Solution to the Mutant Liar.

In the previous installment, I told you about the Mutant Ninja Liar's Paradox. I didn't tell you my solution to it though. I will do that here but be warned: you will probably enjoy it more if you have already spent some time trying to figure out the paradox on your own. If you haven't, don't let me spoil you the fun of exercising those neurons!

Let me remind you, you wrote on your hand sentence Y:

Y = "There is no true statement written on my hand."

And you're asking the

Question: Is there a true statement written on my hand?

It seems like either answer leads to a contradiction. But there's got to be an answer, it's not like the question is meaningless! 


Let's start with this, take a look at this sentence:

Lumi on valkoista.

Is it true? False? Meaningless? Well, in Finnish it means "snow is white", in Fakish it means "light is liquid", and in English it doesn't mean anything. So that one sequence of letters, that sentence corresponds to a true statement in Finnish, to a false one in Fakish, and to no statement in English. I hope that illustrates a crucial difference between a sentence and a statement:

  • Sentence: sequence of characters.
  • Statement: meaning of a sentence, can be true or false.
  • Language: assigns meaning to sentences, translates a sentence into a statement.

So given that, here's a first, rather superficial and unsatisfying, but very important to understand solution to the paradox:

Solution 1. The answer to the Question is No. There is no true statement, or statement of any kind, written on my hand. On my hand there's a sentence. As we just saw, a sentence is neither true nor false, it may correspond to a true or false statement in some particular language, but that depends on the language! In Fakish, for example, the sentence on my hand means "forty two is less than one", so Fakish happens to translate it into a false statement. In English, however, the sentence corresponds to a true statement, since there's indeed no true statement (or any statement) written on my hand, only a sentence. And that's it, no paradox!

Are you satisfied with this answer? I hazard a guess that you aren't. It seems like a very superficial and pedantic way to weasel out of the paradox. Seems like it's pretty easy to fix the statement/sentence issue and bring the paradox back.

So let's do just that and see how easy it in fact is. If the issue is that we don't really write statements but sentences, let's just replace the offending word:

Y2 = "There is no true statement sentence written on my hand."

But wait a minute, didn't we just learn that sentences in and of themselves are neither true nor false? We need a language to translate a bunch of symbols into something with meaning. So "true sentence" doesn't actually make sense. "Ok", I hear you sigh, "could you be more pedantic? Fine, I'll just specify that the language to be used is English." Great, let's do that:

Y3 = "There is no sentence written on my hand that corresponds to a true statement in English."

We got the paradox back now, right? Y3 is a perfectly sensible sentence in English, so English can take it, process it, and translate it into a statement. A statement is either true or false, but either answer leads to a contradiction, just as before. We haven't gained anything, we are back at square one, right? 

Well, remember how in the original post I was talking about how a sentence is meaningless if it uses terms that are not yet defined? For example, "42 is jkiuioster than 1" is a meaningless sentence in English. Although in an extention of English called ReasonMenglish that sentence is meaningful. That's because ReasonMenglish is exactly like English, with just one extra word added: jkiuioster means greater.

So are we sure that all the terms in Y3 have meanings in English? If not, Y3 would be meaningless in English and there would be no paradox. It seems like that's a silly possibility, everything in Y3 seems perfectly meaningful.

But things are not always as they seem. To see why, let's make up a language from scratch. We are lazy, so our language will only have three  words:

  • snuu will mean snow
  • uus will mean is
  • buutifuul will mean beautiful 

Let's call this language Uunglish. Now here's a question: what do the following sentences mean in Uunglish?

  • Snuu uus buutifuul.
  • Uunglish uus buutifuul.
You might think that they mean:

  • Snow is beautiful.
  • Uunglish is beautiful.

But you would be wrong. The first sentence indeed means that, but the second one is meaningless in Uunglish. That's because our new language only knows three words and the term "Uunglish" is not one of them. So Uunglish gets stuck trying to assign meaning to the second sentence because it has no idea what the term "Uunglish" is referring to. 

Ok then, let's rectify this. Let's extend Uunglish and define a slightly bigger language that contains the same three words plus a fourth one:

  • Uunglish will mean the three-word language defined above.
What should we call this extended version of Uunglish? How about Uunglish2 - not terribly creative, I know. Now the sentence "Uunglish uus buutifuul" is meaningful in Uunglish2, all the terms are defined. But what about this new sentence:

  • Uunglish2 uus buutifuul.
This one is again meaningless in both Uunglish and Uunglish2, because both of these languages have no idea what the first term in that sentence means. Are you starting to see the pattern? Uunglish doesn't know what "Uunglish" means, Uunglish2 doesn't know what "Uunglish2" means. The same kind of thing will happen if we define Uunglish3, and so on. 

Can we fix this, can we define a language that can refer to itself? Because if we can't then Y3 is meaningless in English, since Y3 uses the term "English". Let's try to fix this by extending Uunglish2 with the following fifth word:

  • Uunglish3 will mean this very language.
But if you recall what we said when we were talking about the original Liar's Paradox you might see why that doesn't work. The problem with the sentence "this statement is false" was that it was referring to itself. So for it to have a meaning, it would already have to have a meaning! It is as ill-defined as this "definition": a blarfengal is a blarfengal.

Similarly here, in defining our new Uunglish3 language we are attempting to refer to something not yet defined - namely the language we are still defining!

But if it's true that a language can't refer to itself, then how is it that we seem to do this all the time in English? Even the previous sentence referred to English! And the one you just read. To explain this, let me remind you, and elaborate a little on, what we mean by "language" in this discussion. We mean some specific, exact way (algorithm, mapping, correspondence) to take sentences and assign them meanings. In particular, some sentences are translated into statements. A statement is by definition unambiguous: it is either true, if the state of affairs it describes corresponds to reality, or false otherwise. Other sentences, like "How are you?", have meanings but don't correspond to statements.

Given this, I hope it's clear then that English is not one specific language in the above sense, any more than Dmitriy is just one person. If I say "Dmitriy wrote this article" this sentence by itself is ambiguous since there are many Dmitriys. Yet we still often say things like that because sometimes the context removes the ambiguity. By itself  "Jane found Fluffy" is ambiguous, but if a man has a daughter named Jane and a recently lost rabbit named Fluffy, then that sentence would be perfectly clear if he said it to his wife.

It's the same with English. Just like we had Uunglish, Uunglish2, Uunglish3, there are many specific languages under the umbrella of English. Just as with Dmitriys, Janes, and Fluffys, sentences about English have enough ambiguity that we don't end up using one specific language to refer to itself. Often it's more like using Uunglish2 to talk about Uunglish. 

That might seem like too vague of an explanation. Things will get clearer in a minute, when we finally talk about how all this helps to resolve the paradox. 

Recall the new version of the sentence on your hand:

Y3 = "There is no sentence written on my hand that corresponds to a true statement in English."

The Question then becomes:

Question: is there a sentence written on my hand that corresponds to a true statement in English?

Solution 2. For the question to be unambiguous we will assume that, like with Jane and Fluffy, "English" is now the name of some specific variant. Since the only thing written on your hand is Y3, the Question is then asking: does the specific language English translate Y3 into a true statement?

The answer to that is No. The specific language English doesn't know its own name, so it can't decipher the last word in Y3, "English". So it doesn't translate Y3 into any statement, true or false.

But wasn't there supposed to be a problem with the No option? I'll semi-quote what I said in the first article. 

Option No means: there's no sentence written on my hand that corresponds to a true statement in English.

But take a look at Y3. Option No directly asserts Y3, word for word. So if option No is true (in English) then Y3 is true (in English). So there's a sentence, Y3, on my hand that corresponds to a true statement in English. This directly contradicts what option No says. So if option No is true then option No is false.

Doesn't this still demonstrate a problem with the No answer? See if you can spot the problem in this demonstration. The problem is with this part: "So if option No is true (in English)..."  The thing is: it's not true in English! It's meaningless in English because it uses the word "English". The same is true for the Question - it references English, so the specific language English can't assign any meaning to it.

But the Question and the No answer can of course be assigned meanings. But it's like with Uunglish, to talk about Uunglish we have to use Uunglish2, meta-Uunglish if you will. So if we want to ask questions about English, you need a meta-language, English2, that knows what "English" is.

So option No is true, only in English2, not in English. And that's how the supposed demonstration that option No leads to a contradiction fails.

If the Question and option No need English2 to make sense, can't we fix the above demonstration by using English2 instead of English? Then the problem from before disappears: option No is now legitimately true in English2, therefore Y3 is true in English2 (since option No and Y3 are the same, word for word). So there's a sentence, Y3, on my hand that corresponds to a true statement in English2. But that, because of the small change from English to English2, no longer contradicts what option No is saying. Compare: 

Option No: there's no sentence written on my hand that corresponds to a true statement in English.

We just showed: there's a sentence on my hand that corresponds to a true statement in English2.

And presto - contradiction be gone!

Well, this was quite a journey, being annoyingly pedantic finally paid off. There is still much more that can be said about this solution but let's leave that for a future article. Let me know in the comments if you have any questions or objections.

19 Comments - Go to bottom

  1. Hey Dmitriy,

    It seems like you're trying to argue that we can adopt the formal language solution, to solve problems of self-reference in English or any natural language. Naturally this might be a desired result, but I'm not sure how you are justifying this in any way. The only hint of justification I'm noticing here is: "the problem is with this part: "So if option No is true (in English)..." The thing is: it's not true in English! It's meaningless in English because it uses the word "English"."

    But this seems to be begging the question. In other words, something is meaningless because it is self-referential, but that is what must be justified in the first place! You did show that we can construct a logically coherent system in our natural language talk, to make sense of supposed self-referential statements, but this isn't a justification.

    Moving on, while it's generally bad form to define a thing in terms of itself, we have to be careful in any assumption that this is why the sentence about English fails. Firstly, there are many concepts that seem almost impossible to define outside themselves. Secondly, we can easily define the given sentence without using the word English (just substitute the definition for English!). We can say something like: "the sentence x written in this language which has properties x,y,z and is spoken in countries a,b,c, as well as the same language that is used throughout this book etc..."

    Also the topic of meaning is very tricky; I completely disagree that we need a definition for a word in order to understand the meaning of that word. There are many words which we can't seem to agree on a definition for (free will being one of them 😊) but yet which we still roughly understand (at least closely enough to converse about).

    A word can do very little and still be meaningful, for example the word "this" can ostensively refer to an object. Thus, reference is sufficient for meaning. Nor does reference need to be clear, as in the case with mathematical propositions which refer to mathematical entities whose true nature we often know very little about. Further, analytic propositions like "if x then x" can be completely content-less and reference-less and yet still meaningful and true. I think it's pretty clear that the liar paradox at least designates a reference, which is one step ahead of the proposition above.

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  2. Hey Dmitriy,

    To elaborate a bit more. Sentences are things which don’t hold truth values yes, if one just means the words on the page. But a sentence is more than just some words on a page, a sentence is also that which expresses propositions and bear content (even if this a process external to the actual sentence). It is generally understood that when we write down sentences, we are writing about the content of the sentences and not the actual characters of the sentence.

    For instance “snow is white” does not mean the word ‘snow’ is the word ‘white’. If we wanted to denote the actual words (as opposed to their contents) we could write the words in quotes.

    This is important because it’s not actually true (even pedantically true) that one can’t say “this sentence is false” because sentences can’t be false. You are assuming that such a thing is just making reference to the characters on the screen, when in fact it is (also) making reference to the content.

    This is why we can go about saying true things like ‘the sentence “Snow is a gas” is false.’

    Now it is true that the content of a sentence can be stating a fact about just the character (the actual words) of a sentence. For example:
    (A) sentence y is longer than x letters.

    When we say the sentence A is true we mean of course that the contents of sentence A (which are about the character of another sentence) are true. Thus, we must not confuse the notion that sentences cannot literally be true/false things (if we mean just the letters on a computer screen), with the notion that we cannot speak of sentences being true. Or that we cannot speak true facts about a sentences’ character.

    It would be better to say that the character of a sentence is itself not a true thing, but there are true facts about a sentences character. However, a sentence is more than just its character, it is also the bearer of content. Thus, when I am speaking of a sentence I can be talking about many things: the thing which bears content and expresses a proposition, the thing which is composed of words etc... While it makes no sense to say that words on a page can be true/false, it definitely does make sense to say that a thing which has content can be true/false, as propositional content can be true/false.

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    1. Relating this to the liar sentence:

      Going back to the liar sentence, I would say the content of the liar sentence is itself, meaning both the content and character of the sentence. It is referring to the thing that is being written down (the words on the screen) while also referring to the contents of the ‘thing’. The contents being the truth or falsity of x (where x is its own content).

      That’s important because it means that we can’t just say that x’s character is not something that can be construed to be true or false, and thus easily dismiss the issue. The liar sentence does more than just refer to its character. At the same time, we can’t say that we don’t know what the liar sentence is, because we have the words of the sentence as a reference.

      It’s true that we don’t know what the content really refers to (what is the thing which holds the truth value?). We have a label for it, but it appears to lack meaning. So, the content basically amount to "that which holds the truth value". However, we should not take this to mean that the concept is entirely vacuous.

      To hopefully make all this clearer, allow me to bring up a frequent analogy of mine:
      (B): “the sentence 'if x then x' is true”. The contents of "if x then x" appear to be empty, but that doesn’t mean that B is only making reference to the character/words "if x then x", because that would make B false. So a thing can have a truth value without it being anything else. Thus, the content of 'x' is just a thing which holds a truth value, and nothing else. There is nothing logically contradictory about there being a thing which holds only one property

      This follows the general meaning of expressions like “sentence x is false”. We don’t just mean the contents of sentence x, we are also making reference to the character of the sentence itself. Specifically, we mean to express “the contents of this sentence written down here on this paper are false”.

      If I was a teacher, it would be totally valid to reply “which sentence was marked false?”, but this wouldn’t make any sense if I was just referring to the content of the sentence. In which case you would just ask “which fact is wrong?”.

      Best,

      Alex

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  3. Hey Alex,

    You bring up several points but some of them we - later - discussed in the other thread. So could you see if you feel some of them have been cleared up already, and give a quick list of the remaining issues? Then I will have a better idea what to focus on in replying.

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  4. Hey Dmitriy,

    I've made all these points before, but I've decided to make the following post which sums up all my previous remarks for clarity.

    It seems to me there are two big issues here:

    1) This concerns the issue of justification, as I discussed in this thread as well as the last. 2) This relates to your argument against the liar sentence which you brought up in this post, that sentences can't be false (they are not true/false things) and so the liar sentence is meaningless.

    1 is basically the issue that we shouldn't start from the standpoint that the liar sentence is a non-statement, this is the thing we want to demonstrate after all (otherwise we beg the question). The way we would go about this is by coming up with some criterion that would establish that x is a non-statement (e.g. it being neither true nor false) and then realizing x is neither true nor false and thus a non-statement. Then we could successfully exclude the liar sentence from the category of legitimate statements.

    The problem is that I haven't seen you advertise any good method by which we are supposed to determine this criterion for the liar sentence, although you did say we shouldn't use the standard method for determining the truth values of statements (since that leads to contradiction).

    The only serious method that was suggested (that I noticed) was a ban on all self-referential sentences. But I in turn suggested that this was an extreme measure, since there seem to be plenty of self-referential statements that are perfectly okay. From your response I got the feeling that you don't believe all self-referential statements are bad, but just those which are empty of content/meaningless. I presume that your conception of meaningless doesn't simply mean being self-referential (because this also applies to the good statements), but maybe something like devoid of reference.

    I'm still confused about this issue however, because you mentioned how the sentence involving the word 'English' written in 'English' is bad. But of course there is nothing wrong with it except that it appears to be self-referential. It does not lack content/meaning as the word 'English' is well defined. Naturally, one can insist that self-referential statements are not well defined, but this goes back to the earlier point about how there seem to be coherent self-referential statements.

    Also, I say it appears to be self-referential because in reality I do not understand the argument for why it was supposedly self-referential. So, I'm still not clear on whether all self-referential statements are deemed 'bad' or just those devoid of content/meaning.

    Moving forward, I in turn replied that the liar sentence can't be completely devoid of reference, since it refers to the sentence (i.e. the thing being written down). In any case, as I already mentioned, there are propositions which are completely without referent that are nevertheless meaningful (like analytic propositions), an example being "if its great, then its great". It's true that we don't have a definite grasp on the meaning of the liar sentence, but as I also mentioned that's true of many concepts (like free will).

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    1. 2) This brings me onto the topic of 2. You didn't explicitly make this argument, but it seems like you would agree that the liar sentence can't be true/false because sentences can't be true/false in and of themselves. This seems to suggest that since the liar sentence appears to lack content (outside of reference to the words) then it can't be true/false since there is no reference to anything that bears a truth value.

      However, this doesn't necessarily follow. I earlier brought up an example of an analytic proposition (A: "if x then x") which is true despite lacking all reference and being almost completely devoid of content. 'X' in other words, isn't making reference to an object in the world; it's just a thing which appears to hold a truth value (it holds only one property).

      What this hopefully illustrates is that something can being true or false, while being devoid of any real semantic content. That's because the rules of our natural language follow to some degree the rules of formal and logical languages, wherein truth or falsity is established by the relation of the syntactic elements. Of course in natural language, one does need a semantic theory of contents to establish the truth/falsity of most propositions, but as we see this is not always so.

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    2. Sorry for any slight incoherence/grammatical errors exhibited in the above two posts; I posted these while in a rush.

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  5. Hey Alex,

    This will address a part of your remarks here and your last remarks in the other thread.

    >. That follows from your point that the liar sentence can’t be referring to sentences (since it has to refer to statements).

    I haven't made that point, I was saying it can't be referring to sentences because a sentence, by itself, doesn't refer to anything. You think it does, so I think this might help:

    What does the sentence "Lumi on valkoista" refer to? Does it have the property of referring to snow? Notice I'm not asking if it means something about snow in Finnish, which it does.

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    1. Yes it means something about snow in Finnish, and it refers to snow in Finnish.

      We can therefore say that the sentence has the property of referring to snow. This is a conditional property of the sentence, and not a necessary property. It is the former because of course in another language the sentence can be meaningless.

      As an analogy, I possess the property that I can breathe air. But this is a conditional property, one that wouldn’t exist if I were to be in a hard vacuum without a suit. So it’s true that I possess the property of being able to breathe (in an atmosphere). Much like the sentence possesses the property to refer to things (in Finnish).

      It is not appropriate however to say that this extra component (Finnish) is the thing which carries meaning (along with the sentence). Just as we wouldn’t say that I, along with the atmosphere, were breathing. We should not confuse the necessary conditions of properties with the property-holders themselves.

      And I take it that you are saying that the pair (S,L) is the thing which gives meaning to something yes?

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    2. Please let me know if this is true: “ And I take it that you are saying that the pair (S,L) is the thing which gives meaning to something yes?”

      In other words, does the sentence mean x, or does the (S,L) pair mean x? This is distinct from whether a sentence is capable of acquiring meaning by itself (it is not, it needs a bunch of necessary conditions).

      One thing I’m confused about is the fact that you repeatedly make reference to sentences having meaning (e.g. a statement being the meaning of a sentence).

      Additionally, you made mention to possible referents of the liar sentence before finally discarding each referent as being implausible. I never got the sense from the passage that there cannot be a referent because of the fact that sentences are not things which refer, in and of themselves.

      I also found it strange that the option of the liar sentence (or rather, the S,L pair) referring to the (S,L) pair was never considered. Even though that would be the most natural contender for self reference if indeed an (S,L) pair is the entity which expresses meaning. This option doesn’t seem to run into paradox at first glance.

      Let me know if I’ve made a mistake anywhere.

      Best,

      Alex

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  6. Hey Alex,

    >Please let me know if this is true: “ And I take it that you are saying that the pair (S,L) is the thing which gives meaning to something yes?”

    I am not exactly clear on what you mean by something giving meaning to something. What I am saying is:
    - S by itself doesn't have a meaning, by which I mean that there is no canonical (non-ad hoc) map from the set of sentences to the set of propositions/meanings/statements.
    - a meaning CAN be canonically assigned to a pair (S, L), by which I mean that there is a non-ad hoc map from the set of such pairs to the set of meanings. That map is of course:
    M: (S, L) -> L(S)

    From your other comment:
    I wasn't sure what the term "conditional property" meant, I have never seen it and Google didn't help. Did you use it as a synonym for a contingent property? If so, what you were saying wouldn't quite work I don't think.

    For your air breathing analogy, yes, in a possible world where I am in the middle of the Sun I couldn't breathe; in the actual world I can.

    But for the sentence, what is the condition that would have to be true for it to refer to snow vs light vs puppies? It's not "Finnish language exists" obviously. It's not "Finnish language", that's not a condition.

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    1. "I wasn't sure what the term "conditional property" meant, I have never seen it and Google didn't help. Did you use it as a synonym for a contingent property"

      Yes

      "But for the sentence, what is the condition that would have to be true for it to refer to snow vs light vs puppies? It's not "Finnish language exists" obviously. "

      Why not? Exactly what do you think is required for meaning/reference to take place? I would say the existence of the Finnish language and the existence of the above sentence probably still isn't sufficient for meaning yes, we need a few other conditions (like a mind to interpret the meaning perhaps?). But such talk about meaning quickly gets complicated.

      In any case, the standard philosophical interpretation is that meaning is concerned with representation, or 'aboutness'. When we say y is about x, we are expressing a two-way relation between the two. It is unclear exactly what this relation consists of, or if it's even something 'real'. But that's not so important for our conversation today.

      What is important is that we think there exists such a relation between the words in a sentence and the object or facts in the world (what I will term entities). When we say a thing has meaning therefore, we mean that thing participates in the above mentioned relation. In particular, the component that has meaning is the component that is about the other entity (e.g. sentence). This could be a word or a sentence, but can also be a brain state.

      This relationship is strictly between the word and the external entity. Again, this is the standard philosophical interpretation. I am curious to hear why you think that a word or a sentence can't have meaning, or 'assign' meaning as you put it.

      I do wonder what you think about the updated version of the paradox that comports better with your theory of meaning. "This (S,L) pair is false".

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    2. Where 'S' refers to the above sentence of course

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    3. Also, I'm not really sure what exactly the 'map' is. Other than that it is the thing which relates the sentence to the fact and gives it meaning. But it's not clear to me how this relation is supposed to take place, since the mere existence of L is apparently insufficient (by your own statement).

      The only difference is that you have formally designated such a relation, whereas the mere existence of a language does not do this. But it would be absurd to say that such formal designation is required for meaning (do you think that people spoke meaningless phrases before such designation was ever undertaken?). I assume no, and so I'm really struggling to appreciate the distinction.

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  7. >"But for the sentence, what is the condition that would have to be true for it to refer to snow vs light vs puppies? It's not "Finnish language exists" obviously. "
    Why not?

    Because then, in a world where all sorts of languages exist, that exact sentence refers to snow, refers to light (since Fakish exists), refers to puppies (since Puppyish exists, which is a language that assigns the meaning that puppies are cute to every sentence), etc. Basically then it refers to everything under the sun.

    That would be a bizarre and useless notion of "referring" imo.

    >What is important is that we think there exists such a relation between the words in a sentence and the object or facts in the world (what I will term entities).

    There exist many such relations, or maps, which I called languages. In other words, when I say "language" in this article I mean such a relation, in other words:

    Def. A (idealized, or specific) language is a map from the set of sentences to the set of meanings.

    There's no privileged / canonical language. In that sense S by itself doesn't have a meaning. "jhyui fdet" by itself doesn't have a meaning. But Puppyish maps it to the meaning that puppies are cute. Other languages map it to (assign to it, translate it into) different meanings. Does my definition of a language make sense to you?

    >I do wonder what you think about the updated version of the paradox that comports better with your theory of meaning. "This (S,L) pair is false".

    That's pretty much what the second half of the article is about. Because your updated version is essentially Y3.

    > since the mere existence of L is apparently insufficient (by your own statement) [to give a sentence meaning]

    It is sufficient, as I hope the Def above clarified.

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    1. My last sentence was too sloppy. What I should have said is: as I hope Def clarified, L itself is a relation/map that assigns meanings to sentences.

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    2. “Because then...it refers to everything under the sun.

      That would be a bizarre and useless notion of "referring" imo.”

      I completely agreed with the points you made, up until the very last. :)

      What is so bizarre about that? Secondly, not all languages exist, and until they actually ‘exist’ (here let us define language as a cultural medium of communication), the words refer to few actual things. So it might be better to say that words/sentences have the potential to make infinite references.

      “That's pretty much what the second half of the article is about. Because your updated version is essentially Y3.“

      I see a fundamental difference, the Y3 sentence speaks about a sentence referring to something. Whereas the updated version speaks of the (S,L) pair referring to the statement/proposition/fact.

      On the topic of meaning:
      In brief, when we say that ‘snow’ means snow we mean to say that there is a correspondence between the word and the entity of snow; such that the syntactic relations between the words are somehow supposed to capture or mirror the relations between the real life entities.

      In other words, because snow is white, we should expect to see the words ‘snow’ being related to the word ‘white’ in sentences that describe the real world, which we do. Also there are other relations of course, like ostensive gesturing or pointing.

      This is the bare bones definition of meaning. Meaning may just be this (a nominalist version of meaning) or it may be this plus added stuff.

      Thus, we should interpret your claim about L being that which maps the sentence to the fact, as being the thing which does the correspondence. But what does it mean to do the correspondence?

      Presumably it means to partake in the above relationship to other linguistic entities, in a similar fashion to how the real snow entity is related to other objects.

      Expanding on this means that we should think of sentences as themselves corresponding to states of affairs in the above fashion.

      But this doesn’t make much sense if we’re speaking of the (S,L) pair. For example, the state of affairs “I have a heart” is related to the state of affairs that “I pump blood”.

      But the (S,L) pair is not the thing which participates in similar sentence relations. You’ll never see them linguistically related by language users, in the above fashion.

      That said, one can make such an approach workable by adopting what philosophers of language call an ‘unstructured’ theory of meaning. Basically, this means that we don’t have to analyze the words to understand the meanings of the sentence.

      Under such a scheme you could admit that the meaning of a word is the above correspondence, while also believing that the meanings of sentences are different.

      “It is sufficient, as I hope the Def above clarified.”

      The definition clarified the function of L, but I’m still confused about what L really is. We didn’t really discuss my points about it being somewhat incoherent to describe the entire English language as referring to a thing.

      Do you define L as just the thing which performs the stated function, or does L have words, is it spoken etc...?

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  8. There is of course one last thing I haven’t brought up because I assumed it obvious, but I wish to do so now. Your theory of meaning stretches the ordinary interpretation of language statements outside conventional bounds.

    For instance we speak of sentences meaning things, and not (S,L) pairs meaning things. Again there is nothing wrong with doing this if we have some compelling reason to think we should change the standard interpretation. But I don’t see any good justification for thinking that sentences aren’t the entities which directly refer/mean.

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    1. Also about Y3:

      I see what you mean now.

      If I understand this correctly, your solution is to say that self-reference is illegitimate, and that we need a meta-language to speak about a language.

      The idea being that all other ‘self-referential’ statements which seem okay, are ‘okay’ only because they are not in fact self-referential (under your theory of meaning).

      The problem is that you can only eliminate the above issue by first stipulating that ordinary language statements about sentences referring are technically wrong.

      This seems an even bigger issue in my opinion. It would be better to bite the bullet and contend that those self-referential sentences like “x is longer than...” are plainly unacceptable. As opposed to maintaining that all talk about reference and meaning is technically incorrect.

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