A response to "Solution to the Mutant Liar"

This is a response article to a post by my colleague Dmitriy titled Solution to the Mutant Liar.

In a previous article, ReasonMeThis (aka Dmitriy) laid out a two-part solution for circumventing the liar sentence paradox ("this sentence is false" and similar versions). The first part of the solution involved the idea that natural languages "don't know their own name", i.e. can't talk about themselves. This, in more technical terms, is tantamount to adopting a Tarski-like theory of language, which entails that natural languages (i.e. a language like English, as opposed to formal languages of logic) actually exist in a hierarchy composed of lower "object languages" and a higher "meta-language". 

An object language is incapable of talking about itself, whilst a higher-level meta-language can talk about the object language. In turn this meta-language exists as an object language to yet a higher-level meta-language and so on and so forth (all languages can't talk about themselves). Such an approach neatly sidesteps all issues of self-referential paradoxes like the liar sentence.

However, there appear to exist many self-referential sentences which are perfectly acceptable, like the sentence:
  • P = "This sentence is longer than 6 characters". 
The challenge comes in attempting to reconcile the apparent truthfulness of P with the first-part solution adopted above. 

This is where the second part to the solution comes in. ReasonMeThis proposes that we shouldn't think of sentences as being the main bearers of meaning or reference, but rather a sentence that is paired with a language (here defined as a map from sentences to statements), which can be termed an (S, L) pair. This solves the issue of the first solution wrongly designating sentence P and sentences like it as bad, and we can now say that sentences like the above are expressible in the object language of choice. Since a sentence isn't the main bearer of meaning, sentence P is no longer self-referential. 

In the following post I will argue against such a conception of meaning which is entailed by the second part of ReasonMeThis' solution, in favor of the more conventional interpretation that a sentence can have meaning. 

Dmitriy's response: Alex gives a reasonable summary of my solution. But there are important considerations it doesn't quite capture, and there's a bit too much emphasis on the specific object (S, L), which is not as crucial to my solution as Alex believes. Let me give you a condensed, step by step version of my solution. Even if you have read the original article, this version will clarify the logic, I think. You can skip it on the first pass and just read Alex's article and then come back to it.
  1. A (written) sentence is just a bunch of characters. The same sentence (for example "Ya!") can mean different things in different languages. So if we want to talk about what some sentence means, or whether it's true, we need to specify the language. When we don't specify it explicitly we rely on people to guess the language from the context. In short, a language is what assigns meaning to characters.
  2. A sentence like "The sentence on my hand is not true", therefore, doesn't present a paradox because it's ambiguous, it doesn't specify "not true" in what language. The Liar's Paradox relies on there being only two answers regarding whether the sentence is true - yes or no, - with either option leading to a contradiction. But with an ambiguous sentence there's no reason to assume there is a definite yes or no answer: the sentence might be true in one sense and not true in another. Until the ambiguity is removed, the Liar can't get off the ground.
  3. It might seem trivial to remove the ambiguity by writing on my hand "The sentence on my hand is not true in English" and then asking: is the sentence on my hand true in English? But the problem is "English" doesn't describe a specific, precise way to assign meanings to sentences any more than "Alex" describes a specific person. Two native English speakers can easily have a miscommunication because they interpret the same sentence differently. So English has many specific versions; we can say that, roughly speaking, each person's brain implements a specific version of English, slightly different from everybody else's.
  4. There's still no paradox, the sentence on my hand is true in some specific versions of English and not true in others. To completely remove the ambiguity we can replace "English" with "English0", some particular specific version, some definite way to assign meanings to sentences. Now the question "is the sentence on my hand true in English0?" has a definite yes or no answer.
  5. The paradox is then resolved by discovering that the answer "no" doesn't actually lead to a contradiction. The contradiction is supposed to be this: the sentence S on my hand reads "The sentence on my hand is not true in English0". Is S true in English0? I am saying no, which means I am asserting: the sentence on my hand (S) is not true in English0. But this assertion looks awfully like S itself, word for word! So the purported contradiction is: I am saying that S is not true in English0 (A), and in the same breath I am asserting S word for word (B)!
  6. But there's actually no direct logical contradiction between A and B. "S" doesn't contradict "S is not true in English0", both could be true at the same time. The only way I would have a problem is if I, in asserting A and B, was myself speaking the specific version of English "English0". In that case, for B to be true, S would need to be true in English0, which would contradict A.
  7. So then can we resurrect the paradox by arranging it so "English0" refers to the very same specific version of English that I am using? There's certainly no obvious way to do it. I can try to define the term "English0" as whatever specific version of English is currently "in my brain", but at the time of introducing this definition there's no term "English0" in my vocabulary yet, I am only just defining it! If I now start using this new term, then I am now speaking a slightly different specific version of English from before: namely, my original version, the one I called "English0", didn't have the term "English0" in it but my new version does. So when I start speaking about "English0" I am no longer speaking English0! 
  8. Did I prove that there's no other way to refer to English0 while speaking English0? Did I establish there's no alternative ingenious way to resurrect the paradox? No and no, but thankfully I didn't have to. The goal was to resolve an existing formulation of the paradox, not to "pre-resolve" all possible not-yet-existing modifications of it. 
In summary, the pair object (sentence, language) is not crucial to my solution. What's important is that a language is needed to give a sentence meaning. As a result, "normal" versions of the liar sentence don't present much of a paradox. Without a specific language specified, they are ambiguous, they don't present a "yes or no" choice needed to get the Liar's Paradox off the ground. When we fixed the ambiguity problem we discovered that the paradox disappeared - the "no" option doesn't actually lead to a contradiction.

The relationship of meaning.

Let xRy (or R(x,y) if you prefer) stand for "x means y". Here y represents the state of affairs in the world and x is the entity we wish to assign the meaning to. To say that we are assigning a meaning to x is just to say that this is the entity that we think represents or means y.

I am contending that x can be a sentence (let's term this the "sentence theory of meaning"), and ReasonMeThis contends that it cannot (more specifically, that x should be an (S, L) pair). In general, I am saying that sentences are the main sorts of entities that we use to represent states of affairs in the world.

Dmitriy: I contend that x shouldn't be a sentence, provided we want x to have the meaning, a unique meaning. If we are fine with a sentence having many meanings then there's no problem of course. It's just we would then have to be fine with a conception of meaning where x can mean that snow is white and that snow is not white, since different languages can easily assign completely different, even opposite meanings to the same sentence.

Defeaters against Dmitriy's position. 

A. Such a position contravenes standard talk about meaning. We often say that "sentence x means fact y". In response, ReasonMeThis points out that we can be speaking shorthand; meaning that when we say the above we actually want to express "the (S, L) pair refers to fact y". 

Dmitriy: or that we want to express "S means y in language L (the choice of L being guessable from the context)".

But one must do more than simply point out this possibility. For the issue here is not that it's implausible that we are not speaking literally, but that we have no good reason for thinking that this is so. Thus, it's not enough to know that there are cases where people use shorthand to describe something, one must instead show that we have reason to believe that this is so for ordinary statements involving sentence meaning. 

The default position, in the absence of evidence, is surely the one that adopts a literal interpretation. Otherwise, one could argue that a sentence can be shorthand for anything (e.g. "I like bunnies" is actually shorthand talk for "I like bunnies and there is cheese on the moon").

Dmitriy: I don't agree that that's the default position, since we speak shorthand all the time. For example, "Alex disagrees with me" or "It will rain on Thursday" are all shorthand - we rely on the listener to guess from the context which Alex or which Thursday we are talking about. If Alex's argument depends on the premise that there's no such shorthand going on when we say "S means y" he has the burden of demonstrating that premise.

B. Such a position contravenes our (most people's) intuitions. It's just natural to think that a sentence is the entity which represents the state of affairs being described. Similarly, when we point to a bottle and utter "bottle", the most natural interpretation is that the word means the thing.

Dmitriy: I don't have a problem with that, since I assume people are implicitly holding the choice of L fixed. Given that, any word, such as "bottle", certainly represents whatever that word means in L.

C. Such a theory is needlessly complex; the sentence theory of meaning is much simpler in comparison. It's far more straightforward to describe the meaning relation in terms of sentences. One can make a correlation that connects syntactic (linguistic) relations between sentences and the relations between the states of affairs those sentences are thought to represent. If, on the other hand, an (S, L) pair is the standard representation of a state of affairs then we need a more complicated relation than that which is captured by the above.

Another way of putting this is to note that if an (S, L) pair is the 'x' entity in the xRy relation, then meaning becomes a complicated second-order relation (a relation about relations). That's because ReasonMeThis describes L as the map between the sentence and the state of affairs in the world. This L relation is itself a sufficient encapsulation of the sentence theory of meaning, and so
 it's not clear why we need meaning to include a second-order relation. His theory of meaning, in other words, is the sentence theory of meaning + a second order relation. So, it is appropriate to place an additional burden of evidence on his part.

Dmitriy: It just seems complex, but we don't need to say "The pair object (S, L) means x". We can just say "S means x in language L" and everything looks perfectly fine. Mathematicians are used to pair objects, but we don't have to impose that notation on everybody else. The important part is that to get a meaning out of a bunch of letters we need two ingredients: the sequence of letters S and the language L we are using to interpret it. 

The theory of meaning becomes more complex if we try to get away with only one ingredient, S. Then S can have a gazillion meanings, including opposite meanings, and it thereby becomes murky or vacuous to say "S means y" or "S is true".

Advantages Dmitriy cites for his solution.

A. Adopting such a theory solves the problem of self-referential paradoxes.

Counter: In fact there exist multiple ways (arguably better ways) of doing so. First, we can dismiss the issue that drove us to adopt the second-part solution in the first place; this being the fact that certain self-referential sentences (e.g. sentence P) appear acceptable at face glance. Writing off such sentences (and all self-referential sentences) as illegitimate may seem problematic, but it is more than traded off with the advantage that we can keep literal interpretations of ordinary language statements involving meaning.

Alternatively, we can abandon the Tarski-like approach to natural language interpretation. There are alternative theories of truth, like Kripke's theory, which don't require that we separate natural languages into component object languages and meta-languages. Kripke's theory still solves all paradoxes of self-reference, and is thus arguably preferable because it comports better with our natural languages (we don't think of English as being an object language that is incapable of talking about itself). 

Also, Kripke's theory of truth is compatible with certain sentences, like sentence P (for further reference see: Outline of a Theory of Truth). Finally, there are different ways of resolving these paradoxes that don't adopt the approach of designating the act of self-reference as problematic.

B. We avoid the problem of infinite possible reference, described here by Dmitriy:
If we tried to assign a meaning just to S, then you would have a situation where S means P1, but also means P2, P3, etc. So when we say "lumi on valkoista" means that snow is white that, taken literally, be vacuous, since it would also mean that light is liquid, that puppies are cute etc.
Counter: Unfortunately, it's not really clear (to me at least) why this so-called problem is an issue. Dmitriy appears to believe that a sentence having the ability to mean multiple things according to the language one speaks makes sentence meaning 'impotent', but I don't quite grasp why this should be such an issue. However let me take a stab at this nonetheless.

Issues of pragmatism: "If a sentence could mean anything, how are we supposed to know what it means?"

I see no issue here as context makes clear what a sentence is supposed to mean (i.e. what language we are supposed to adopt).

Dmitriy: So then Alex and I seem to be in agreement that a second ingredient, a language, is needed to get a meaning. Of course most of the time this ingredient is guessable from the context - as I pointed out before, we speak shorthand all the time, relying on the listener to guess the missing ingredients from the context. Speaking in a completely precise, self-contained way would be awfully tedious.

Issues of complexity: "Introducing an infinite (or huge) number of relations violates Ockham's razor"

First of all it isn't an infinite (or huge) number of meanings, but an infinite/huge number of potential meanings. That's because there exist a bunch of necessary conditions that are required for a particular sentence meaning to be actualized (like the existence of the English language). Since we don't have an infinite/huge number of actual natural languages that use similar sentences, there's not an issue of huge complexity in practice.

Secondly, Ockham's razor is really more applicable to ontology, but the sentence theory of meaning doesn't propose a more complicated ontology than what ReasonMeThis' theory requires. That's because additional ontological entities (e.g. natural languages) are necessary prerequisites of such relations. So it's not that the relations bring extra entities into the world, but vice versa.

Addendum: To be clear, I am here assuming a natural language is the actual thing (i.e. an entity that has words, is spoken etc...).

Dmitriy: I think Alex and I can agree that if we want S to have the meaning then a second ingredient, L, is needed. If we are fine with S having many, including opposite, meanings, then such a conception of meaning doesn't need a second ingredient. 

I prefer the former conception of meaning for reasons I sketched above. The specific formalism of the pair object (S, L) is not an essential feature of it, the less mathy, more everyday way to express it is "S means blah blah in language L".

5 Comments - Go to bottom

  1. Hey Dmitriy,

    I just had a look at this. I think there might be still some slight confusion over the nature of our (my) positions.

    The phrase "S means x in language L" is ambiguous. It can mean (in the weaker form), "S means x and the existence of L is a necessary condition for S to mean x". Or, "use L when deciding the meaning of S".

    In the stronger form, it means "the (S,L) pair means x". The sentence theory of meaning (my theory) supposes the former, whereas yours depends on the latter being true. Notice that the stronger interpretation implies the weaker one, and this seems to be the very crux of the issue. In general, your theory supposes more and therefore has a much greater burden of proof to establish. It must suppose that there is an additional hidden meaning to phrases like "S means x in language L", as well as suppose that there is an additional relation inherent to meaning.

    To quote myself, "Another way of putting this is to note that if an (S, L) pair is the 'x' entity in the xRy relation, then meaning becomes a complicated second-order relation (a relation about relations). That's because... (the) L relation is itself a sufficient encapsulation of the sentence theory of meaning... His theory of meaning, in other words, is the sentence theory of meaning + a second order relation."

    So again, I totally accept that we need a language L to extract a meaning from a sentence, but that just reduces to the weaker interpretation. The problem comes when adopting the stronger interpretation; it comes in other words with demanding that the (S,L) pair be the main referent involved in the relationship of meaning. There are all kinds of semantic issues that arise from this (like meaning ending up being a more complicated second-order relation), and it's why I don't like it.

    If, in the end, you accept the weaker reductive version of ordinary language statements like "S means x in L", then we are in complete agreement. But let's take a step back here and see why the issue of the stronger interpretation comes about in the first place.

    Your argument solves the liar paradox by invoking a Tarski hierarchical account of natural languages like English. We ignore of course, any potential problems of interpreting natural language in this way; when we proceed we can see that we still encounter a second issue.

    This issue is (as I mention in the article) that perfectly reasonable sentences like "this sentence has more than 3 words" are problematic under the Tarski hierarchy. If we adopt your solution we can't say that the above sentence written in English is true in English (instead it must be true in English0).

    Whether this problem is an issue for you is of course a matter of debate, but the only way to avoid it is to adopt the stronger interpretation (SI) of the S-L pair description from above. We need our (S,L) pair to be the main referent of the relationship of meaning, otherwise the sentence becomes potentially 'self-referential" and therefore falls afoul of the Tarski criteria.

    When we introduce SI, the sentence can be clearly expressed in English, and it can also be evaluated as true in English. That's because the sentence isn't the main referent of meaning, and so there's no possible issue of self-reference that could even arise in the first place (so we don't need Tarski or English0 here).

    I hope all of this makes sense. This is the entire reason I wrote up the article in the first place; otherwise there's not much point to introducing the SI version of meaning except as an idiosyncratic preference (i.e. meaning shouldn't be "vacuous") which has no direct bearing on the problem at hand. If you don't accept SI, then of course we have no disagreement.



    1. “Whether this problem is an issue for you is of course a matter of debate”

      I meant of course a matter of personal opinion. Unless, that is, your left and right brain hemispheres happen to be in conflict over this.

  2. Hey Alex,

    I like this format, a structured minidebate. I am hoping a curious reader would be able to read the whole thing through, including these comments, and form their opinion. So I will just give a superquick reply and "rest my case", and give you the last word.

    >The phrase "S means x in language L" is ambiguous. It can mean (in the weaker form), "S means x and the existence of L is a necessary condition for S to mean x". Or, "use L when deciding the meaning of S". In the stronger form, it means "the (S,L) pair means x".

    I disagree, I think the phrase simply means L(S) = x. Remember, L is by definition a way of assigning meanings, such as x, to sentences, such as x. So the, I think, clear meaning of that phrase is: L assigns meaning x to the sequence of characters S, or in math notation L(S) = x.

    1. Correction: Remember, L is by definition a way of assigning meanings, such as x, to sentences, such as S.

    2. Actually one of your formulations is pretty close. ""use L when deciding the meaning of S". I would agree if you changed it to "if you use L when deciding the meaning of S, you will get x".


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