kodalai | too much thinking

I'm feeling slight logic rage.

In one of my textbooks, they were talking about recursion paradoxes, and they brought up the example of the cretan paradox. The one that one goes, A traveller on the road means a man from Crete who says, "All Cretans are liars." Supposedly this is a statement with no solution, because either the man is telling the truth and all Cretans are liars, in which case he would also be a liar and thus all Cretans are not liars, thus he would be telling the truth, etc.

Except that that's NOT TRUE. Because the negation of a universal positive is not a universal negative. If the statement "All cretans are liars" is false, then it means "Not all Cretans are liars," it does NOT mean "All cretans are not liars."

In other words, SOME Cretans are liars. This one is. (He must be, because according to the logic of the statement: if the statement "All Cretans are liars" was true, then he would be telling the truth, and as a Cretan who was telling the truth, would disprove the statement. If it was false, however, then not all Cretans are liars, but this Cretan is one, because he just lied.)

It drives me crazy because I see people making that linguistic mistake all the time. Someone makes, or implies, a universal statement: "all A are B." Then someone else contradicts it by saying "that's not true, all A are not B," when what they MEAN is that "not all A are B."

THE TWO STATEMENTS MEAN VERY DIFFERENT THINGS. "All fish are blue" and "All fish are not blue" are two entirely independant universal statements that bear no logical connection to each other whatsoever.

The textbook gave another example, which was also wrong. The example it gave was this:

The statement below is false.
The statement above is false.

It cited that as an example of a logical paradox, ie, the statements cannot accurately describe one another... but they CAN. It simply means that one of the statements is false and the other is true, and it doesn't really matter which is which.

Suppose you say "The statement below is false" is statement X, and "The statement above is false" is statement Y. (You can even reword it, for ease of reading, to say "Statement Y is false" and "Statement X is false.")

If statement X is true, then statement Y is false. If statement Y is false, then statement X is true. Conversely, if statement Y is true then statement X is false, and if statement X is false then statement Y is true. So they can indeed simultaneously accurately describe one another.

The CORRECT version of that paradox is as follows:

The statement below is false.
The statement above is true.

These CANNOT simultaneously accurately describe each other. If you reword it to: "Statement Y is false" and "Statement X is true," then you get this:

If statement X is true, then statement Y is false. However, it statement Y is false, then statement X cannot be true. And if Statement X is false, then Statement Y is again false, and thus the paradox.

Anyway.