In my previous post
I raised some questions about the "Law of Excluded Middle" (LEM), which states that
(LEM) For any proposition p, either p is true or p is not-true.
The gist of my concern was that if LEM applies across the board then this implies that reality is discrete all the way down
, and thus that genuine continuity is really an illusion.
But subsequently I've come to see that my argument trades on a possibly illicit shift from propositions to reality. As defined, LEM applies to propositions
. Propositions, I gather, are usually taken to be abstract
representations of conceivable states-of-affairs that are true when those states-of-affairs obtain and false otherwise. It's this abstractness that I failed fully to grapple with.
An abstraction always leaves something behind. For example, when I notice that my car needs a washing I temporarily ignore or abstract from
other details of the car (its location, directional orientation, level of gas in the tank, etc.).
Now, the question of whether reality is fundamentally continuous is not a question about abstractions but about what's concrete, about reality-in-itself. Hence, if we think of propositions as invariably abstract, then continuity in reality can't pose any problems for LEM.
Of course, there is the issue of whether propositions are invariably abstract. I think they are so for us
, because as finite beings situated in space and time we can only approach reality in bits and pieces from one perspective or another. So the only way we can get clear on anything is to focus on some things to the exclusion of other. In other words, human thought requires abstractions.
A transcendent, omniscient being, however, wouldn't know reality in bits and pieces or from any particular perspective but rather all-at-once. Such a being would have, I think, a non-abstract
representation of the maximal state-of-affairs that is reality. If such a being is so much as possible, then propositions (understood simply as representations of conceivable states-of-affairs) are not necessarly abstract.
But all that means is that LEM may not apply for beings like God.
There's another way in which the abstractness of propositions (as they occur in human minds) that is relevant to LEM, though. Consider my car again. When I say to myself that it needs a wash, I make a mental judgment that it is a car and that it needs a wash. In those two respects I've mentally categorized it as this
. Hence, LEM applies to my thought with respect to those categories
. But I've made no judgment about other features of the car. As far as my conception goes at that moment
, whether the car is moving or not is indeterminate, whether it needs gas or not is indeterminate, whether it is pointed east or west is indeterminate, and so forth. In other words, the determinateness of my conception of the car as needing a wash does not entail anything about how the car is to be determinately categorized in other
respects, much less whether the car can in principle be determinately categorized in all
respects. The proposition in my mind is neutral on all that.
Well, I've gotta run now. So to abruptly conclude: (1) Vagueness concerns what's not
brought into focus in our conceptions. With regard to what is brought into focus, LEM applies. So insofar as there is a determinate abstract proposition at all, it obeys LEM. (2) LEM implies nothing, however, about how things are in reality. So one can't draw any grandiose metaphysical conclusions from LEM, like "apparently continuous space and time are really
just sets of discrete points". LEM is merely a law of human thought
insofar as those thoughts are determinate.