Instead of saying “This sentence doesn’t have truth value 1, nor 1⁄2, nor 1⁄4, …” (which, even if infinitely long, would only work for countably many truth values), you could simply say “This sentence has truth value 0″, which is just as paradoxical, but the paradox also works for real-valued or hyperreal truth values.
Yeah, the logic still can’t handle arbitrary truth-functions; it only works for continuous truth-functions. To accept this theory, one must accept this limitation. A zealous proponent of the theory might argue that it isn’t a real loss, perhaps arguing that there isn’t really a true precise zero, that’s just a model we use to understand the semantics of the logic. What I’ll say is that this is a real compromise, just a lesser compromise than many other theories require. We can construct truth-functions arbitrarily close to a zero detector, and their corresponding Strengthened Liar will be arbitrarily close to false.
Not sure exactly what you’re getting at? I suppose you’re imagining extending the sequence to ordinal values? We can do this, sure, and still can have the same issue with sup/inf not being well-defined.
The story of 1⁄2, 1⁄4, … could be continued immediately to the infinite case, right? (And all the way up the ordinal ladder.)
Instead of saying “This sentence doesn’t have truth value 1, nor 1⁄2, nor 1⁄4, …” (which, even if infinitely long, would only work for countably many truth values), you could simply say “This sentence has truth value 0″, which is just as paradoxical, but the paradox also works for real-valued or hyperreal truth values.
Yeah, the logic still can’t handle arbitrary truth-functions; it only works for continuous truth-functions. To accept this theory, one must accept this limitation. A zealous proponent of the theory might argue that it isn’t a real loss, perhaps arguing that there isn’t really a true precise zero, that’s just a model we use to understand the semantics of the logic. What I’ll say is that this is a real compromise, just a lesser compromise than many other theories require. We can construct truth-functions arbitrarily close to a zero detector, and their corresponding Strengthened Liar will be arbitrarily close to false.
Not sure exactly what you’re getting at? I suppose you’re imagining extending the sequence to ordinal values? We can do this, sure, and still can have the same issue with sup/inf not being well-defined.