe that P.
I believe that I exist.
If I believe that P then I believe that I believe that P.
If I don't believe that P then I believe that I don't believe that P."

If I believe that P then from that alone I can derive by reiteration that I believe that P. What may be intended are the following three statements:

1. For all P if I believe that P then I believe that I exist.
2. For all P if I believe that P then I believe that I believe that P. [modal axiom 4 and normal reasoner in doxastic logic or principle of positive introspection]
3. For all P if I don't believe that P then I believe that I don't believe that P. [principle of negative introspection]

If that paraphrasing actually means what one intended to say, then it would be a way around the circularity.

• Your set of propositions doesn't entail that I believe anything because all of its propositions have "if I believe that P" at their start and it could be that for any P I don't believe that P. I'm going to assume that in proposition number 1 you meant to write: "I believe that I exist.". Even then your set of propositions doesn't entail that I don't believe that cats exist, whereas my set of proposition entails that I don't believe that cats exist (Subtitute P by "cats exist" in the contrapositive of my first proposition.) Commented Sep 14, 2019 at 13:44
• @avraham I think what you have in the first line is "I believe that P". If I added that then you would get by reiteration "I believe that P". I agree that I don't think the three propositions would entail "I believe that P" nor would the assumption "I believe that P" entail these three propositions. Also my first one included "I believe that P" because I assumed that was the antecedent of all of these propositions. I am not sure what it is the argument is trying to show so I offered a paraphrase of what I thought it meant. Commented Sep 14, 2019 at 14:16