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Started reading about beliefs and propositions, and the working definition I have of X believes P is that X thinks P is true. So suppose I don't believe P. Would it be more accurate to say I think P is false, or that I think ~P is true?

I know beliefs can be thought of as structured mental representations of the world that support other beliefs, which is why I would tend to the latter idea that my not believing P means that I think ~P. Since I can then infer or support other beliefs from this true belief of mine. So I may not believe in Santa Claus and if one night I see a man in a red suit climbing out of a chimney, the fact that I think the proposition "Santa Claus doesn't exist." is true is what actually supports my inferring the man I see is not in fact Santa Claus, but a regular human being or burglar or something.

Whereas if I merely took P (Santa Claus exists) to be false, I could not actually infer anything from this false proposition.

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You have a scope of negation problem, if "I don't believe that P". You could be denying having any belief, meaning that you are relatively agnostic as to P, which is appropriate if you haven't seen enough evidence. You could also be persuaded that it is actually false to assert P (you believe ~P). Taken as a whole, if P is false, then ~P is true -- assuming you recognize the law of the excluded middle. Then you would have to scrutinize the structure of P to see what else you can learn from knowing that ~P is true i.e. P is false.

  • I don't believe that P does not mean I don't have any belief in P, it means I think P is false or alternatively ~P. Not having a belief or denying having a belief P is true or false is being agnostic about P which is the attitude one has when they think the evidence doesn't support one hypothesis over the other. – Charlie Brown Jun 5 '15 at 11:35
  • My understanding was that (a)gnosticism was regarding knowledge, not belief. If I'm wrong, correct me please. – Joe Moeller Jun 12 '15 at 8:21
  • @CharlieBrown, I would have thought that's exactly what "I don't believe that P" means. We wouldn't hold someone to be in internal logical contradiction were they to say "I don't believe that P, but then again I don't believe that ¬P either". We would simply say they were agnostic or ambivalent as to the truth of P. – Paul Ross Jun 12 '15 at 9:57
  • That is the problem with the scope of modal negation in English. Note the weird disconnect between "You may not do that" which limits you, and "I may not do that" which expresses my freedom. Clearly any rule that allows for that is grammatical and not a logical convention. – jobermark Jan 25 '16 at 17:40
1

In English, negation of a modal statement naturally rises to the most controlling interpretation of the broadest phrase into which it might fit: "A does not believe P", without any quirky inflection, expresses "A believes P to be false" -- A believes (not P). Likewise "A does not want P" means A is actively opposed to having P, not that he lacks the wish to have P, but might not be opposed. On the other hand "A does not need P" really means that A is independent of P, rather than that A is dependent upon the absence of P. Whatever form gives the agent the most power is meant by default. So the vocabulary, and not the form, determines the interpretation.

(There are obvious exceptions, but the last statement remains true. Most ludicrously "You may not do that" forbids you, whereas "I may not do that" indicates my own freedom.)

But that grammatical fact is arbitrary, and the shifting about of the negation makes it hard to proceed in English. So to the degree that is part of the question, it is about English grammar/usage and not about logic.

If we step outside of English it is clearly possible to disbelieve P, or to fail to be convinced of P. In a sort of pseudomathematical form

not(believes(I, P)) != believes(I, not P)

In the broader symbology of logic, at this point one often whips out the 'box' which indicates some special interpretation of 'necessarily', in this case 'is believed'. ('Necessarily' because it is what seems necessary for this to be true in the mental world of the person in question.)

So there is a distinction between

A: ~[] p

vs

A: [] ~p

The former meaning that from A's point of view p is not proven and the latter meaning that from A's point of view p has been proven false

For every 'box' [] there is a 'diamond' <>, indicating some variant of 'possibly', in this case 'would consider'

So the corresponding phrases

A: ~<> p

and

A: <> ~p

Indicate respectively that A is not open to considering P, (which means he believes ~P) and that A is open to considering ~P.

There is an entire formalism for clarifying such things, the doxastic variant of Modal Logic, which you can look up anywhere.

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"Don't believe" can have at least two meanings other than believing the negation. First you might have no evidence, i.e. don't know: "The color of Napoleon's horse was white. Yes, No, Don't know?" Secondly, the question or statement might be such that even "don't know" would be incorrect or highly misleading. E.g. the classic "You have stopped beating your wife." is in this category. A possible answer/opinion might then be then be the Zen "Mu", which is in the direction of having no opinion (not even "don't know"), or unasking the question.

  • "I don't believe that the color of Napoleon's horse was white." seems to be quite a separate attitude or disposition than "I don't know that the color of Napoleon's horse was white." The first is an attitude to the proposition N "Napoleon's horse was white" which privileges the hypothesis N is false, while the 2nd is not an attitude to N nor privileges one truth-value of N over another. I don't think humans use the word belief to describe the 2nd attitude. – Charlie Brown Jun 4 '15 at 23:43

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