On second thought, I'm not a hundred percent sure that the statement "it is not unlikely that this statement is unlikely" is a contradiction. Someone help me... .

Can a statement be un/likely? I'm just going to start with that, because I'm not sure in this case what that means. Do you mean that: "This statement is unlikely to be true?" Sentences or statements can be true or false, or provable or not, but I'm not sure about "un/likely", other that its "un/likely to be true/false". So in this case, is the sentence unlikely to be true or unlikely to be false? Because that might change the resulting statement. (The other option of "un/likeliness" I thought of was "this statement is unlikely to be in existence" and I'm not sure what to do about that, or if its nonsense)

As for: if "it is false that "this statement is unlikely (to be true?)"" resulting in "it is not unlikely(to be true?) that this statement is unlikely(to be true?), I'm not sure they are equivalent. One seems to be a statement about another, self-referencing statement, and the other is a related but different self-referencing statement. That may result in some confusion. Also, in this case, its not clear that the two "unlikely's" mean the same thing: again, you'll have to clarify. Is it "unlikely to be true" that this statement is" unlikely" or "unlikely to be true"? Or is it unlikely to be false that this statement is unlikely to be false? (or any other combination)

Whether it is a contradiction or not depends on what you mean by "unlikely". If a self-referential sentence is attributed a truth value, it must reference its own true value to be able to lead to a contradiction. "This sentence is green" can equal true or false without resulting in a contradiction.

The final "its is unlikely(to be true) that this sentence is unlikely(to be true)--(I'm just going to assume you mean this) does seem to pose some sort of problem, but I'm not sure, as you are, that it leads to a contradiction, because the "unlikeliness" has some sort of "degrees of true and falseness" that aren't as black and white as traditional truth values. So you don't exactly have A and not A, but more of a probability that there is a probability of A.

It seems like you are dealing with some form of fuzzy logic, but I'm not well versed in that topic, though I've read a little. I'd love to tackle the logic of this further, but currently in that I am a novice. Hope this helps some.

I think you need to try your luck with many valued logics, since "unlikely" has no true/false interpretation.

Typically probabilities are ascribed to events. Can the statement be seen as an event? Maybe not. So in this sense it could be a category error.

There are attempts to ascribe probabilities to statements in general, like one by Carnap. In these contradictions are given probability 0 and tautologies are given probability 1. Assume the statement is false, then the statement is unlikely, then the statement is true - a contradiction. Assume the statement is true, then the statement is unlikely and likely, but that's a contradiction. Assuming it's true or false allows you to derive a contradiction, so this is essentially a mapping of the statement onto the liar statement. Whether you dismiss it as 'just another contradiction' at this point also depends on your philosophical taste (see Arthur Prior).

But with probability there are more than just true or false (seen as 1 and 0), so the logic need not derive a contradiction from the principle of excluded middle and analysis of the statement. It seems plausible that 'This statement is unlikely' could be ascribed a probability which isn't 0 or 1 without contradiction, but I don't know how to ascribe a probability to it.

Couldn't "unlikely" be another truth value in a 4th-valued logic in a logic with truth values: 0,1,2,3; where 0/1 are the usual true/false and 2/3 as unlikely/likely.

Spell it out, how do you formally reason with true, false, likely and unlikely?

Well, 0/1 are dual to each other as always with negation; and so 2/3 are dual to each other with negation.

Now for the conjunction,disjunction truth tables:
A| B | A^B
1 1 1
0 1 0
0 0 0
1 2 2
1 3 3
0 2 0
0 3 0
2 2 2
3 3 3
3 2 3

Where you can take '2' as 'likely' and '3' as unlikely.

As you can see you can expand on any 2n-logic which is really based on binary logic.

The difficulty arises with odd valued logics.

Iterated conjunction of 'likely' should produce 'unlikely' in some cases. Iterated disjunction of 'unlikely' should produce likely in some cases. I don't think it fits so neatly while representing how likely and unlikely events work together.

Iterated conjunction of 'likely' should produce 'unlikely' in some cases. Iterated disjunction of 'unlikely' should produce likely in some cases. I don't think it fits so neatly while representing how likely and unlikely events work together. — fdrake

eh?! Suppose something likely with 90% probability. Iterate it a bunch of times. .9 x .9 x .9.... The more you do it, the smaller the number, never larger! I'm sure you know this as well or better than I. So what have I misunderstood?

That's what I meant, partly. If 1000 events that are 99% likely happen, and they're independent, the probability that they all happen (iterated conjunction) is about 0.00004 - very unlikely. This shows that A and B can be likely individually but not together.

Consider 100 events with 10% chance to happen, and they're independent and disjoint, then the probability that at least one happens (iterated disjunction) is 1-0.1^100 = 0.9999... 99 times...9, very likely. This shows that A or B can be unlikely individually but that at least one happens is likely.

Disjunctions require one of their events to happen in order for the disjunction to happen. It's the 'at least 1' where the magic lies, since that's the same as 'not 0 times', if that makes it clearer. Saying A happens or B happens is always more likely than saying A happens alone.

Conjunctions require all of their events to happen in order for the conjunction to happen. So they're always less likely than their individual items and their disjunction.

First one of these is A and B, second one of these is A or B. The sizes of the shaded areas are a direct representation of their probability.

Thank you.

But where does the line between what is likely and what is unlikely lie?

Is every probability between 0 and 1/2 excluding the endpoints is "unlikely" and the complement is "likely".

Or do you have some other subjective criteria?

I think it's a Sorites paradox thing really. If I had to choose I'd say >0.5 is likely and <0.5 is unlikely. Really all this shows is what's expected: it's hard to map probability - something you could think of as an uncountably infinite valued logic - onto one with finitely many values.