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.
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.