Hi Avi.

I would like to know if there any difference in meaning between these two propositions -

1) I do not believe in X
2) I believe in no X

This discussion has been moved over to the comments facility at:
http://www.thelogician.net/2_future_logic/2_chapter_20.htm

Something about you (optional) Logician

Hi Avi,

I know this is an old thread but I would be interested in reading your reply to zahc's question, but when I enter the url given in your reply I get a 'not found' error. Does the comments facility you mention still exist?

Hi Joe. The link given to zahc is now the following page:
http://thelogician.net/FUTURE-LOGIC/Credibility-20.htm#4.%20%20%20%20%20%20Opinion%20and%20Knowledge
The text referred to is Future Logic, chapter 20, section 4.
However, I do not remember what the comments facility mentioned was about.

But to answer the question, how to compare: 1) I do not believe in X, and 2) I believe in no X.

As I write in the said page of FL, "believing something does not imply disbelieving its contradictory, since people sometimes do (however ‘illogically’) believe both a thesis and its contradictory; therefore, disbelieving and not-believing are not identical. Also, ... believing something does not imply that it is true, and the true may be disbelieved."

Thus, for example, a person may at once believe in God (or a god), and disbelieve in same, implying some uncertainty or confusion within him or her. People's thinking in practice is not always 'logical'.

I am not sure whether zahc meant: I do not believe in some specified X versus I believe in no X whatever. Or maybe there was a typographical error, and he meant: I do not believe in X versus I believe in not-X.

I believe in some X, and I do not believe in that same X, are (to a logical mind) contradictory; although in practice they can exist in the same mind. Similarly, I believe in X, and I do not believe in X. That's because the copula's polarity (believe, do not believe) differs, while the predicate is the same (some X; or X).

I believe in some X and I believe in no X seem, on the other hand, contrary. Likewise, I believe in X and I believe in not-X seem contrary. In these cases, the copula (believe) has the same polarity, while the predicates (some X, no X; or X, not-X) have opposite polarities.

So, one might say regarding 1) I do not believe in X, and 2) I believe in no X, that 'logically' the latter implies but is not implied by the former. But to repeat, there is no firm logic in this context.

Something about you (optional) logician-philosopher

What is meant by 'logically'? - in accord with the norms, rules of logic - i.e. of opposition, eduction, syllogism, etc.

This signifies a distinction between the descriptive and the prescriptive. Beliefs CAN be adopted very illogically - BUT they SHOULD be made to follow logical criteria and rules.

If you think logically, then belief in X and disbelief in X are in principle incompatible. But the behavior of people in practice is sometimes to mix these beliefs. Even if contradictory beliefs may in fact be mentally impossible simultaneously at once, they are possible in the same person over time and place. The left hand may not be allowed to know what the right hand is doing...

Something about you (optional) logician-philosopher

Hi Avi, it reminds me of the distinction between 'absence of evidence' and 'evidence of absence', which can be quite subtle. I suppose you could say that 'absence of evidence' is an argument from ignorance, although it may not be necessarily fallacious. For me, the statement 'I do not believe in X' seems more natural because it is a negation, whereas 'I believe in no-X' (assuming that was what zahc meant), suggests a positive belief in everything other than X, which doesn't really make much sense if what you're intending to convey is a lack of belief (regardless of the evidence for that lack).

Yes I think we all hold contradictory beliefs because we haven't really thought through what we believe. It's common these days to hear about 'cognitive dissonance', which is supposed to be a kind of psychological stress which occurs when someone holds contradictory beliefs.

We agree.

Something about you (optional) logician-philosopher

Joe, to return briefly to the ambiguity of zahc's formulation of belief statements.

I think the best way to express them is: subject S believes that P, where P is any proposition (of categorical, conditional, or whatever form). With this we can better look at the various possibilities. As already said, in practice (i.e. descriptively) anything goes; people CAN and do believe just about anything they like, without any logic. But in principle (i.e. prescriptively) they SHOULD, i.e. OUGHT TO, follow logic, inductive and deductive, because only that will lead them to the best possible (i.e. most realistic, true) beliefs.

We can then say in relatively general terms (where P, Q, R, etc. are propositions), If S believes P, and logically P implies Q, then S should believe Q. Or If S believes P, and logically P is incompatible with Q, then S should believe nonQ. And so forth for all oppositions, eductions, syllogisms, etc. E.g. a syllogism: If S believes P and Q, and logically P and Q together imply R, then S should believe R. And so forth for any type of argument.

Now, what about the status of negative belief statements? S believes P and S believes nonP are contrary positive belief statements. Then their contradictories, S does not believe P and S does not believe nonP, are subcontrary positive belief statements. In short, if we want to be logical with our beliefs, these four propositional forms will make up the usual square of oppositions. But to repeat, in practice people are not necessarily that consistent.

Something about you (optional) logician-philosopher

Joe, upon reflection, one more detail that I'd like to add to my last post (which I assume you read).

There are, of course, degrees of belief - belief is not just an either-or matter. Thus, I may strongly believe something, and more weakly not-believe that same thing, without self-contradiction. and this is true not only 'in practice' (irrespective of logic) but also 'in principle' (following logic).

Thus, beliefs based on inductive logic, with reference to probabilities of truth, will have degrees. If something has a probability of only 70%, then my belief in it should have a 70% degree (strong), and my non-belief in it should have a 30% degree (weak).

Something about you (optional) logician-philosopher

Hi Avi, sorry about the late response. I've just moved house and have been without internet for a couple of weeks.

Regarding your last post, I think you're referring to so-called 'epistemic' probability, i.e. the interpretation of probability as a degree of belief, as opposed to other interpretations such as probability as an objective frequency. There has been a lot of heat generated over the years about which interpretation is the 'correct' one, but it seems to me that the differences have been exaggerated. To my mind, a degree of belief (say 70%) in a proposition must take into account the frequencies involved if it's to have a firm rational basis. Nor do I think there is any basis for a hard dividing line between probability and deductive logic.

For example, take the classic deductive syllogism :

All men are mortal
Socrates is a man
So, Socrates is mortal

This is valid and so it is impossible for the premises to be true and the conclusion false. An example of a 'statistical syllogism' is :

99.9% of men are mortal
Socrates is a man
So,Socrates is mortal

In this case, while it's possible for the premises to be true and the conclusion false, it's not likely. But what is so special about 99.9%, as opposed to 100%, which makes the first argument part of logic but the second not?

Anyway, I'm going off-topic.