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FAWS comments on Raising the forecasting waterline (part 1) - Less Wrong
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That's only reasonable if some agent is trying to maximize the information content of your answer. The vast majority of possible statements of a given length are false.
Sure, but how often do you see each of the following sentences in some kind of logic discussion: 2+2=3 2+2=4 2+2=5 2+2=6 2+2=7
I have seen the first and third from time to time, the second more frequently than any other, and virtually never see 2+2 = n for n > 5. Not all statements are shown with equal frequency. My guess is that the percentage of the time when "2+2 = x" is written in contexts where the statement is for a true/false logic proposition rather than an equation x = 4 is more common than all other values put together.
That's surely an artifice of human languages and even so it would depend on whether the statement is mostly structured using "or" or using "and".
There's a 1-to-1 mapping between true and false statements (just add 'the following is false:' in front of each statement to get the opposite). In a language where 'the following is false' is assumed, the reverse would be actual.
I'm not sure your statement is true.
Consider:
The sky is blue.
The sky is red.
The sky is yellow.
The sky is pink.
The sky is not blue. The sky is not red. The sky is not yellow. The sky is not pink.
Anyway, it depends on what you mean by “statement”. The vast majority of all possible strings are ungrammatical, the vast majority of all grammatical sentences are meaningless, and most of the rest refer to different propositions if uttered in different contexts (“the sky is ochre” refers to a true proposition if uttered on Mars, or when talking about a picture taken on Mars).
The typical mode of communication is an attempt to convey information by making true statements. One only brings up false statements in much rarer circustances, such as when one entity's information contradicts another entity's information. Thus, an optimized language is one where true statements are high in information.
Otherwise, to communicate efficiently, you'd have to go around making a bunch of statements with an extraneous not above the default for the language, which is wierd.
This has the potential to be trans-human, I think.
But whether a statement is true or false depends on things other than the language itself. (The sentence “there were no aces or kings in the flop” is the same length whether or not there were any aces or kings in the flop.) The typical mode of communication is an attempt to convey information by making true but non-tautological statements (for certain values of “typical” -- actually implicatures are often at least as important as truth conditions). So, how would such a mechanism work?
But, on the other hand:
The sky is not blue. The sky is not red. The sky is not yellow. The sky is not pink.
You need to be more specific about what exactly it is I said that you're disputing - I am not sure what it is that I must 'consider' about these statements.
On further consideration, I take it back. I was trying to make the point that "Sky not blue" != "Sky is pink". Which is true, but does not counter your point that (P or !P) must be true by definition.
It is the case that the vast majority of grammatical statements of a give length are false. But until we have a formal way of saying that statements like "The Sky is Blue" or "The Sky is Pink" are more fundamental than statements like "The Sky is Not Blue" or "The Sky is Not Pink," you must be correct that this is an artifact of the language used to express the ideas. For example, a language where negation was the default and additional length was needed to assert truth would have a different proportion of true and false statements for any given sentence length.
Also, lots of downvotes in this comment path (on both sides of the discussion). Any sense of why?
It's true of any language optimized for conveying information. The information content of a statement is reciprocal to it's prior probability, and therefore more or less proportional to how many other statements of the same form would be false.
In your counter example the information content of a statement in the basic form decreases with length.