How understand abstraction when some cases can’t be abstracted?

Question

Like the liar sentence “this sentence is false” is said not to be a proposition.

So not all sentences can be abstracted into props. Can infinite sentences be abstracted into propositions. Can infinite not be? I think so to both.

So what we really have is some infinite set of sentences which can and another which can’t be propositionalized. The abstraction doesn’t seem to reduce the size of the infinity or measure of English sentences.

I’m not saying abstraction is not useful, but I understand the overall process of propositional abstraction more as a function or transformation. I am struggling to accept abstractions actually occur.

You might just say well abstractions are just special functions which generalize by omitting unimportant details, what’s the problem if there are edge cases? My problem is abstractions establish a new domain, they aren’t actually recursive or reducing. We can’t just go work in prop logic and be done with English sentences. Abstracting is actually increasing the domains of study.

An analogy from biology is that nothing self-replicates or is completely autopoetic (Kauffman). Not DNA, RNA, any species nor individual. DNA is not an abstraction of us because it has helper molecules not part of it.

I hope I’m picking out something that at least makes sense even if no one agrees. Is this still abstraction even if it does all the things and has the problems I’m claiming it does above?

I didn’t bring up math so maybe there are better abstractions.

Answer 1

Abstractions throw away details for tractability. For example, (regular two-value) logic throws away all sentences that don't have a binary truth value so that we can define logical operations, rules of inference, and the like. There are actually two important costs of this abstraction. First is the cost that we cannot deal with sentences like the Liar within the formalism. The second is that we loose distinctions. For example when you reduce "The sun is hot" and "The moon is wet" to predicate logic, they both become p(x). You have lost the distinction between them. You can recover the information that they are distinct by using different symbols, p(x), q(y), but you have still lost a lot of content. Because you have lost content, you can no longer infer things from the abstracted form that you could have inferred from the original. For example, you can't infer that x is more massive than y, although you could infer that the sun is more massive than the moon.

So the end result is that we gained some abilities and lost some abilities. More detailed abstractions might let us gain more and lose less; for example, there are three-valued logics that can handle the Liar as a proposition. However, in the end every abstraction (by definition) throws something away. It's a tradeoff, but when done right it's a very useful tradeoff.