2014-07-16

Referential transparency

This is another language post, since referential transparency is a given in mathematics. It is more general than the previous ones, and doesn't mention my new language or Lojban at all. This serves as a consolidation of my opinions from a conversation with la .az. about Quine's Quantifiers and Propositional Attitudes. I use Death Note for examples throughout, but I only reference events up to episode/chapter 2. No spoilers!

The essence of referential transparency in logic is that if two entities are identical to each other, there are no functions which can be applied to them that give different results. All languages, I believe, can achieve referential transparency under the right interpretation. Spoken languages and non-transparent programming languages will often rely on implicit information to tell apart predicates or arguments, but once all of this information is added, things will work out.

To see how referential transparency can work, consider “I believe that Kira has killed people” and “I believe that Light hasn't killed anyone”. Na├»ve interpretations relying on referential transparency would claim that, since Kira and Light are the same person, these beliefs are contradictory, and the speaker is being irrational. However, that's not the case. The speaker doesn't know that they are the same, so can legitimately hold both views.

At this point, Quine gives up on referential transparency, inventing the idea of “opaque contexts”, in which values can't be substituted freely. But I think that this is a bit extreme. If Yagami Souichirou is speaking, he can substitute “Light” for “my son”. Matsuda can also substitute “Light” for “Yagami's son”. The key problem in their reasoning is that they don't know that Light and Kira are the same. Words like “believe” do induce a context, but not an opaque one.

So, we are forced to concede that “Light” and “Kira” refer to different entities, since there exist functions to tell their referents apart. But they are, in some sense, the same. To the reader/viewer's logic, and to Light's own logic, they are indistinguishable. To formalise this, I use the notions of “referrers” and “phenomena”. The words “Light” and “Kira” refer to distinct referrers. We make all of our claims on the referrer level, since this is as much as we can know. Referrers contain information about claims people have made about them. Some of these claims are that they are identical to another referrer. If a person believes that two referrers refer to the same phenomenon (i.e, believes that they are “the same”), they can unify their knowledge about both referrers. Light knows that the referrers “Light” and “Kira” both refer to the same phenomenon, but Souichirou doesn't. Hence, Light knows that he's killed people, but his dad doesn't.

Furthermore, early on, Light believes that Lind L Tailor is L. According to him, both of those referrers have the same referent, so he believes that killing Lind L Tailor will kill L. When he realises that they are, in fact, different, he is able to separate all of his claims about L from his claims about Lind L Tailor. The obvious one is “Lind L Tailor is dead. L isn't.”.

Naturally, unification of knowledge about different referrers is done via fuzzy logic. In this post, I've used clear-cut examples to illustrate the points. There are many cases of this in real life, like aggregating experiences with friends. Each time, you're pretty sure that they're who they were before. But L, thinking that Light might be Kira, would be rash to completely unify all of his knowledge from those two referrers, but would be overly cautious to not do it at all.

1 comment:

  1. 0) Thanks for sharing your thoughts with us! It's refreshing and helpful to see these problems approached from an intellectual background different from my own. Is your theory of reference a novel creation? Does it have any points of contact with the existing academic literature?

    1) As a reader who only partially remembers Death Note, it would be helpful if you gave a recap of the relevant facts of the story as far as your examples go.

    2) If I understand correctly, you are giving a three-part theory of reference in which noun phrases are related to objects by an intermediary abstraction called a referrer. The nature of these referrers depend partially on the speaker. Under what precise conditions are two referrers identical?

    3) It's not entirely clear to me on what grounds you object to referentially opaque objects; I can see a mark against them on grounds of metaphysical extravagance. They appear to be objects fabricated to solve a philosophical problem which comes up when trying to formalize ordinary reasoning into certain logical systems, and are probably not in a typical language user's pre-philosophical ontology. I think a good way to say this is that referentially opaque propositions are distinct from ordinary ones in that they include certain contextual information baked into them which ordinary claims lack, but which is very helpful for analysing certain important ideas, such as belief, in terms of truth-functions.

    It is, I think, reasonable to view nonce fabrications of strange new abstractions with suspicion. But this also seems to apply to your intermediary reference layer. A common motivation for semantic theories is that they explain the relationship between our language and reality, but here you seem to have built a wall, or at least a metaphysical bureaucracy, around reality. It seems somewhat vulnerable particularly considering that you also posit the existence of an independent objective reality to which we do not have direct access. I'd like to see some more argument in favor of the intermediary layer along those lines.

    4) If our noun phrases always touch referrers instead of phenomena, how do you articulate this question?

    5) Structurally, your three-part theory results in the property that certain 'that-clauses' in ordinary language lack substitutivity with respect to all(?) terms we can actually use in speech. In other words, the sentences "Yagami believes that Light killed people" and "Yagami believes that Yagami's son killed people" claim different things, as in Quine's view. (I may be getting the people wrong in this example, as I don't really remember the plot of Death Note.) Isn't that the very property you were trying to eliminate?

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