Further Thoughts on Excluded Middle
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 and not-that. 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.
posted by Alan Rhoda @ 12:53 PM
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33 Comments:
Alan: Hence, if we think of propositions as invariably abstract, then continuity in reality can't pose any problems for LEM.
Tom: Well, arguably one could still say there’s a problem for LEM because it’s that reality which is supposed to ground the truth of propositions describing it. If reality is fundamentally continuous (which I take to mean “in process”) then one could possibly argue that no matter how fast the shutter speed of the camera can go in taking a picture of reality, the picture will always be out of focus because reality is at every level in process. It’s never “still” enough, never “stops long enough,” to ground propositions that are meant to describe concrete realities, the idea being that reality has to be discrete enough to serve as grounds to which propositions corresopnd or not.
I don’t think this is a problem (for me at least). One can still hold (as my comment to your previous post notes) that reality is continuous through and through but that LEM holds if propositions assume this continuity (even if they appear to posit fixed states of affairs). In other words, given that reality is continuous, and given that propositions are true virtue of correspondence to reality, it follows that truth grounds are invariably continuous. This continuity must transfer by means of correspondence to propositions, and so to LEM. Given a continuous worldview (a process worldview), what we essentially say when we affirm LEM is not “For any proposition p, either p is true or p is not-true,” but something like “For any proposition ‘p is in process of…,’ p is either in said process or not.”
Your partner in crime,
Tom
Tom:
If reality is fundamentally continuous (which I take to mean “in process”)
HammsBear:
Not necessarily "in process" but non-discrete. Where does "shade" end and "sunny" begin? When does short end and tall begin? When does A# end and Bb begin? And what of E# and Cb?
Though if the quantum loop gravity guys are right and space is discrete at the planck scale, one could argue that all analog is discrete, at that level.
Alan,
I think by vagueness you are trying to get across the idea that we cannot think about all aspects of a concept at once. But I am not sure this is valid. At least the example you use fails. When I think about 'car' it includes the engine, etc. So all this information is entailed in the word car when I think about washing the car.
Maybe the relation between abstraction and reality in time can be covered by what I call fullness. Fullness encompasses the fact that there is both a 'static' reality (an ideal) and a process that grounds each metaphysical truth. Both a 'being' and a 'becoming implicit in reality.
Let me explain. consider the proposition 'My husband loves me.' This proposition was true before My husband and I got married, and it is true now, but it is more fully true now because my husband's love for me now is closer to the ideal of God's love then it was then.
Also between bald and not bald there is a fullness of baldness. The true/false dichoctomy exists timelessly, (at a given t something either is or is not) thus in 'two dimensions' so to speak. Hoever, it cannot account for unity over time or the recognition of an ideal. For this we need a 'third dimension'.
This fullness is only relevant for propositions that deal with the metaphysical- Those ideals which are grounded in God's essence rather than our own conceptions of the physical world. (it seems absurd to me to talk about an ideal car for instance)
I realized that above I talked about baldness as having fullness then said that conceptions of the physical world don't count. Sorry for the contradiction. This is just some exploration, I haven't thought everything through.
c grace, an example
Bill is six feet in height, he walks into a room where some others are, all five feet five inches in height or less.
All perceive Bill as being "tall".
Now Will walks into the room, he's seven feet in height. He perceives everyone else in the room, Bill included, as "short", i.e. "not tall".
Is Bill "tall" or "not tall"?
Any propositions about Bill's height could not claim LEM as unassailable, imho. In this case, Bill can be "tall" and "not tall".
This is because reality is analogue/continuous and not discrete at most levels we perceive. We see water, not the discrete hydrogen and oxygen molecules that make it up.
Many propositions that concern reality cannot hold to LEM, imho.
Tom: If reality is fundamentally continuous (which I take to mean “in process”)…
HammsBear: Not necessarily "in process" but non-discrete. Where does "shade" end and "sunny" begin? When does short end and tall begin? When does A# end and Bb begin?
Tom: Propositions about “height” (and other such examples) all concern ‘comparative’ states of affairs and as such depend on two or more discrete entities being compared. There doesn’t need to be a specific height that absolutely defines “tall.” Bill is only “tall” when compared to someone shorter than him. But of course he’s “short” when compared to someone taller than him. There’s nothing here to undermine LEM. How many grains of sand mark the difference between “mound” and “pile”? Well, it all depends on those making the observation. Everybody will judge the difference a bit differently. The point is that LEM holds universally within a sufficiently defined society’s semantic, not across different semantics. What “tall” is will differ from Holland where a 6’ girl isn’t particularly out of the ordinary to China where a 6’ girl would be highly unusual. But all semantics agree on the notion of “tall” and “short.” They agree that any person is either taller than, shorter than, or the same height as any other person. LEM holds.
Musical notes as well. Sound is a mechanical disturbance or wave produced by a vibrating object and transported through a medium from one location to another. Sound waves possess length, frequency, speed. Each musical note has its own pitch that differs from the next. Middle C = 261.6226 hertz; C# = 277.183 hertz. LEM isn’t undermined by a note of greater than 261 and less than 277. Would that note be C or C#? Well, that’s a semantical/defintional issue, not a metaphysical one. We in the west can come up with new names for notes at such frequencies. Other cultures have. There is continuity after all. But in spite of the continuity, any sound wave will have its own hertz measurement. LEM would apply to that I should think. Though LEM might seem to be undermined by the fact that there are sound waves with hertz measurements that (when measured technically by western standards) that overlap, the vagueness issues (in my view) from imprecise definitions and finite prehensions of reality. It still remains (as far as we know) that every sound wave has its own frequency, speed, and hertz measurement and that any other sound wave is either shares the same measurement or is of either greater or lesser hertz.
As Alan says: "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."
Tom: I’m cool with this. It seems compatible with the fundamental doctrine of a process metaphysic—“Existence cannot be abstracted from process.” Alan would disagree with this doctrine (since he holds to God’s atemporal existence sans creation), but we would still agree that LEM is a rule of human language as it describes experienced reality (and that our experience is always limited; we ‘prehend’ but we cannot ‘comprehend’). For me, so matter how deep you go into analyzing reality, you never get to static realities devoid of process.
Tom
Tom:
Middle C = 261.6226 hertz; C# = 277.183 hertz. LEM isn’t undermined by a note of greater than 261 and less than 277
HammsBear:
1. Is the proposition pC = (hertz of 269.4028 is C) true or false?
LEM holds that if pC is not true then necessarily it is false. Would you say that, universally/necessarily, anyone hearing a hertz of 269.4028 would recognize that as being C? or not C?
You would actually get some people asserting pC is true, "that's C" and some asserting it's false, "that's a C#". (pC V ~pC) would not be necessarily true.
Fuzziness of reality doesn't need LEM to be never true, only that it doesn't necessarily/universally hold, which is what Alan was saying.
2. There are numerous logics that don't hold to the universality/necessity of LEM:
http://plato.stanford.edu/entries/contradiction/
"In those systems that do embrace truth value gaps (Strawson, arguably Frege) or non-classically-valued systems (Lukasiewicz, Bochvar, Kleene), some sentences or statements are not assigned a (classical) truth value; in Strawson's famous dictum, the question of the truth value of “The king of France is wise”, in a world in which France is a republic, simply fails to arise. The negative form of such vacuous statements, e.g. “The king of France is not wise”, is similarly neither true nor false. This amounts to a rejection of LEM, as noted by Russell 1905. In addition to vacuous singular expressions, gap-based analyses have been proposed for future contingents (following one reading of Aristotle's exposition of the sea-battle; cf. §2 above) and category mistakes (e.g. “The number 7 likes/doesn't like to dance”)."
What is being rejected, he says in his notes, is not (P V ~P) but the necessity of it.
3. LEM is used in non-constructive proofs in mathematics but not allowed in constructive proofs.
What's a non-constructive proof? For example to prove that U = (unicorns exist) is true, you assume that U is false and then show how that is absurd (reductio ad absurdum).
Once you've proved that it's absurd to hold that U is false, ~(unicorns exist), then you've proved U is true (unicorns exist) via LEM. But you've still not produced an actual unicorn.
One example used to show the 'weakness' of LEM is this:
Assume there exists irrational numbers a and b such that a^b is rational {a^b means a raised to the b power}
q = a^b
LEM says that either q is rational or q is irrational (r v ~r) but not both.
Given, the square root of 2, sqrt(2), is irrational and 2 is rational.
A. Let a = sqrt(2) and b = sqrt(2), then q = sqrt(2)^sqrt(2). q is irrational
B. Let a = sqrt(2)^sqrt(2) and b = sqrt(2), then q = [sqrt(2)^sqrt(2)]^sqrt(2)
But (a^b)^c is a^(b*c) so q = sqrt(2)^[sqrt(2) * sqrt(2)] = sqrt(2)^2 = 2, a rational number so q is rational
So q = a^b has been proved to be rational or irrational (r V ~r), showing how LEM doesn't universally hold.
Bear: Is the proposition pC = (hertz of 269.4028 is C) true or false?
Tom: I don’t mean to be obtuse or difficult, but 'it all depends'. It depends upon the scale of measurement one employs, and one MUST employ some scale. Is pi 3.14? Well, yes and no. It’s also 3.14159. But it’s more. See? Is the carpenter measuring his cut to within an inch or within 1/100th of an inch? If we’re in a recording studio with high-tech sound detection equipment wanting to tune a grand piano for a performance by John Williams, I’d say a sound wave of 269 hertz is not the “C” we want. But if we’re just tuning my piano at home for my youngest and we don’t have high-tech equipment and can only measure the hertz to within a range of accuracy +/- 5% and we got it a bit higher than 261 hertz, I’d say yeah, that’s C.
The point of all this is that “C” is never posited without some relevant context that permits or forbids the employment of certain ranges. Is pi 3.14? Well, yes, that’ll do for some applications. It’s true for those applications. But for some other applications (one’s requiring 3.14159265) it would be false I would think. This isn’t a denial of LEM. It just means that language is never employed absolutely (i.e., irrespective of contextual concerns—how the terms used in the propositions get their meaning as intended by the one making the assertion).
Bear: LEM holds that if pC is not true then necessarily it is false.
Tom: Right.
Bear: Would you say that, universally/necessarily, anyone hearing a hertz of 269.4028 would recognize that as being C or not C?
Tom: Depends on how good an ear they’ve got, or whether the person listening is a scientist in a laboratory with sound equipment.
Secondly, the failure to recognize it as C isn’t equivalent to recognizing it as not-C. C the difference? There’s ‘not-recognizing it as C’ and then there’s ‘recognizing it as not-C’. These aren’t the same thing. People who don’t know English for example would not recognize that particular pitch as “Middle C.” There’s no trouble here for LEM. LEM means that when a person asserts “C = 269 hertz” that assertion is true or false. But there are no assertions apart from shared language, context, and history. These are all there in the term “Middle C.” “C” is just the English letter in a particular context with a shared history of musical notation use to describe a sound wave of 261 hertz (or a hertz range, depending on the demands of context). In China they may call it something else or nothing at all.
Bear: You would actually get some people asserting pC is true, "that's C" and some asserting it's false, "that's a C#".
Tom: Probably, because some have better ears than others and some may more rigorous than others in limiting the hertz range.
Bear: (pC V ~pC) would not be necessarily true.
Tom: It would be true if C is constant, as it should be. Take any one of the hearer’s understanding of C and put it in for (pC v ~pC) and the formula will hold universally.
Bear: Fuzziness of reality doesn't need LEM to be never true, only that it doesn't necessarily/universally hold, which is what Alan was saying.
Tom: I got that, yeah. I’m still looking for an example of fuzzy reality that isn’t in fact just an example of semantic confusion. Musical notes, baldness, and tallness aren’t examples so far as I see.
Bear: …in Strawson's famous dictum, the question of the truth value of “The king of France is wise,” in a world in which France is a republic, simply fails to arise. The negative form of such vacuous statements, e.g. “The king of France is not wise”, is similarly neither true nor false.
Tom: I remember our dealing with this a long time ago. In the event France is a republic (without king or queen) both “The king of France is bald” and “The king of France is not bald” are false. LEM doesn’t mean one of this is true and the other is false. It would mean this IF these were contradictories. But they’re not contradictory props, they’re contrary props. The contradictory of “The King of France is bald” is not “The King of France is not bald” but rather “It is not the case that ‘the King of France is bald'."
Bear: This amounts to a rejection of LEM, as noted by Russell 1905.
Tom: He just mistook “The King of France is not bald” as the contradictory of “The King of France is bald.” Bad move.
Gotta go.
Tom
Tom:The point of all this is that “C” is never posited without some relevant context that permits or forbids the employment of certain ranges.
HammsBear:That's what propositions about mathematics have, a severely limited domain. But our propositions are about the world (sea battles, baldness, etc.)
If C = 261.6226 hertz and C# = 277.183 hertz and nothing inbetween is allowed, it's easy to state that 277.183 is not C and (C v ~C) holds. But in the "real world", it'd be awkward to hold that 269.12 hertz is (C v ~C) when the audience is split down the middle. LEM is useful in severely restricted domains but it's claim to universality breaks down in more real/analogue/continuous/fuzzy domains.
HammsBear: If C = 261.6226 hertz and C# = 277.183 hertz and nothing inbetween is allowed, it's easy to state that 277.183 is not C and (C v ~C) holds. But in the "real world", it'd be awkward to hold that 269.12 hertz is (C v ~C) when the audience is split down the middle.
Tom: Why would the audience be split down the middle? Not because they don’t think there’s a definite hertz line between C and C#. It’s because half have one definition of what C is and the other half have another definition of C. They're drawing the lines at different places. If C as defined by group A as limited to 261hertz stands in for the first C in (C v ~C) and C as defined by group B as 277 hertz stands in for the second C in (C v ~C), then of course (C v ~C) would be false because C wouldn’t represent the same thing in each even though by rule it should (they’re both labeled C). But if C is constant, if you define you terms and others know how you’re defining things, LEM holds. Even if hertz ranges between 261 (C) and 277 (C#) are figured into the definition of those notes (that is, C is defined a 261 hertz +/- half the distance to the note above and the note below C) LEM would still hold. What denying LEM requires is that there be no definite hertz line possible between 261 and 277 hertz, that the metaphysical (and not definitional or epistemological) line between the two is fuzzy. But of course there are clear hertz measurements possible all up and down the scale from 261 to 277.
Tom
I said: "Even if hertz ranges between 261 (C) and 277 (C#) are figured into the definition of those notes (that is, C is defined a 261 hertz +/- half the distance to the note above and the note below C)"
Take away that final C. It should read: "...and the note below 261 hertz)"
Just to be clear.
I'm saying what Alan said: "LEM is merely a law of human thought insofar as those thoughts are determinate."
I would only add that its a law of human thought as that thought proposes to express the 'corresponding relationship' between reality and language.
Tom ;o)
Tom:C is defined a 261 hertz +/- half the distance to the note above and the note below C
HammsBear:To what level of accuracy? That's the problem with our analogue/continuous/non-discrete reality. We could demand accuracy down past what even your instruments could measure.....
This is why static laws with coarse-grained accuracy only universally hold in severely limited, artificial/abstract domains.
Mathematical proofs need to be proved (a tautology) and that's the problem with non-constructive proofs, they aren't really proved, only inferred by LEM (RAA).
Is pi a rational number or irrational number? They don't know yet because they haven't solved it yet. So pi is (R v ~R) has no proof yet. geddit?
Tom: C is defined a 261 hertz +/- half the distance to the note above and the note below C.
HammsBear: To what level of accuracy?
Tom: To whatever level one specifies. That’s just it, one must specify what one means by C when one talks about C, spelling out that this or that hertz measurement or range is or isn’t C. It’s simple. If we say C is 261 hertz plus half the distance to the note above (C#) and minus half the distance to the note below (B), C would be a hertz range and LEM would hold for propositions about C and a particular note with a particular hertz measurement would fall within that range or not.
But we may wish to be more specific and limit C to 261 hertz, period. And we could then define C# as 277 hertz, period. You’re saying that in this case a sound wave of, say, 270 would be a case of fuzzy reality since there’s obviously no middle ground between a C and a C#. But you’re confusing language/semantics with reality. Of course there would be middle ground, i.e., sound waves with hertz measurements of greater than 261 and less than 277. It would just be the case that our language (definitions) wouldn’t have any middle ground between 261 and 277, so sound waves of greater than 261 and less than 277 would be neither C nor C#. What musical note would such sound waves be? They wouldn’t be musical notes at all per our definition. We’ve defined them out of our musical vocabulary. But this has nothing to do with LEM.
HammsBear: That's the problem with our analogue/continuous/non-discrete reality. We could demand accuracy down past what even your instruments could measure...
Tom: That’s not a problem for LEM, which applies to propositions that carry intended meanings.
HammsBear: Is pi a rational number or irrational number? They don't know yet because they haven't solved it yet. So pi is (R v ~R) has no proof yet.
Tom: Assuming pi is infinite, propositions positing p would still fall under LEM. Any given number you toss out there is either pi or not. Assuming pi is finite (which I don’t think it is), the same holds true.
Come on, gimme something to chew on! ;o) Where’s that fuzzy reality—something that both is and isn’t something concrete and which isn’t just a semantic show?
Tom
Bear, it looks to me like you (and all constructivists) take LEM (p v ~p) to mean (either p is provable or ~p is provable). Forgetting for the moment whether this is true for God or not (one could argue God could prove whatever is provable), everybody agrees it’s not true. This would seem like a denial of LEM. But where constructivists study what is provable, classical mathematicians study what is true. And whether or not p or ~p is provable or not, it’s self-evidently the case that p is either true or not true.
Remember too that LEM doesn’t entail bivalence. Bivalence means p is true or false. LEM only claims that p is true or not-true. But the latter is consistent with multivalued systems (in which every proposition possesses some truth value).
Hope that helps clarify things a bit. If you’re taking LEM in constructivist manner to mean provable or not-provable (you’ve mentioned proofs several times and I keep thinking, “As if that matters…”), then I’ll agree with you that LEM doesn’t hold. Not everything is provable. But that’s not the understanding of LEM Alan is working with. Hopefully he’ll be back from his atemporal visit to Maui and set us straight.
Tom
Tom:Come on, gimme something to chew on! ;o) Where’s that fuzzy reality—something that both is and isn’t something concrete and which isn’t just a semantic show?
HammsBear:
Let a and b be irrational numbers.
Is q = a^b rational or irrational? necessarily(R v ~R)
I'm sure I don't know. Are you suggesting it's both rational and irrational?
Tom
Tom:I'm sure I don't know. Are you suggesting it's both rational and irrational?
HammsBear:Sorry, I should have been more clear.
It's a classic example of why intuitionists and constructivists have rejected LEM because it proves q = a^b is rational without producing a number for q.
Like proving unicorns exist without producing one.
RAA, non-constructive proofs are seen to be weaker proofs because actual answers aren't produced.
That's why some have rejected LEM.
It seems clearer to me now Bear that you're taking LEM (p v ~p) to mean (either p is provable or ~p is provable). If that's how you want to define LEM, then I agree, not all things are provable, therefore LEM doesn't hold universally.
But that's clearly NOT how LEM is usually understood. LEM as traditionally understood implies simply this: Either unicorns do exist or they don't exist. LEM says THAT disjunction is TRUE. There's no middle ground between existing and not existing. Whether or not we're able to prove WHICH of these is true (whether, say, unicorns exist or not) is what you seem to be applying LEM to, but that's a different issue.
Applied to your math question, LEM does not mean q = a^b is provably rational or irrational (so that if it's not provable LEM fails). All LEM says is that (q = a^b) is or isn't true (regardless of whether or not there's a proof for its being true or not true, or rational or irrational, etc.).
Tom
Bear-
What in your view are the relevant applications and/or issues that require denying LEM? As far as I can see the real applications only may be in developing proofs for claims involving irrational numbers. Isn’t this a little ‘out there’ anyhow?
More to the point, why believe LEM wouldn’t apply to propositions about:
(a) comparative states of affairs (taller than, shorter than, etc.),
(b) states of affairs whose descriptions vary definitionally from person to person (baldness, the difference between a mound and a pile, where C ends and C# begins), and
(c) future contingents (is “he will freely choose X” neither true or false or is it false and “he might/might not choose X” is true)?
In other words, what problems do you feel are avoided in these cases by denying LEM? Still differently, what problems do you think I face in describing these states while assuming LEM? What is there to be said about (a), (b), or (c) that you feel can’t be said by propositions under the rule of LEM?
My feeling is, if so far as we are able to determine there appear cases (irrational numbers) for which LEM does not apply, it still remains to be argued that (a), (b), or (c) are examples of such cases. Can you offer some reason for why LEM should not apply to propositions describing (a), (b), or (c) realities?
Tom
Tom:
Either unicorns do exist or they don't exist. LEM says THAT disjunction is TRUE. There's no middle ground between existing and not existing.
HammsBear:
But there is 'middle ground' in other examples.
Either Bill is bald or is not bald. Removing which single hair changes Bill from not bald to bald?
You could arbitrarily define a definite boundary, say exactly 50% hair loss is the boundary between bald and not bald, but again, doesn't point out the un-universality of LEM? Do you think people see the change when you say "Bill is not bald", remove one hair and say "Bill is bald"?
HammsBear: But there is 'middle ground' in other examples. Either Bill is bald or is not bald.
Tom: I love it! I'll give this one more go from my perspective. Yes, Bill is bald or not bald (assuming Bill exists). But the word ‘bald’ references some particular state of affairs held in mind by the person making the claim about Bill. And nobody makes the claim without SOME understanding of baldness in mind. For sure ‘zero hairs’ is bald. Nobody doubts that. But we almost invariably say of a man with very little hair on his head that "He's bald." Why? Because reality (the guy's HEAD) is fuzzy or indeterminate? Hardly.
What undoubtedly happens is people say a man is bald or not based not on a particular hair count but rather based on some overall impression they have about his head. Who bothers to count hairs? Nobody. What we invariably do is employ ‘bald’ to describe the effect a man’s appearance has on us, and so ‘bald’ comes out referencing any state of hair loss in which the overall impression a person gets from observing a man’s head (assuming he has some hair) is practically the same as the impression they’d get if he had no hair; that whatever hair he has doesn’t significantly differentiate him from a man with zero hairs on his head.
What nobody does is demonstrate the failure of LEM by looking at a man with very few hairs on his head and admitting they have no idea whether he’s bald or not since they haven’t done an actual hair count. Language doesn’t map reality with such mathematical precision all the time and that’s because WE don’t map reality so precisely all the time. Sometimes we cut corners, and this shows up in our language. One person will say of a man with very little hair, “See that bald guy at the next table?” while his wife will reply, “He’s not bald. He’s got SOME hair.” On your view, what accounts for the difference between these two appraisals of the man’s head is fuzzy reality, some indeterminate nature of reality (the man’s head), something which both is and isn’t the case, an indeterminacy that necessarily shows up in our language with the failure of LEM.
But in my view this gets things backwards and leaves out one very important point about truth, namely, that the words and phrases we use get their meaning from US, the ones making the claim. And nobody makes claims about baldness without making some measurement and basing the choice to use the word ‘bald’ accordingly. So the person who says the guy at the next table is bald is obviously using ‘bald’ to include men who have as little hair as the man at the next table (viz., not enough hair to matter, not enough hair to make one’s appearance significantly different from a man with zero hairs on his head). Based on that measurement (a measurement that defines how the person is employing language), the person decides to just call the man bald and save himself the energy of having to insert ‘nearly’ or ‘very nearly’.
But the man’s wife who objects “No, he’s not bald. He has SOME hair,” is employing the word ‘bald’ based on a different measurement. Hence, ‘bald’ is NOT constant. It doesn't carry the same meaning, which is different from saying it doesn't carry any distinct meaning. So their disagreement over whether the man is bald or not has nothing to do with fuzzy reality or the failure of LEM. It’s nothing more than semantics, which is shown to be the case when the husband replies, “Well, sure Hunny. Yes, if by ‘bald’ you mean ‘having not a single hair’ then you’re right, that guy’s not bald. But look at him! He’s got a tiny patch of hair growing out the back of his head. What difference does that really make? None. He might as well be bald." And if in the man's judgment the guy might as well be bald, then it's meaningful for the man to SAY the guy is bald. But this has to be kept in mind when trying to apply LEM to the man's description of the guy at the next table versus his wife's description of the same man; because to her it's not the case that he might as well be bald. Different impressions and judgments. That's all.
Hammsbear: You could arbitrarily define a definite boundary, say exactly 50% hair loss is the boundary between bald and not bald, but again, doesn't point out the un-universality of LEM? Do you think people see the change when you say "Bill is not bald", remove one hair and say "Bill is bald"?
Tom: The line that marks the boundary in a person’s understanding of ‘bald’ and ‘not bald’ doesn’t have to be a particular hair count and almost certainly can't be that. Nobody can go around counting the hairs on a guys head. The line is drawn based on the impression one gets from a man’s appearance relative to the absolute absence of any hair (the one measure of hair loss we all agree is consistent with a mans’ being, strictly speaking, ‘bald’). A man might have some hair. We (subconsciously mind you) ask ourselves, "Would he look all that different if he had NO hair?" If we judge that he wouldn’t, we say "He’s bald." If we judge that he would, we say "He’s not bald."
This is not an example of there being a middle ground between existence and non-existence. It’s just that our judgments about appearance compared to what we both agree is undoubtedly ‘bald’ (that is, zero hairs) are determining where we draw the line and how we use words. LEM doesn't require us to make the same judgments. It just says that once we render a judgment, the reality we reference either qualifies or not under our particular judgment.
And that’s not at all arbitrary. How the appearance of man’s head impresses us is anything but arbitrary! If it were, then ‘bald’ would never mean anything. But it always means something even if we feel free to attribute slightly different meanings to it. That’s no denial of LEM. LEM just requires that once a particular meaning for or judgment about ‘bald’ is determined, THEN the disjunct “Bill is bald or not bald” is true (assuming of course the existence of Bill). Given MY employment of the word (where 'bald' = any state of hair loss on a man’s head that leaves me feeling about his head pretty much what I’d feel about it were he perfectly shaven), Bill qualifies as bald even though he has a very small patch of hair on the back of his head which you can see only if you stand in the right place behind him. In this case LEM applies to the proposition “Bill is bald or not bald.” That is, Bill’s head either leaves me feeling this way about him or not.
But say you reserve ‘bald’ to describe the total absence of any hair on man’s head. You scan his head with a high-tech hair detector and discover his head is as smooth as a billiard ball. You've got hard empirical evidence now so you can draw the line. Well, given YOUR employment of the term, the particular state of hair loss YOU have in mind which you call 'bald' either describes Bill or it doesn’t. LEM is right there. Neither of us has escaped it. LEM applies to my claiming “Bill is bald or not bald” when 'bald' is employed by me with my understanding of ‘bald’ in both of its occurences. But it also applies to your making the same claim when employed by you with your understanding of ‘bald’ in both its occurences. What LEM can't apply to is the claim "Bill is bald or not bald" where my understanding of ‘bald’ stands in for the word's first occurrence and your understanding stands in for the second.
This is the mistake you make with 'bald' (and other similar adjectives). You observe that people employ the same TERM differently and conclude that the REALITY being referrenced (some guy's head) is fuzzy. But that doesn’t follow.
Tom
Tom, I think your excessive word wrangling points out how the "continum" of reality doesn't map so neatly to the severely restricted domain that LEM requires. If you think the guy at the next table is bald and your wife thinks he isn't bald, that shows how (bald v ~bald) is useless in describing the man.
If LEM is in the mind of the beholder, then it's not universally applicable.
p = [triangles are generally equilateral]
(p v ~p)?
Tom, let me ask you, why do you think some logicians have abandoned LEM? Do you understand any of their arguments?
Bear: Tom, I think your excessive word wrangling...
Tom: I get wordy. Nobody's perfect.
Bear: If you think the guy at the next table is bald and your wife thinks he isn't bald, that shows how (bald v ~bald) is useless in describing the man.
Tom: Then you're missing the entire point. Where two people employ 'bald' with different meanings (bald v ~bald) won't be true under LEM. But that's no failure of LEM (which requires 'bald' mean the same thing in b v ~b). You see different employments of the same letters "bald" to reflect an indetermiante reality. But that's just mistaken in the case of baldness and other comparative states.
Bear: If LEM is in the mind of the beholder, then it's not universally applicable.
Tom: I never claimed LEM was in the mind of the beholder. It's not. It's the meanings of words and phrases that are in the minds of those making truth claims. LEM applies to those claims where the variables are consistent within claims.
Bear: p = [triangles are generally equilateral] (p v ~p)?
Tom: If p is true, then I'd say the disjunct is true, yes. Either triangles are generally equilateral OR they are not generally equilateral where "generally equilateral" refers to the same thing in both instances. If you disagree, then which would you affirm: that (a) triangles are NEITHER generally equilateral NOR generally equilateral, or (b) that they are BOTH generally equilateral AND generally not equilateral?
I think I understand the arguments against LEM which are based on language well enough. I think those logicians who feel 'baldness' is an example LEM cannot apply to are mistaken. The arguments from irrational numbers I'm afraid escape me. I guess that means I can't know what I'm talking about then when it comes to 'baldness'. But as I said, if it's really the case that LEM doesn't hold for some very obtuse mathematical equations about which there still is some disagreement, it still doesn't follow that LEM can't hold for propostions about comparative states (like baldness) or future contingents.
Tom
Tom: Then you're missing the entire point. Where two people employ 'bald' with different meanings (bald v ~bald) won't be true under LEM. But that's no failure of LEM (which requires 'bald' mean the same thing in b v ~b).
HammsBear:LEM is useful when the domain is severely restricted, as in mathematics and 'classic' logic.
It's not useful when boundaries are 'fuzzy'. My point all along.
The boundary between 'bald' and 'not bald' is fuzzy partly because reality reveals people having different ideas of 'baldness'.
If LEM were absolutely true in all cases, it could be applicable in this example, but it ain't.
It's not a universal principle, outside of severely limited domains.
HammsBear: p = [triangles are generally equilateral] (p v ~p)?
Tom: If p is true, then I'd say the disjunct is true, yes. Either triangles are generally equilateral OR they are not generally equilateral where "generally equilateral" refers to the same thing in both instances. If you disagree, then which would you affirm: that (a) triangles are NEITHER generally equilateral NOR generally equilateral
HammsBear: Trianlges have three sides is an essential/necessary property of triangles.
Some triangles' three sides are equilateral, some are not. Equilateralness is non-essential for triangles, hence (p v ~p) is false.
Call equilateralness an indeterminate property, such as propositions about the future are.
Oops, I mean (p v ~p) is indeterminate for triangles.
Will Alan weigh in and slap some sense into us?
Tom: Then you're missing the entire point. Where two people employ 'bald' with different meanings (bald v ~bald) won't be true under LEM. But that's no failure of LEM (which requires 'bald' mean the same thing in b v ~b).
HammsBear: LEM is useful when the domain is severely restricted, as in mathematics and 'classic' logic. It's not useful when boundaries are 'fuzzy'. My point all along.
Tom: And my point is that in normal daily speech people do NOT employ words and phrases with unrestricted meanings. The very notion of an “unrestricted meaning” is self-contradictory. People invest the words and phrases they use with meaning, and there’s nothing fuzzy about how an individual employs “bald.” That people employ the term with different restrictions (meanings) does not establish the “unrestricted” nature of meaning when it comes to the word “bald.” It only establishes that people restrict the word's meaning differently (not that it's unrestricted). No fuzzy ontological boundaries whatsoever.
--------------
HammsBear: The boundary between 'bald' and 'not bald' is fuzzy partly because reality reveals people having different ideas of 'baldness'.
Tom: Different ideas about baldness or different understandings of when employing “bald” is appropriately meaningful has nothing to do with LEM. LEM applies to a proposition once a person has determined how they’re using “bald” and utters their sentence.
HammsBear: If LEM were absolutely true in all cases, it could be applicable in this example, but it ain't.
Tom: It most certainly is applicable in this example.
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HammsBear: p = [triangles are generally equilateral] (p v ~p)?
Tom: If p is true, then I'd say the disjunct is true, yes. Either triangles are generally equilateral OR they are not generally equilateral where "generally equilateral" refers to the same thing in both instances.
HammsBear: Trianlges have three sides is an essential/necessary property of triangles. Some triangles' three sides are equilateral, some are not.
Tom: This is getting to be funny. Look, Bear, I realize triangles are not all equilateral. (Almost all are not.) But YOU defined p as “triangles are generally equilateral.” ‘Generally’ doesn’t mean ‘universal’ or ‘necessarily’. It just means ‘generally’. So I was just assuming the truth of p as you posited p as true. Triangles are ‘generally’ not equilateral in fact. But I was going with YOUR definition of p. But it doesn’t matter. The proposition “Triangles are generally equilateral” is still subject to LEM. Either triangles are generally equilateral or they are not.
HammsBear: Equilateralness is non-essential for triangles…
Tom: I wasn’t assuming it was. That’s got nothing to do with it.
HammsBear: …hence (p v ~p) is false.
Tom: I don’t think you understand LEM at all, Bear. If p = the proposition “triangles are generally equilateral,” and if triangles are either equilateral or not (and I'm assumign they are), THEN all triangles are in fact either equilateral or not. And of course most triangles are either equilateral or they aren't. Of course not all triangles are equilateral. Of course equilateralness is not an essential feature of triangles. But the disjunct is true: "Triangles are either generally equilateral or not."
HammsBear: Call equilateralness an indeterminate property…
Tom: I don’t think you understand indeterminacy either Bro! ;o) There’s nothing indeterminate about equilateralness per se or equilateralness in triangles. A triangle either is or isn’t equilateral.
We should wait for Alan to jump in 'cause I think we're at an impasse.
Tom
Well he might not be jumping in!
I say this Fall we drive out to Stillwater and buy each other a bee...a sodi pop, and work this out.
Tom
There's a German Gasthaus out that way and they celebrate Ocktoberfest in September.
Is "There's a German Gasthaus out that way that celebrates Ockoberfest in September" true or false?
Couldn't resist.
Tom
Bring the wife, eat a brat, dance the polka, drink a stein or two, trust me, as the evening advances, all logic will become fuzzy..... ;@)
E'en better, laddie:
http://www.irishfair.com/
Tom and HB,
I'll try to jump in for a bit Saturday evening. I've been busy wrapping up a paper the last few days and tomorrow I'm heading down to LA for a day. Ciao.
Alan
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