Has there been much work done on the notion of approximate truth, for example under what rules of inference approximate truth is preserved, or what kind of metric one could use to say that proposition X1 is 'truer than' proposition X2?

Yes, there's quite a good deal of work on this kind of thing. It tends to go under the name "fuzzy logic" or "degree theoretic logic". The Stanford Encyclopedia entry is a good place to start. There's a fairly recent paper by Brian Weatherson called "True, Truer, Truest", if I remember right, that does some work on the philosophical foundations, which have always been a bit unclear.

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