• T Clark
    7.2k
    Banno and I had a short discussion about self-reference in another thread. Whenever I think about self-reference I have two responses. First - it’s fun and a bit exciting. You get the feeling that you’ve stumbled on something profound and important. Second - once you look into it, it’s still fun, but it’s clear it’s neither profound nor important.

    Here's what the Stanford Encyclopedia of Philosophy (SEP) says:

    In the context of language, self-reference is used to denote a statement that refers to itself or its own referent. The most famous example of a self-referential sentence is the liar sentence: “This sentence is not true.” Self-reference is often used in a broader context as well. For instance, a picture could be considered self-referential if it contains a copy of itself (see the animated image above); and a piece of literature could be considered self-referential if it includes a reference to the work itself. In philosophy, self-reference is primarily studied in the context of language. Self-reference within language is not only a subject of philosophy, but also a field of individual interest in mathematics and computer science, in particular in relation to the foundations of these sciences.

    The philosophical interest in self-reference is to a large extent centered around the paradoxes. A paradox is a seemingly sound piece of reasoning based on apparently true assumptions that leads to a contradiction. The liar sentence considered above leads to a contradiction when we try to determine whether it is true or not. If we assume the sentence to be true, then what it states must be the case, that is, it cannot be true. If, on the other hand, we assume it not to be true, then what it states is actually the case, and thus it must be true. In either case we are led to a contradiction. Since the contradiction was obtained by a seemingly sound piece of reasoning based on apparently true assumptions, it qualifies as a paradox. It is known as the liar paradox.

    Most paradoxes of self-reference may be categorised as either semantic, set-theoretic or epistemic. The semantic paradoxes, like the liar paradox, are primarily relevant to theories of truth. The set-theoretic paradoxes are relevant to the foundations of mathematics, and the epistemic paradoxes are relevant to epistemology. Even though these paradoxes are different in the subject matter they relate to, they share the same underlying structure, and may often be tackled using the same mathematical means.
    The text references the importance of self-reference to the foundations of mathematics. I assume it is talking about Russell’s paradox. In computer science, there was a brief discussion of the importance of programs that can modify themselves.


    So, my impression is that most self-reference is useless. It seems cool because it’s about us thinking about ourselves, but there is little of substance there. It has never seemed to me that the liar’s paradox has anything interesting or important to say about truth or language. I’d be interested in hearing about situations where self-referential ideas actually contribute rather than obscure.
  • Miller
    92
    Reflection. Initially the mind gets all its content from reflection of the senses.
  • the affirmation of strife
    39

    I’d be interested in hearing about situations where self-referential ideas actually contribute rather than obscure.

    You're probably asking about philosophy, and I can't really help there. However, as someone who knows a bit of programming and mathematics, self-reference can certainly be interesting in those spheres and even sometimes useful (recursive functions provide concise ways to code certain things).

    As the article alludes to at the end, things get even more interesting when thinking about self-modification of programs or self-specialising compilers (I've lost a bookmark to an interesting and not too technical blog post about this, maybe I can find it again...)

    In terms of mathematics, the book "Vicious Circles" by John Barwise and Lawrence Moss seems to be a good reference for what they call "hyperset" theory, an extension of set theory that allows for self-referencing and circularity. I haven't read much, and it's very dense. Working understanding of set theory required. I wonder if there are any mathematicians here that could break it down for us.


    Reflection is not recursion. I can reflect on the past, but I can't change it. Imagination and reflection are closely linked, it's true. And that's an interesting topic in its own right.
  • the affirmation of strife
    39
    From chapter 4 of that book (Circularity in Philosophy):
    Descartes' "Cogito, ergo sum" was intended to be an irrefutable argument from undeniable premises. Descartes could not doubt the fact that he thought. [...] The reason is that Descartes' act of doubting itself requires thinking [...]. Basically, Descartes' famous dictum is shorthand for something more like: I am thinking this thought, and this I cannot doubt because my doubting requires my thought.

    Seems like a different approach to that dictum than the usual, ontological one.
  • the affirmation of strife
    39
    An example from linguistics (Chapter 4.4). The authors talk about how the following sentences are well-posed:

    Professor Hill denounced the judge who had harassed her.
    The law school professor who had worked for him denounced Judge Thomas.

    Whereas this variant involves a "vicious circle":

    The law school professor who had worked for him denounced the judge who had harassed her.

    Interestingly, we still seem to understand it.
  • T Clark
    7.2k
    As the article alludes to at the end, things get even more interesting when thinking about self-modification of programs or self-specialising compilers (I've lost a bookmark to an interesting and not too technical blog post about this, maybe I can find it again...)the affirmation of strife

    Yes, I am primarily talking about philosophy. I tried to be careful not to be too dismissive of self-reference. I had read that the kind of programing uses you describe are valuable. I guess I'm trying to separate the wheat from the chaff - uses with real value as opposed to just a bunch of gee whiz stuff.

    In terms of mathematics, the book "Vicious Circles" by John Barwise and Lawrence Moss seems to be a good reference for what they call "hyperset" theory, an extension of set theory that allows for self-referencing and circularity. I haven't read much, and it's very dense. Working understanding of set theory required. I wonder if there are any mathematicians here that could break it down for us.the affirmation of strife

    My attitude toward self-reference in math is ambivalent. First off, I'm good at the math required to be an engineer. That's really different from what we're talking about here. When I look at Russell's paradox, for example, it seems like a trick, yet many mathematicians seem to think it undermines math as a whole. We had a discussion about a conversation between Wittgenstein and Turing a week or so ago. Turing proposed that Russell's paradox undermined math to the point that it might lead to a bridge falling down. That seems goofy to me, but my level of expertise is too limited for me to have any confidence in my judgement.

    But yes, the goofiest part of self-reference for me is its use in philosophy. The liar's paradox seems like a little joke that people have decided to take seriously. I can't see how it gives any insight into meaning or truth, as some propose.
  • the affirmation of strife
    39
    I'm just flipping through the book a bit. I realise it's not the kind of thing people would buy just for the sake of an online discussion, so that's why I'm putting some snippets up here. Let me know if it gets too much.

    This paragraph right at the end of the book gives an idea of the conclusions they draw from their maths shenanigans:

    The tongue in cheek title of our book is intended to suggest that circularity
    has an undeservedly bad reputation in philosophical circles. On the other hand,
    we certainly do not think that every proposal or argument using circularity
    bears close scrutiny. For example, one of the morals of our resolution of
    the Hypergame Paradox is that certain kinds of circular definitions really are
    incoherent.

    So I think that matches your intuition and it at least gives confidence that the kind of separation you talk about should be possible. I'll need to look into it more to give better examples of "useful self-reference".
  • T Clark
    7.2k
    I'll need to look into it more to give better examples of "useful self-reference".the affirmation of strife

    I appreciate your input. I didn't start this discussion because I have a particular end in mind. I just want to see where it goes.
  • TheMadFool
    13.7k
    I don't understand one thing with so-called self-referential sentences. Let me explain.

    Human self-reference
    Sarah says:

    1. I am a bad, bad girl! (1st person)

    2. You are a bad, bad girl, Sarah! (2nd person)

    3. Sarah is a bad, bag girl, isn't she? (3rd person)

    Linguistic self-reference

    4. I am false (1st person) ???

    5. This sentence is false (2nd/3rd person?)

    Why are self-referential sentences like the liar sentence (5) only in the 2nd/3rd person while we humans can do the same in 3 different ways (1, 2, 3)

    Another issue:

    If I say "this bag is black", I have to actually point at the bag in question. That is to say we need another piece of information ( :point: ) to clarify what "this" refers to.

    Consider now the liar sentence "this sentence is false". How do I know "this" refers to the liar sentence itself? Where's the :point: ?

    4. This :point: "Paris is on the moon" sentence is false.

    5. This :point: "This sentence is false" sentence is false.

    Ambiguity?
  • the affirmation of strife
    39

    "This" is not second person.

    Your other point is about incomplete information, which is indeed the first hurdle for most "silly" kinds of self-referential paradoxes.
  • TheMadFool
    13.7k
    "This" is not second person.the affirmation of strife

    Third person?
  • the affirmation of strife
    39
    There is no "person" AFAIK these are called "demonstratives" or something like that.
  • TheMadFool
    13.7k
    There is no "person" AFAIK these are called "demonstratives" or something like that.the affirmation of strife

    :ok:
  • the affirmation of strife
    39
    Forgive the mathematics, I want to relay (again from that book) what the authors think of as a useful consequence of self-reference in logic. Consider the Russel paradox (a non-reflexive set is a set that doesn't contain itself):

    There is a set R which consists of all and only non-reflexive sets:
    R = {x | x is non-reflexive}
    But then we see that R belongs to R iff R is non-reflexive, which holds iff R does not belong to R. Hence either assumption, that R belongs to or R does not belong to R leads to a contradiction.

    They say that in later chapters they prove that circularity is not the villain here... I'm way out of my depth though.

    But, consideration of that set is useful because:
    Suppose we have some set b and form the Russell set using b as a universe.
    That is, let R_b, = {c ∈ b | c is non-reflexive}
    There is nothing paradoxical about R_b - The reasoning that seemed to give rise to paradox only tells us that R_b ∉ b. In other words, the Russell construction gives us a way to take any set b whatsoever and generate a new set not in b.
  • TheMadFool
    13.7k
    How should I correct my post?
  • the affirmation of strife
    39
    Well, I was just responding to say that your question
    Why are self-referential sentences like the liar sentence (3) only in the 2nd person while we humans can do the same in one additional way viz. in the 1st person?
    didn't make sense to me. Humans can use whatever grammar they like, so I'm not sure what you are confused about here.
  • the affirmation of strife
    39
    The liar's paradox seems like a little joke that people have decided to take seriously. I can't see how it gives any insight into meaning or truth, as some propose.T Clark

    So, it looks like the value of the liar's paradox or Russel's paradox etc. comes from the insight into how we can or can not formulate truth. The authors give a plain-language summary of Tarski's Undefinability Theorem for Truth:

    There is no single first-order formula that serves to define the truth of all sentences of first-order logic in the universe (of sets).
    parentheses added
  • T Clark
    7.2k
    There is no single first-order formula that serves to define the truth of all sentences of first-order logic in the universe (of sets).
    parentheses added
    the affirmation of strife

    I kind of get that, but it seems like a joke. A meaningless technicality. I can't see how it tells us anything useful about truth for any other propositions.
  • the affirmation of strife
    39
    I tend to agree. My very uninformed first impression is that it could have some implications for certain normativist viewpoints but as I say I'm not really well versed enough in the philosophy.

    For completeness (chuckles), I've just found that the SEP also has an article on non-wellfounded set theory (aka hyperset theory). They have "Vicious Circles" in their references, and a lot of the same topics seem to be briefly covered. Fairly technical, but maybe something useful is there.
  • StreetlightX
    7.8k
    The perplexity about self-reference among philosophers has always stuck me as hilarious. As if language ought to be up to the artificial standards of philosophers, rather than philosophers themselves dealing with artificialities. It's like that joke about the guy who only ever looks under the lamplight to find his keys, and then complains that because they aren't there, his keys must have winked out of existence. Self-reference is everywhere and perfectly ordinary. If philosophers would like to make them into anomalies, that says more about the failings of the philosophical imagination than language.

    As for the relevance of self-reference: it draws attention to the event of language, it's taking place. It's the institution, at the level of the proposition, of the what is extra-propositional in language. When language takes itself as an object, the separation of language 'here' and object 'there' evaporates: language becomes enthinged, enworlded. Or rather, the always-already enworldedness of language shows itself and stops being obscured, for the briefest of moments. Self-reference is puzzling only to those who want to treat language as a pure, self-enclosed system, sterilized from any imbrication in the world.
  • the affirmation of strife
    39

    I think think the fascination with self-referential paradoxes specifically comes from their use as a way to refute arguments, especially in epistemology.
  • StreetlightX
    7.8k
    True - but I take this to be a commentary on relative aridity of most epistemological approaches rather than anything.
  • TheMadFool
    13.7k
    Descartes' "Cogito, ergo sum" was intended to be an irrefutable argument from undeniable premises. Descartes could not doubt the fact that he thought. [...] The reason is that Descartes' act of doubting itself requires thinking [...]. Basically, Descartes' famous dictum is shorthand for something more like: I am thinking this thought, and this I cannot doubt because my doubting requires my thought

    :up:
  • TheMadFool
    13.7k
    The word "this" is not a self-referencing word, is it? Typically, it's used to denote spatial proximity. in contrast to the word "that" which expresses spatial remoteness. Good enough for government work; after all "this" is closer than "that" - closer to the self.
  • Mww
    2.8k


    Cool what you did with the title.
  • T Clark
    7.2k
    Cool what you did with the title.Mww

    Aww... shucks.
  • T Clark
    7.2k
    There is a set R which consists of all and only non-reflexive sets:
    R = {x | x is non-reflexive}
    But then we see that R belongs to R iff R is non-reflexive, which holds iff R does not belong to R. Hence either assumption, that R belongs to or R does not belong to R leads to a contradiction.
    the affirmation of strife

    So, it looks like the value of the liar's paradox or Russel's paradox etc. comes from the insight into how we can or can not formulate truth.the affirmation of strife

    I'm interested what you and @StreetlightX have to say about the Russell paradox as opposed to the liar sentence. From what I have seen, mathematicians and philosophers of mathematics claim that the Russell paradox undermines the credibility of mathematics in general. We had a discussion a few weeks ago about a discussion between Wittgenstein and Turing where Turing claimed the inconsistencies in math might cause a bridge to tall down. That seems silly to me, to believe that an anomaly in number theory could contaminate calculus.

    What are you guys thoughts?
  • the affirmation of strife
    39
    Ah yes, the elephant in the room. I finally had time to read a little bit about that debate. It now seems silly that I have not read Wittgenstein before.

    credibility of mathematicsT Clark

    I struggle with this idea. I think of mathematics as a concise language for encoding models of reality[1]. The symbols and rules are invented, but what they describe is discovered[2]. Would it make sense to talk about, for example, the "credibility" of the Japanese language?

    I think W. has it right: there are only two causes for the bridge to fall down. Either the model (physics) is wrong, or the mathematical rules were not followed. The same reasons for a failure in communication: either you misunderstand what I am talking about, or I am talking gibberish. The first of those problems has nothing to do with language, so we'll move on to the second.

    The problem: what should we do if we are presented with contradictory mathematical rules. For the language analogy, this is like finding a contradiction in your Japanese grammar book. On page 24 it tells you to say X in situation Y, but on page 135 (it's not an easy language, you understand) it instructs you to say the opposite i.e. (not X) in situation Y. Solution: buy a new grammar book.

    In addition to what @StreetlightX said about the "enworlded-ness" of language (arising from the fact that it is invented by humans), I would like to then add a second point: language is dynamic. It will evolve. We didn't have mathematical rules for talking about circularity in set theory, so we invented hyperset theory. It just takes a bit of coffee and head-scratching.

    That's not to say that contradictions are completely harmless (and circularity is hard to think about, so it can easily lead to contradictions). I think some of Turing's fear was justified. It's not nice to end up in a situation where the rules are contradictory. You have to go back to the drawing board and maybe throw out a lot of work. But I fail to see how someone could even construct a bridge, or anything else, based on contradictory instructions. The best that I could offer would be a stream of colorful language directed at whatever theorist had handed me the instructions (actually, it's more likely that I would be the theorist...)

    ---

    [1]: Is this still controversial? I mean, Einstein called it a language. My first year lecturer did the same.

    [2]: Without getting bogged down in ontology, I just mean to say that there is some kind of distinction between these processes.
  • Philosophim
    777
    To add to the great contributions in this thread, I think one of philosophy's tasks is to examine assumptions we take for granted and see if they hold under scrutiny. Liar's paradoxes show us that certain assumptions we make lead to illogical conclusions. That's incredibly important, because what if you are making those assumptions in arguments that are not liar's paradoxes? That means you are holding faulty assumptions that are not clear in another argument. Can we learn from the liar's paradox? I think we can.

    "“This sentence is not true.” I'll just change it to "This sentence is false" for less typing.

    Lets look at this from a logic perspective. We could say, "If this sentence is true, then its false"

    A -> ~A
    If A is true, then we get A is not true.

    A = (A -> ~A)

    Now negate the formula, and assume the sentence if false.
    ~A = (A or A)

    ~A = A
    (If I did my logic right, its been a while)

    So if the sentence is false, its true, and if its true, its false. We definitely have a contradiction.

    As we can see, there's something weird going on. But why? Our intuitions feel like the sentence makes sense, but logically, it doesn't. Because we're being too general. We realize we've said nonsense by being too implicit. That's the lesson we can glean. Just because we can say or posit an idea in language, doesn't mean it makes sense. You've previously posted the question, "What is metaphysics?" Many times people use metaphysics to disguise liars paradoxes. Terms that are ambiguous are great ways to hide nonsense terms and conclusions within them. If you can pick them out, you can ask for clarification.

    Solving the liar's paradox can give us a tool to solve other nonsense points while keeping within the spirit of the discussion. Nonsense arguments are often unintentional, and often times hide an underlying meaning that wasn't quite nailed with the language. So I could propose this to someone instead:

    "I don't think we're being specific enough with our words. Do you mean perhaps, "This sentence is a false sentence"? Because at that point, we can look at the sentence and see, "No, that is a viable and correct sentence. It is false that that is a false sentence.

    Or

    Proposal:
    A = a sentence
    ~A = not a sentence

    A therefore
    A = ~ A

    And we can see that its a contradiction right off the bat, and that A must be a sentence.

    Liar's paradoxes are a great teaching tool about the ambiguity of language, but also about seeing through the intentionality of a person's argument. When discussing philosophy with others, we should be generous towards the other person's argument. Sometime we're not just trying to show that a person's argument is viable, we're also trying to see if we can use language correctly to better cast what they are intending to argue as well.
  • T Clark
    7.2k
    Either the model (physics) is wrong, or the mathematical rules were not followed.the affirmation of strife

    That's the heart of the argument. Many people, I guess some really great mathematicians and logicians, don't agree. I have a feeling it has something to do with mathematicians being natural idealists. You can't futz with the ideal world. It's perfect. If it's not, somehow the whole thing falls apart.

    The problem: what should we do if we are presented with contradictory mathematical rules. For the language analogy, this is like finding a contradiction in your Japanese grammar book. On page 24 it tells you to say X in situation Y, but on page 135 (it's not an easy language, you understand) it instructs you to say the opposite i.e. (not X) in situation Y. Solution: buy a new grammar book.the affirmation of strife

    I don't think this analogy applies. Seems like with the Russel paradox, we start with what appear to be consistent rules and get contradictory results.

    In addition to what StreetlightX said about the "enworlded-ness" of language (arising from the fact that it is invented by humans),the affirmation of strife

    Is this the issue, that mathematicians and logicians don't believe math was invented by humans? That they think it is intrinsic to the world?

    I think some of Turing's fear was justified.the affirmation of strife

    I don't get it. I'm not sure I can even see the connection between number and set theory and calculus. But then, my math is of the practical, engineering sort.

    [1]: Is this still controversial? I mean, Einstein called it a language. My first year lecturer did the same.the affirmation of strife

    There are certainly people who believe that the Russell paradox says something profound about math and logic.
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