While I'm on a roll, I figure I'll jot down what I have so far for my views on the relation between logic and reality. I'm trying to find some way of arguing for the position, or at least making it clearer; I'm attempting to balance listening to my intuitions with a desire to communicate clearly to others and be sufficiently critical of myself. So, here's the argument.
As stated earlier in my post on individuation and logic, logic requires individuals: things which are undivided in themselves (law of identity) and divided from everything else. I mentioned that this entails a partition of the world. I intend this in a mathematical sense: any application of logic would entail a partition P, which is a set of elements {a, b, . . ., z} such that for every x and y in P, the intersection of x and y is the null set, and the union of everything within P is the universe (everything, pretty much). The disjointment thesis I take to follow readily from the law of non-contradiction (there is an exact isomorphism between logic and set theory, from which I get the idea of a partition, where "and" -> "union", "or" -> "intersection", "not" -> complement, "false" -> "null set", and "true" -> "universal set").
The universality thesis I take to be a bit harder to prove. It's easy if you allow the law of the excluded middle, but as that is contentious I propose a different proof. Let's say that the union of all the elements in P does not contain everything. Then, there is something in the universe which is not in P (P', or ~P, or something of the sort). But then we can form the partition R from the union of P and ~P. ~P will be disjoint from everything in P, and so everything in R will be disjoint from everything else. Further, the union of P and ~P now does contain everything and by definition is the universal set. Therefore, R is the partition which we were looking for. I guess this ends up being an argument for the law of the excluded middle if one adopts the strict logical starting point.
So, my first argument is simply this: I don't see why I am forced to believe that this accurately describes the universe. Therefore, I am entitled to pursue a different research project. This I take to be the solidest, though of course least persuasive position.
The second argument is that it is possible that the universe does not hold up to such a partition. If this is possible, it is possible in some possible world. But if it is possible in some possible world, then (assuming our modal logic is at least reflexive) from that world it would be possible that the universe could be partitioned. If it is possible that the universe could be partitioned, then this would create a partitioning from the standpoint of that world. So the universe would be partitioned from the standpoint of that world, and it would not be true there that the universe would be non-partitioned. Therefore, if it is possible for the world to be non-partitioned, then it is non-partitioned. Ergo, etc. I think I'm playing fast and loose with the modal stuff here, and I need to work out details at some point, but that's at least the broad outline.
The third argument is the least developed. I'm not sure whether it confirms or denies my senior paper (which was on the inapplicability of Goedel's Theorem to reason; the key ambiguity being the definition of reason). More or less, here is how it would go: the elements in our partition would be isomorphic to some formal system which can express arithmetic. This is due to the fact that any decent partition actually deals with arithmetic and other mathematical facts. However, in this case Goedel's Theorem applies, and there exists some truth in the universe which has not been accounted for. But the partition was supposed to be universal. I think that I may be equivocating on "universal" throughout the various definitions, and this gets into the problem of the one and the many (is "universal" the most general thing underlying everything, or is it the collection of all the particulars?). Of course, I think this problem gives a pretty good impetus to the rejection of partitioning itself.
So, not really any fully developed ideas, but some which I have been batting around.
2 comments:
This will take some processing, but just a couple of brief initial comments:
In your proposed argument for the universality thesis (the one you start off by rejecting in the next paragraph), you begin by saying that it would be easy to prove given the law of the excluded middle, but as this law is contentious you suggest this alternate proof. Then you begin your alternate proof by assuming the principle that for every class (P) there is a complement class (non-P). I see this principle as equivalent to the law of the excluded middle. Perhaps however there is a significant conceptual difference, but the two are necessarily materially equivalent--hence your point that this proof could be taken as a proof for the law of excluded middle.
Also, about the law of excluded middle, this law is not intended to apply to everything--just to every (descriptive) statement or proposition, or Tarski sentence. Now statements/propositions/Tarski sentences are very narrowly defined things--in fact, it is essential to a logician's definition of whichever of these concepts she uses that the law of excluded middle applies to them. But (as Austin argues in his 1950 paper "Truth" in a suppl. vol. of Aristotelian Society which I was reading yesterday) there are "statements" which are not statements in this sense--assertions that are proper in ordinary discourse but which do not aspire to a bivalent truth value. I think I can come up with a response to Austin (which probably will entail contextualism of one form or another, which will put me in agreement with some cognitivist-expressivists--Mark Timmons--and I will need to further distinguish my view from theirs). But he is certainly pointing to an important phenomenon about ordinary language here.
The logical universe is a universe of statements or propositions (or, I think, statements of propositions, or propositions that can be stated). There is lots more to the furniture of the universe, linguistic and otherwise, than propositions (you could argue).
I suspect however that there is a descriptive statement to be made about every non-propositional item in the universe; and this is roughly how we import everything into the logical universe of discourse.
I am out of my league here, and I am really on a rambling mode tonight... Thanks for posting, and for reading my comments!
Yeah, I started writing that argument for the universality thesis, and realized that it was pretty much equivalent to LEM. I think, though, that at least some reason can be given for why it should be allowable in this case, even if not in general; basically, if there is not a clearly delineated non-logical sector, there cannot be a clearly delineated logical sector either, and so the partition breaks down.
I'm sure quite sure what to do about descriptive statements vs. other kinds, or whether I would be talking about a particular ontology rather than our statements about it (in which case, I would go entirely with first-order set-theoretic terminology). At any rate, I'm fine with Austin's conclusion; I would interpret it as meaning that we don't actually operate within a strictly logical framework normally.
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