I want to use some notion of the power of logic/synonymy relation here. We can always have a vacuous synonymy relation—then all valid substitions preserve truth. Something like propositional logic would be more powerful, and quantificational logic more powerful yet.
Consider how much we can do in Peano arithmetic because of the power of the quantifier. We can use induction to deduce many things about divisibility, the solutions of Diophantine equations, and onward to large parts of mathematics.
For a less precise example, consider reasoning only in coordinates, or with the ability to perform a change of coordinates. A change of coordinates gives a synonymy relation, which we may find simplifies an argument significantly, or lets us “really understand” what’s happening. The picture that I have is that of a weaker and a stronger logical system—in the weaker system, we can’t express the notion of a change of coordinates, but in the stronger system we can. In fact, it’s doubtful whether there’s a reasonable language in which we can express many other thoughts, but not changes of coordinates, not even by some “coding trick”. So, we can take this example as a metaphor, or as a premonition of a generalization of the idea struggling to come into existence. I hope it is still able to shed light on what I mean by the power of a synonymy relation.
Anyway, here you want a normative direction pointing from vaguer languages to crisper languages. The crisper languages are better because they have synonymy. As you say “you don’t really understand something unless you can translate it into a transparent-context description”.
I want to construe this as about the power of synonymy relations. Crisp languages are better because we can use them to have interesting sequences of thoughts, because their synonymy does something for us. And so not just any synonymy relation are better—powerful ones are better.
I strongly doubt, as may be clear, the possibility of a “simple” definition that could measure power in terms of e.g. elementary syntactic criteria, but I think that we still have to admit that it’s what we want here.
I have a feeling that something is missing from the Łukasiewicz stuff you’ve been doing because I don’t think that assigning a quantitative degree of truth is all that vagueness really ought to be about. Vague sentences shift their meanings as the context changes. Vague statements can become crisp statements, and accordingly our language can gain more powerful synonymy. As a slogan perhaps, “vagueness wants to become ambiguity”.
I want to use some notion of the power of logic/synonymy relation here. We can always have a vacuous synonymy relation—then all valid substitions preserve truth. Something like propositional logic would be more powerful, and quantificational logic more powerful yet.
Consider how much we can do in Peano arithmetic because of the power of the quantifier. We can use induction to deduce many things about divisibility, the solutions of Diophantine equations, and onward to large parts of mathematics.
For a less precise example, consider reasoning only in coordinates, or with the ability to perform a change of coordinates. A change of coordinates gives a synonymy relation, which we may find simplifies an argument significantly, or lets us “really understand” what’s happening. The picture that I have is that of a weaker and a stronger logical system—in the weaker system, we can’t express the notion of a change of coordinates, but in the stronger system we can. In fact, it’s doubtful whether there’s a reasonable language in which we can express many other thoughts, but not changes of coordinates, not even by some “coding trick”. So, we can take this example as a metaphor, or as a premonition of a generalization of the idea struggling to come into existence. I hope it is still able to shed light on what I mean by the power of a synonymy relation.
Anyway, here you want a normative direction pointing from vaguer languages to crisper languages. The crisper languages are better because they have synonymy. As you say “you don’t really understand something unless you can translate it into a transparent-context description”.
I want to construe this as about the power of synonymy relations. Crisp languages are better because we can use them to have interesting sequences of thoughts, because their synonymy does something for us. And so not just any synonymy relation are better—powerful ones are better.
I strongly doubt, as may be clear, the possibility of a “simple” definition that could measure power in terms of e.g. elementary syntactic criteria, but I think that we still have to admit that it’s what we want here.
I have a feeling that something is missing from the Łukasiewicz stuff you’ve been doing because I don’t think that assigning a quantitative degree of truth is all that vagueness really ought to be about. Vague sentences shift their meanings as the context changes. Vague statements can become crisp statements, and accordingly our language can gain more powerful synonymy. As a slogan perhaps, “vagueness wants to become ambiguity”.