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It seems to me that the proposition: "I am skeptical of everything" is synonymous with "I don't believe in anything".

But obviously in Doxastic logic there must be some form of axioms that say that the above statement results in contradiction.

I mean if you say the following:

  1. I am skeptical of everything.

and assume that "everyone believes in something" is an axiom in doxastic logic, then we can infer that

  1. I believe that I am skeptical of everything

results in a contradiction since if "I am skeptical of everything" is the same as "I don't believe in anything" then we get a contradiction with 2, which we inferred.

Now, to my question how would you symbolize this line of reasoning in Doxastic Logic?

Appreciate your input.

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    The statement may contradict the axioms of doxastic logic, but that does not make it contradictory, a skeptic is under no obligation to accept the axioms. Also, a consistent radical skeptic declines to assert (or deny) anything, literally, which is subtly different from asserting "I don't believe in anything". That is a sort of surmise by outsiders on their behalf.
    – Conifold
    Sep 11 '16 at 21:43
  • If you trully skeptical of everything then wouldn't it mean that you don't believe in anything? I mean what will amount of you being skeptical?, skepticism is a sort of belief assertion. You either believe or you don't believe, the law of excluded middle. Sep 11 '16 at 23:34
  • As you can guess, a skeptic is not bound by the excluded middle either. If you are asked whether say the string theory is true, yes and no are not the only options. You could also say that the jury is out at this time, and judgement should be suspended. This is skeptic's position on every issue, and indefinitely. Radical skeptic can really function only in a company of non-skeptics, by taking up their claims and showing them inconsistent or unfounded, otherwise he should remain mum in eternal suspension. This is rightly criticized as impractical and insincere, but it is not self-contradictory.
    – Conifold
    Sep 12 '16 at 18:24
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Skepticism and denial are two separate things. "I don't believe in anything" is denial, whereas a skeptic would more likely say "I am doubtful of everything." Skeptics have beliefs, but have reservations about those beliefs, and are willing to change those beliefs given the right proof/compelling argument.

So axioms may be accepted not necessarily as self-evident, but more as strongly believed assertions, while maintaining that they may still be wrong.

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Axiom: Everyone believes in something.

P1: I am skeptical of everything.

Conclusion: I believe that I am skeptical of everything

This conclusion cannot be drawn.

Even if we assume that "I am skeptical of everything" is synonymous with "I don't believe in anything", we cannot infer "I believe that I am skeptical of everything". The axiom ensures only that the individual making the assertion has some belief or other. It does not ensure that every statement within L is a belief.

I think you're right to assume that the axiom above would prevent someone from coherently asserting that they don't believe in anything. But to be honest, I'm a little unsure why you're attempting to rely on the inference (the conclusion), because we can obtain a contradiction prior to inferring anything if we assume that the two statements you consider equivalent are actually equivalent.

With unclear terms such as "skepticism", we'd be much better off talking about more precise phenomena such as withholding judgment, doubting the truth of some statement, or asserting that every statement is potentially falsifiable.

Also, I'm not sure how mainstream philosophy or logic would respond, but I think we have good reason to refrain from considering perceptual reports (There is a red car) or reports of outlook (I am skeptical of everything) as being on par with those things we normally think of when we refer to "beliefs": our attitudes toward some subjectively unverified assertion. If this is right, then it would be senseless to say "I believe that I am skeptical of everything"; you'd be, in a sense, unnecessarily tacking an attitude onto another attitude.

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