Universal Logic and the Form of Transcendental Arguments

Universal Logic and the Logical Form of Transcendental Arguments;
                   A Possible Link Between My Dissertation and Kant

Before I get into the meat of the argument, there are a few table preparations to be made. Firstly, since this essay is meant to draw a line between transcendental philosophy (particularly, with a bent towards Kant), and my dissertation. Also, since I have just begun, at the suggestion/questioning of James, to attempt to weld my conception of philosophy to a Kantian framework, or at least investigate such a possibility… I suppose it would be appropriate for me to develop certain portions of my arguments revolving around Universal Logic, following Kant, to see what benefits, and problems, naturally arise.

S1. Chomsky’s Universal Grammar
If Chomsky is right about Universal Grammar – that there is a cognitive apparatus which serves as the foundation for language acquisition – then that bit of the brain also likely contains the ability to apply recursion. I say likely, because although the jump from signal-based communication to full-blooded human language requires the integration of the recursion (Chomsky), it is also possible that that bit of the brain that applies recursion, developed separately, and only afterwards ‘linked-up’ with symbol-based language (and so is not just a part of UG). In any case, the idea is that either:

(a) Some bit of the brain that had the recursive function, linked up with signal-based communication, to create language.
(b) Some bit of the brain developed the recursive function with the signal-based communication to create language.

I’m going to accept (a) prima facie, for reasons outlined in section 2, but also for the Kantian implications, explored, very briefly, in section 4. So, why accept (a)?

S2.Kripke’s Problem of Adoption
It strikes me that since (a) refers to the recursive function before its use in language, it suggests that the recursive function already had a use before its integration. That is, that it was not selected originally, in the Darwinian sense of the term, for the purposes of language. But also, I happened to attend a lecture given by Saul Kripke on the foundations of logic, about a year ago. And in it, he argued that there were certain logical laws, that it would be impossible to adopt; because their adoption would require their adherence. He called this the adoption problem, which, roughly sketched, was thus (taken from my website): How does one grasp for every instance of ‘x’ apply UI… without UI? The rules very adoption requires its adherence. The rule presupposes itself. This was put brilliantly by Romina Padro, who gave the example (call it S):

1.All Animals in the movie Madagascar talk.
2.Alex is a lion in the movie.
3.Therefore, Alex talks.

Now, if we imagine a person called Harry who doesn’t grasp UI, and who goes to see Madagascar. Well, one could explain the law to him, sure, but can he adopt it? What logical rule could he use to apply UI to UI in order to acquire it? Although he may read, say, a review of Madagascar containing 1., and may also have had the rule explained, he still would not be able infer 3., He might say: “but the review doesn’t say whether the lion talks or not.” But the inference would be impossible for Harry, because the jump from 1. to 3., would require the adoption of UI.  And you need to have the use of UI to apply UI to all instances appropriate for UI. Otherwise, UI could not be applied to all such cases fitting its application. Romina suggested that this may lead us to consider such ‘basic’ logical laws as foundational. Whether or not the other ‘basic’ logical laws contain similar paradoxes of adoption, that’s yet to be seen. But it is interesting that it appears as though certain logical laws, must, in some way, be innate, or at least this one. Because, if it cannot be taught, what else could it be?

So, accepting S1. and S2., we can now define, roughly, a case where a logical law, and also, the recursive method for structuring concepts, objects, variables, whatever, are innate features of the human mind. And so, we now have a very fragile case for the foundations of a number of conceptual tools. Those tools we can use to derive axioms concerning the nature of synthetic a priori judgements, but also, it may aid our understanding of the logical structure of transcendental philosophy, more generally.

S3.Universal Logic and the Nature of Synthetic A Priori Judgements
From this point, we have a method by which information from the world might be ordered by the brain, such that we have:

(1) A rule about how concepts can be ordered, innately.
(2) A logical law, or a rule of inference, to apply to those concepts, innately.

Although, to state one possible problem with Kant’s conception before I continue; his suggestion that a priori knowledge is “absolutely independent of all experience” strikes me as partially, (though not entirely) nonsensical. Kant appears to recognise this, when he says: “all knowledge begins with experience. …. In the order of time, … we have no knowledge antecedent to experience ….” There are many cases, where, if this point were true, his method of distinguishing types of knowledge, say, by measuring the degree in which the rules we derive can be based on experience, and so, the limits of possible extrapolation from one system to the other, would still not be trivial. But this case would not allow Kant to justify his claim of absolute independence. And that is because, if the input from the world (a) shapes our brains development (it does), and (b) shapes the method by which our brain organises information (it seems likely), as with the case of UL, for example, then even if the mechanism had the ability to be used independently from experience, the point remains that it was developed by virtue of past events/experiences, and their relation to the world and its structure. Whether they are our experiences, or our ancestors, is another important question. Whether experience, in that sense, can be passed down the evolutionary ladder, is another. Which connects, conceptually, to the fact Kant defines mathematical propositions as synthetic a priori truths. But it is possible, that UL can derive, say, mathematical principles if, for example, mathematics folds to logic, and logic to certain, cognitive principles. Another problem that relates to this: If maths was originally invented (Wittgenstein), for which there is still a strong case, then the basis for its axioms are also linked to the world. If we derive certain simple equations, such as “2 + 2 = 4” from the way in which it is possible for us to order objects, then that connection remains true, in virtue of the previous connections regarding each new derivation. The foundations of all the bricks of my house are still the earth; although what links them can also be seen as foundational in a different sense. Or simply, experience may be what forms the basis of the rule, from which we extrapolate further cases of that rule. And if that applies to mathematics, then perhaps that is synthetic, but simply a different kind of synthetic than is normally understood. Continuing, if we accept Kant’s basic idea, that information from the world is ordered by our mind, and we also accept that recursion, and the rule of Universal Instantiation, are both innate features of the brain – some bit dedicated to the ordering of information – then we have two conceptual tools to apply to those judgements to try and find out something more about their nature. Namely:

(a) The function of recursion: the ability to place variables, concepts, whatever, at different orders. &
(b) The rule of Universal Instantiation: the ability to apply to whatever is placed in the recursive system, a rule by which the variables, concepts, whatever, are defined.

(Note: there are some possible, interesting implications still to investigate. For example, the possible derivation of set-theory from the two, which, considering maths, or a lot of maths, is reducible to set-theory, begins to give us an interesting perspective concerning the nature of the application of mathematics, and also its basis as synthetic judgement, as mentioned above.)

S4. Final Remarks
Before I make some remarks about a few possible implications regarding these two principles (as an investigation would be impossible to present in this essay); I wanted to widen the scope of the point, slightly, about the distinction between a priori as partial, or absolute independence from experience. Moore, for example, defines Kant’s transcendental position as not being about the knowledge of objects, but about “our knowledge of how we know them.” This could be a false distinction. Connecting to this his use of Kant’s spectacles, he seems to take the faculty which orders the brain, and the world it interprets, as requiring a further step, connection, or lens. He is assuming a connection beyond the fact that the world exists, and our mind was formed within the world to understand it. It’s an ontological mistake. His point about the spectacles is thus: he thinks Kant is saying that in order to view an object, we require this cognitive ordering (and, in a sense, he is). Therefore, we can’t really observe anything but the structure which orders those things, itself. But: If an axe hits a tree, and causes a cut; there is no explanation required beyond what was just given. Perhaps, that is a poor analogy. But the point I’m trying to get across is, Moore’s problem seemingly originates from his understanding that Kant considers the input as being sent from one object to another, computed in some way as to order it, and then presented as a final picture; and that in some instances, this process occurs without empirical grounding (which would cause the problem he raises). On the other hand, if the brain is formed by/in the world, then one first has to explain why there need be a third-intermediary between its operation and reality, at all. The impact of W on B shares the link of impact. The logical rule: iff A then B. A, therefore B., shares the application of the rule. It is not absurd to suggest that the structure of the mind, shares with the world, the impact of the latter on the former regarding its structure. And so, in this case, what links (1) Kantian a priori synthetic truths, (2) UL and (3) the world, is that the latter is the bedrock of the former in both cases. Or, those rules are the grooves which direct the flow of input from reality; cut out from thousands of years of past experience. Kant’s question concerns the relations between the structure of the world, and the structuring capacity of the mind; and wouldn’t one expect the two to be consistent with each other? Otherwise, what would be the use of the mind in the first place? (Why would this feature evolve?)

These are just my initial ideas, and of course, they could be very wrong, and very misguided. In any case, below are three different examples of a priori judgements, which I will present some cursory remarks on (it would take a full investigation, to consider the implications of the above):

(1) “Nothing can be red all over and green all over at the same time.”
(2) “2 + 3 = 5”
(3) “If person A is taller than person B and person B is taller than person C, then person A is taller than person C.”

Qualitative, spacial, and mathematical concepts may all be equally foundational to the computations of UL. In fact, it seems that they are among the most basic concepts, when it comes to considering what constitutes experience, intentionality, logical laws, etc., (the really difficult stuff).  But at least regarding maths, there are some possible methods to answering how it might be the case that we have such knowledge. If, for example, UL contains basic logical principles, enough to, say, form a set-theory, in which, maths can be compressed; then it makes sense that the extension of those rules, in the form of mathematical propositions appear to be synthetic (and in a sense, they are). Colour is another subject, and I find this example at odds with Wittgenstein’s elucidations of colour, and language as a way of life. But he also discusses, at length, the logical relations between colours, and colour concepts, which I think pertains in some way to this point, and the emphasis Kant places on structure. It strikes me that something could be called red all over, and green all over, if for example, that’s what the people of some particular culture called “purple”. And in the same sense, “2” might mean something entirely different to some other culture. So, I suppose this is a point about (1) conceptual necessity, and (2) how the logical relations between colours prohibits their use in such a proposition; and that appears to be a very interesting problem. Still, the axioms of mathematics, and of the rules of inference, both have possible linkages to the system of UL being proposed in this essay. One might need only continue down a similar line, with causality, time, and inference in mind, to come to a sensible conclusion to other basic, logical laws. And with such logical laws to hand, one may be able to make a case to explain colour concepts, and the reason we need terms like a priori synthetic, or conceptual necessity, at all. But I suppose, this is enough to discuss. But also, it appears there’s lots more to do, regarding Kant.

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