First I need to write; I am not an expert. The things I write here are tries at exploring my own intuitions around this subject. I try to dash my brain at the limits of his ideas, not to find the truth in itself, but to maximally expand my intuition. With that caveat, here are two limits to Carnap's epistemology.
First, bring into your mind a coherence theory of truth, and then say that questions external to this network of statements and ideas make no sense. The reason they don't is that one simply cannot go beyond them, as long as one wants to use this language explain it. How, in other words, can I explain with words what cannot be said in words? Think of this view as being inherited from early Wittgenstein, and I think you are approaching the right idea.
Incidentally I'm not a great believer in precision in language. The reason is that my theory of meaning basically says that words function by awakening cognitive capacities, and that new meaning is made by the deliberate connection in consciousness by two priorly unconnected "rooms." Meaning is private in the sense that each meaning- structure has a history particular to that person, and collective insofar that this corresponds to other peoples notions. The prior paragraph is then as good as it gets if it brings out in you the relevant places where meaning resides, not matter if Carnap used those words or not. I would therefore like you to trust me in that the ideas you have in your mind after the last paragraph correctly connects the right clusters of concepts. But this is not an exact science, of course.
For didactic purposes, I here need to introduce two interlocutors.
A: In introducing this network of declarative statements and beliefs Carnap will run in to two limits.
Z: Limits of what?
A: Merely limits, I think. If there are no external questions, then the limits will be in the ability of the internal language to represent itself.
A: Then I'm guessing these limits will be limits in how the language is able to represent itself.
Z: Yes exactly. I want you to envision this network of truths very literally, as a weave of white wool, completely round. Since, on the classical view, every belief must be justified, every thread connects to something else; these are the justifications, and the following are sort of... conclusions.
A: I suggest we use some alternate terminology. Let all parts of threads be "lemma", and justifications here are only "pre-lemma", and following them are "post-lemma", relatively.
Z: Okay, I can work with that. Now, with this intuition we are able to better describe some of the limits of Carnap. The first question is; what do the edges of this textile look like?
A: Let me think. I can envision two versions. One of them is that the strands are just roughly cut off, and a second is that every thread neatly wraps back into the fold.
Z: And what would those two versions represent?
A: The first is that there is a limit to human cognition, and to the society of knowledge in the sense that; we could have made more lemmas to tack onto the end, had we just thought about the edge cases. As soon as we reached those lemmas, new one's would be produced.
Z: So, for instance, in a mathematical framework, as soon as a new problem presents itself, there is the area ripe for expansion?
A: Right, that is one version of it. The second one is that, in a sense, this system is internally closed off. You don't phenomenally experience any edges to your thinking because even fringe ideas, in this case those that are not connected to many other ideas, but kind of exist almost on their own, refer back to som other lemmas that support them.
Z: How, on this view, would new knowledge arise?
A: This question must be answered on two levels. Either internally to one person through a synthesis of prior lemmas that would create a new representation of earlier facts, or make a new connection between existing threads, or, on the societal level, this same thing happening but between two minds, such as in a research collective.
Z: With this I think we can approach one of the limits raised in class, namely that Carnap should really be careful when he is even talking of there being external questions that we should not take seriously, like the very existence of things or a framework.
A: Yes, but, I think this is difficult. Carnap says that existence is a presupposition in every internal declaration of facts. In our terminology this would merely be a pre-lemma attached as a substrate to every lemma in the fabric of statements, but, I think maybe Carnap didn't have this clear idea of it. To throw his own ideas back at him, I think that in the idea of the internal the external is presupposed. There cannot be internal statements if there is no difference between the internal and the external. But, this last sentence is in fact a pleonasm, a direct analogy, a tautology to his statement of existence as presupposed.
Z: Is he begging the question?
A: I think so. If you are to presuppose external reality, I think Peirce did so more elegantly, because he didn't erect this dichotomy of outside and inside, and so he gets around all the problems of representation which was clearly on Carnap's mind.
Z: Yes, I think I see what you mean, but, I have a second intuition which we don't have time to explore so deeply, and that is that any internal framework that presupposes existence will have a problem of dealing with negative statements. How do you say that something is not true, when both the not and that which is true has the presupposition of existence? You get into al sorts of questions such as, "How can the not being be?" and all those parmenidean conundrums.
A: Yes, but moving on; I think you had a second limit you wished to discuss.
Z: Yes. There is an internal dimension too, which has to do with the nature of language. The problem is, can Carnap describe the nature of the system, if he is not able to put himself outside of it?
A: Okay, I think I understand you, but I sense you are employing the Munchhausen Trilemma, and I'm not sure our readers are familiar with it, so could you perhaps introduce it?
Z: Certainly. Under this name it was introduced by Albert, but it really has it's roots in the Sceptical school of philosophy, and is treated of in Sextus Empiricus "Pyrrhonic Sketches". It is a threefold critique of the possible truth of a lemma. There are three possible critiques it says; One; the statement is circular, or it rests on something which must rest on something ad infinitum, or the whole system of statements only float in a vacuum, without external reference.
A: And our mat of woven text would at once have all properties I think?
Z: Indeed. Put your finger on a tread and follow it; the outcome must be one of these. It reaches an edge, it comes back to itself, or you trace every tread, but can never go to an external vantage point from which you can view everything. And, I'd love to go on right now, but first I need to introduce three new concepts; 'generativity', 'embedding', and self- similarity.
A: Actually, you speak too much, let me do it. Generativity is the feature of language by which elements of meaning combine to create new meaning, 'Embedding' is the reduction of a prior statement to a flexible word like "which", that takes as it's reference the prior sentence or meaning as a whole, and brings it into the next sentence. Self- similarity is the idea that structures repeat their structure at different scale, like how a branch of a tree, if put on the ground, kind of look like a whole tree. Now, what are you trying to say with all those ideas?
Z: Yes... through a juxtaposition of a few terms now I have tried to open up many rooms having to do with spacial reasoning, complexity and depth. Language is a complex phenomenon which builds on itself; as Munchausen did in the story, it pulls itself out of the bog by it's own hair. Through generating new meaning, in a social as well as a private sense, and through using parts of itself in new connection it builds, so to speak, outwards and enlarges itself. Roughly speaking there is a constant synthesis going on, which reaches out to embrace even new phenomena..
A. You have lost the thread haven't you...
Z: Maybe a litte, but exploring one's intuitions is not an easy thing. Well. Opposite to this expansion, one can analyse, that is, take apart one's ideas; the ungenerativity, to make a neologism, the decompositionality of language. And this sense of going inward is then, I would argue, something different to going along.
A: So thinking like this isn't covered by the trilemma of necessary connections in a network of statements?
Z: It is. but, at the same time, it is running towards the simples of simple statements, those which all others are supposedly created from.
A: I get two immediate associations when you speak like this; one is the logical atomism of Russel, and the other is the simplest elements of impression one gets in Hume from which all other things are made.
Z: Parts of my intuition may come from there, I don't know; but truly I am thinking of morphemes..
A: The smallest unit of meaning...
Z: Yes. A concept which I'm not really sure of yet. My initial intuition is that, at least from the viewpoint of cognitive science the idea of the morpheme is a bit like the idea of the atom; a useful idea from which it makes sense to start if one aims to research the combinatoriality of matter, or in this sense, meaning, but in truth merely a theoretical construct.
A: If it is only a construct, how is it useful in this analysis?
Z: It is used precisely because it is a construct. The very idea of the morpheme is to establish a smallest unit of meaning. But if, as I believe, there are smaller units, but that the morpheme is only the smallest analyzable part of meaning, then Carnap has found a second limit; which is the limit of the resolution of language with reference to itself. I seem to think that, because of embedded ness and generativity, there are hierarchical structures of language, and that these get their meaning in virtue of their structural self similarity; and that this self similarity rests on a prior cognitive structure of standing in a kind of isomorphic relationship to the linguistic least meaningful things; morphemes. I think I'm only trying to say that there are these structures, and that therefore there is a limit to be found here.
A: I cannot fail to notice that your argument works by analogy. That is usually not a strong argument.
Z: No, you're right. But, I have a second argument by analogy which I think can strengthen it. When you define a word you do so by using a phrase which generative meaning is supposed to make up this word, correct?
A: Let us say for the sake of argument that you are.
Z: OKay, this process can be repeated for all the words of the definition, and, presuming that your definitions don't change during this process, you will in principle find all the words of your vocabulary.
A: I don't entirely agree, but, go on.
Z: Well, then you should be able to find the morphemes, because these will be the bottom concepts, since these are undefinable on account of them being a simple as simple can be.
A: I see where where you're going; How do you understand what these mean unless you can define them?
Z: Exactly. I can only see two solutions to this problem. Either they rest on extra- linguistic phenomena, as I believe, or they are subject to one of the mechanisms in the trilemma, which I kind of also believe. These two ideas are not opposed, and I think are roughly analogous to our earlier statements about the edges of the woven textile.
A: Yes, and Carnap expressly stated that the internal statements did not refer to mental states...
Z: ...which for him only leaves the ways of the trilemma. But all in all, I think this is a weakness in him, and why the resolution of the internal language will remain a limit on his view.
I think we can safely leave our interlocutors here. To sum up, there are two limits. One is the reference internally of things external, which is so to speak trying to break through the edge of the system of statements, and the other is the problem of the resolution of the language system. There are other compelling arguments to make in favor of this latter point of view, such as the jain concept of anekantavada, and the idea of the map that cannot cover the entire world, but these remain versions of the same concepts brought forward by the interlocutors.
This text has made many references to intuitions. This is to highlight that the opinions in this journal entry were written entirely from my remembrance of Carnap, and should not be taken a authoritative.