“when you have the statement refer to itself, you get a paradox” is not necessarily true. For example, the statement “this statement has five words” is self-referential and true. No paradox. Even a self-referential statement that includes its own truth value can be non-paradoxical: “This statement is true and has two words” is merely false.
By the way, this leads me to consider Prior’s resolution as somewhat problematic:
“This statement is true and has eight words”
“This statement has eight words”
The first statement is true and the second false, hence they cannot be equivalent. Nevertheless, adding “This statement is true and ” to any statement should not change the statement’s truth value if we accept that every statement implicitly states its own truth.
That’s not really a problem with Prior’s resolution. Rather, it’s a different problem with self-reference, which appears whether we adopt Prior’s resolution or not.
Compare: “P” and “P and P” are usually equivalent. But
“This sentence has five words.” and “This sentence has five words and this sentence has five words.”
don’t have the same truth value. The problem seems to be that the meaning of “this sentence” isn’t the same in the two ostensibly equivalent sentences. Whatever your favorite solution of this problem is, it seems that Prior could just graft that solution onto his own.
Prior’s solution to the liar paradox needn’t solve all paradoxes of self-references. As long as his solution is compatible with other solutions to other paradoxes, Prior has still contributed something of value.
Just a minor point:
“when you have the statement refer to itself, you get a paradox” is not necessarily true. For example, the statement “this statement has five words” is self-referential and true. No paradox. Even a self-referential statement that includes its own truth value can be non-paradoxical: “This statement is true and has two words” is merely false.
By the way, this leads me to consider Prior’s resolution as somewhat problematic:
“This statement is true and has eight words” “This statement has eight words”
The first statement is true and the second false, hence they cannot be equivalent. Nevertheless, adding “This statement is true and ” to any statement should not change the statement’s truth value if we accept that every statement implicitly states its own truth.
That’s not really a problem with Prior’s resolution. Rather, it’s a different problem with self-reference, which appears whether we adopt Prior’s resolution or not.
Compare: “P” and “P and P” are usually equivalent. But
“This sentence has five words.” and “This sentence has five words and this sentence has five words.”
don’t have the same truth value. The problem seems to be that the meaning of “this sentence” isn’t the same in the two ostensibly equivalent sentences. Whatever your favorite solution of this problem is, it seems that Prior could just graft that solution onto his own.
Prior’s solution to the liar paradox needn’t solve all paradoxes of self-references. As long as his solution is compatible with other solutions to other paradoxes, Prior has still contributed something of value.