Sunday, July 16, 2006

Is the "Law of Excluded Middle" Really a Law?

The so-called "Law of Excluded Middle" (LEM) is often taken to be one of the most fundamental laws of logic. It may be expressed as follows:
(LEM) For any proposition p, either p is true or p is not-true.
LEM is to be distinguished from the Principle of Bivalence (BV), which states:
(BV) For any proposition p, either p is true or p is false.
BV entails LEM, but the converse does not hold because "not-true" may not be equivalent to "false" (That they are equivalent is what BV claims. LEM makes no commitment on the matter.)

Anyway, it occurs to me that LEM works whenever things admit of sharp classification into either this or not-this, and fails insofar as things do not admit of sharp classification. In other words, LEM works insofar as things are precise and fails insofar as they are vague. To say that a concept is vague is to say that it has fuzzy boundaries, that there is a penumbra or "grey area" in which it is simply not clear whether a given item should fall inside or outside the boundary, or whether it should simply stay on the boundary.

For example, under normal conditions a given lightbulb must be either ON or OFF, with no in-between state. Consequently, in this case we have an instance where LEM applies. If p is "This lightbulb is on" then p is either true (the lightbulb is ON) or it is not true (the lightbulb is not-ON, i.e., OFF). Contrast that example with a man who has lost most, but not nearly all, of the hair on his head. Is he bald or not-bald? Well, it may not be so clear. Depending on how his remaining hair is distributed across his scalp we might lean more toward one answer or the other, but this case just isn't as sharply defined as with the lightbulb. Perhaps we should say that he's partly bald and partly non-bald or bald in some places and non-bald in others.

Anyway, I think my point is clear that vague cases create difficulties for LEM. The crucial question is whether all such cases merely pose difficulties in the application of LEM, or whether some of them reflect a genuine limitation on LEM itself.

To say that LEM ought to hold always and everywhere is tantamount to saying that reality is perfectly sharp and determinate all the way down and that vagueness is always merely epistemic, a function of the inexactness of our language and of our concepts. This is not obviously correct. In fact, there is evidence from physics that when we go down far enough things really do get "fuzzy" - think of wave-particle duality and such. Now this fuzziness may in the final analysis turn out to be merely epistemic, but it's going quite a ways beyond the available evidence at this point to suppose that this must be the case.

Continuity is another issue where LEM naturally comes into question. Our natural conception of space, time, and motion is that they are smoothly continuous and that points, lines, events, and such are interruptions of sorts in an already existing continuum. The arithmetization program in mathematics, however, proposes to analyze such continua into sets of discrete points. If we adopt this latter course as being the metaphysical truth of the matter, then we can apply LEM to such continua all the way down. But this is a large metaphysical assumption to make. It essentially amounts to saying that continua are not metaphysically basic and can, therefore, be reductively analyzed in terms of discrete non-continua (i.e., points). Conversely, one who takes our ordinary conception of continuity to be closer to the metaphysical truth of the matter - in other words, one who holds that continuity is metaphysically more basic than discontinuity - thereby commits himself to a denial of LEM at the continuum level.

One can't have it both ways. If LEM holds across the board, then continua cannot be metaphysically fundamental. If continua are metaphysically fundamental, then LEM can't hold across the board. Since I am partial to the latter view, I say that LEM is not really a law.

5 Comments:

At 7/19/2006 1:19 PM, Blogger Tom said...

I'll definitely wanna say something about this! But I know I'll only get one good shot at it and Alan is mercilessly logical and consistent, so I'll have to get some steam up first.

Tom

 
At 7/19/2006 5:07 PM, Blogger Macuquinas d' Oro said...

Hi Alan,

Courtesy of a misspent youth --studying logic-- my first reaction to both the LEM and BV is to say that if they are not blatantly tautological, then they are clearly false in any domain beyond that of a few elementary formal languages. You know the formal results I am alluding to : it is decidable whether a wff is a theorem or non-theorem in the standard PropC or in the monadic 1st order PredC, but in any richer languages we lack decidable algorithms for determining whether a wff obtains or does not obtain.

My comment is relevant, I must concede, only on the assumption that we can regard “propositions” as wff’s or sentences in some formal or natural language. Recently you seemed to reject that view. If propositions are not wff’s or sentences, then I’m sure I don’t understand what it is that the LEM or BV is saying. To speculate: is the BV is saying that all “propositional entities” come in just two “flavors”, true and false? But that is metaphysics way over my head.

Whether a linguistic version of the LEM or BV is going hold in some interesting language L is clearly going to depend on how carefully and precisely we define “proposition” for L. If we want even the elementary propositions in L to be decidably bivalent, we are going to have to severely restrict the kind of predicates/predications we allow. Any fuzzy and “gray” predicates are probably going to put paid to bivalence immediately. Natural languages are awash in the fuzzy and gray. Even Richard’s infamous 1200 word fragment of English—remember BASIC ENGLISH?—is going to be full of undecidable sentences.

It’s interesting that English doesn’t seem worried that a sentence like “Mac is bald” may not have a clear truth value under certain circumstances. Even a choice between “Mac is bald” and “Mac is balding” may not be clear ( if for example just a few tuffs remain ). But remember we have the clearly true “Mac is nearly bald.” The underlying reality here is not fuzzy ( pardon the pun): Mac has precisely x healthy hair follicles. But “bald” in English does mean precisely that x=0, or “balding” does not mean x < half the normal number. “Bald” means x is very small, and intrepretations of very small will differ. I almost want to say that bivalence is not an important concern even in the purely declarative parts of natural languages.

 
At 7/19/2006 5:29 PM, Blogger Tim said...

Alan,

I tried to post something on this last night but apparently it didn't make it through.

Short version: Someone might respond thus.

Propositions are bearers of truth or falsity. If there's doubt about whether some piece of language is subject to excluded middle, that is ipso facto reason to doubt that P expresses a proposition.

This seems plausible to me. I think the contrary intuition is driven by the fact that we seem to make true and false statements all the time using vague predicates like "bald." But it is possible, and not (I think) dreadfully implausible, to view such a statement as a façon de parler. Alternatively, we may just plump for the epistemic conception of vagueness. (Actually, there's a way of joining these two responses, I think.) And neither of these seems to me nearly so implausible as giving up LEM.

 
At 7/20/2006 12:52 PM, Blogger Alan Rhoda said...

Good comments Mac and Tim. Unfortunately, I'm short on time right now. (My wife and I are leaving for a week in Maui tomorrow morning.) So I'm going to put up a short follow-up post. I'm also going to turn off comment moderation so that if y'all want to interact in my absence you can.

 
At 7/20/2006 3:47 PM, Blogger Tom said...

(Didn't get to read any of the comments yet; I'm on the run...)

Alan-

I stink at clarity, so be patient. But here are my thoughts.

I would say the question you’ve posed “Is continuity or discontinuity the metaphysical truth of the matter?” is a false either/or and that the truth about the essence of existence is found in process thought’s conjoining of the two. Whitehead’s doctrine is “Existence cannot be abstracted from process.” Entailed in this notion is the thesis that “the notion of a point in process is fallacious” (where ‘point’ here implies that “process can be analysed into compositions of final realities themselves devoid of process.”) My first reaction to this was that it seemed to amount to the denial of individualities or actualities, i.e., that an entity ever IS anything discrete at all. But that’s not his point. We have to take seriously the qualification “…themselves devoid of process.” It is static, fixed, realities devoid of process that Whitehead means to deny. Hence, the notions of “process devoid of individualities” and “individualities devoid of process” are both erroneous. But the tension is there—between “individualities in process” and “process involving individualities” (or your “continuity vs discontinuity”—though I think the “versus” has no basis in ontology).

Whitehead again, “The form of process…derives its character from the individuals involved, and the characters of the individuals can only be understood in terms of the process in which they are implicated.” So there’s no thought of ‘continuity’ being the metaphysical truth of the matter (over and against ‘discontinuity’). Both together describe existence as a dialectic of ‘individualities in ‘process’. The assumption that one of these has to describe the metaphysical truth of the matter is a false assumption for Whitehead. And I’m coming to agree.

It’s only if you take one to be the metaphysical truth of the matter that you run into either of the problems you mentioned. Regarding LEM, I think a process theist (or let’s say a trinitarian process theist) could uphold its legal status universally, but you’d have to apply it carefully. I couldn’t say, for example “Either p or not-p.” I’d have to say, “Either p-in-process-of-this or p-in-process-of-that). Continuity, yes, but continuity en route, and movement is always definable in terms of orientation and ideals. LEM applies to that movement. It would hold universally that individualities are always in process of becoming this or that.

Have more in my head, but I can’t put it into words! It’s baking.

Enjoy Maui! Don't get atemporal on each other now.

Tom

 

Post a Comment

Links to this post:

Create a Link

<< Home