Let's assume the following:

1. If you understand a proposition, then you also understand its negation.
2. It is necessary, if you are to believe a proposition, to understand it.
   
3. It's perfectly possible to believe a proposition and not understand its negation.

It follows from these that understanding a proposition is not sufficient for believing it. So there's an argument.

One might wonder why we should accept (1)-(3), of course.

I think most people would take (2) to be obvious enough. What's meant here by "understanding" is something like: being able to take mental attitudes towards. Belief is just such an attitude.

Regarding (1), this just seems to follow from your understanding what negation is. For detailed discussion, however, one might look at Frege's last essay "Negation".

Regarding (3), one would hope that it is true! Even if we sometimes believe contradictions, one would hope we didn't always have to do so!!

I confess I'm puzzled by Prof. Heck's reply. He defends the following three assumptions:

1. If you understand a proposition, then you also understand its negation.
2. It is necessary, if you are to believe a proposition, to understand it.
3. It's perfectly possible to believe a proposition and not understand its negation.

I interpret those assumptions as follows:

1*. Understanding P entails understanding not-P.
2*. Believing P entails understanding P.
3*. Believing P doesn't entail understanding not-P.

1*-3* imply a contradiction: Believing P does and doesn't entail understanding not-P. If so, then 1-3 imply everything (if I've interpreted them correctly). I also don't see how the falsity of 3 implies that we would always have to believe contradictions. If 3 is false, then believing P entails understanding not-P; I don't see how any unwelcome consequences follow from that.

I would say that there is, in any general sense, a right or wrong way to use a word. There are various generalizations about how people actually use words that are captured in dictionaries. Dictionaries tell us how people do use words, not how they should use them. There is no such thing as the correct use of the term 'game', or any other term. If a person uses a word intending to mean something by it that is not in line with the dictionary, there is nothing wrong with that per se. We do it all the time. for many reasons, many of them very good. We just need to be careful not to be misunderstood. And we might use a word thnking we are using it in line withe dcitionaries and be wrong about that.

Nobel Prize-winning physicist Richard Feynmann said "People often complain of the unwarranted extension of the ideas of particles, paths etc. into the atmomic realm." He responds to the complaint that the extension is unwarranted "Not so at all; there is nothing unwarranted about the extension. We must, and we should and we always do extend as far as we can beyond what we a;ready know, beyond those ideas that we have already obtained ... it is the only way to make progress" (The Character of Phusical Law). He could have said the same thing about using the words 'particle' etc. in a new way. Calling something 'a particle' when it isn't a particle in the usual sense, can be a vital way to make progress.

Extending the use of terms in the way Feynman discusses is vital to the progress of knowledge. But of course we have to try our best to be carful when we do this, so as not to create confusion.

I haven't looked at the wikipedia article, but the view it expresses is VERY old-fashioned. Since Wittgenstein's "family resemblance" concept, and especially since cognitive scientists such as Eleanor Rosch's work in the 1970s, it's far more fashionable to think of concepts (and categories) as constituted not by "necessary and sufficient conditions" but by prototypes and similarity relations, far more befitting "fuzzier" concepts -- for example the way we think about "dogs" is not to generate an "essence" of necy/sufficient conditions, but by having in mind some prototypical dog at the center and then linking it to less familiar, less central, other examples ... This is far more in keeping with your nice suggestion, that what we seek (as I'd put it) are some "characteristic" properties of the paradigmatic members of the set, recognizing that other creatures may share these properties to various degrees and still count as members of the set in question ... So unless the wikipedia article was specifically using the word "essence" to mean what it used to mean (necy/suffic condns), and to distinguish essences from concepts, then I'd say your view is far more on the right track than the wiki's view ....

hope that's useful--

ap

Great question, though I might worry that words like "useful" are about as vague as any of those in your examples, and thus your question may suffer from the same problems! ... I like your point that "the debate itself is informative" -- assuming that's true, which seems plausible, why couldn't that be "useful" enough? We learn an awful lot about our own concepts and beliefs when we grapple over what constitutes a person, or a work of art, or fairness ... So I'm a bit curious why, after recognizing that as a value for even these vague terms, you seem to demand something significantly more. Perhaps you would like these terms to take on refined and precise meanings like those in natural science, at least paradigmatically -- but then again, these terms DO often take on such precision, at least once they're in the hands of philosophers debating the issues. Bertrand Russell has a nice bit where he observes that if scientists are entitled to develop their own terminology and refine ordinary words to make them "useful to science," then philosophers should do the same .... But then, lastly, one more point: even if the examples you mention don't have "clear" or "precise" or "agreed upon" semantic boundaries, all of them seem to display the archetypal properties of vague terms: they have central, "clear" cases, at the same time as they lack clear peripheral borders. (eg "tall" "mountain" -- Mt Everest is definitely a mountain, even if we can't say precisely when smaller things stop being mountains ...) ... So perhaps the primary "usefulness" of such expressions comes from the agreed upon "central" cases, and then the other kind of "usefulness" comes from the informative debate about the boundaries?

hope THAT is useful ... :-)

ap

I think you understand at least one aspect of the problem quite well. As you say, words like 'heap' are useful only if they're vague. Indeed, their vagueness seems built into their meanings: they're essentially vague; they wouldn't be the words they are if they weren't vague. The problem is that their vagueness seems to imply the contradiction that is the sorites paradox (see the two SEP entries that I cited here). And it's not just 'heap', a word we might not care too deeply about. Practically every concrete noun and ordinary adjective we use ('car', 'fetus', 'child', 'person', 'tall', 'rich', 'unjust', 'toxic', 'honest', 'safe', and on and on) is essentially vague and hence apparently implies a contradiction. Yet we can't help thinking that plenty of things do answer to those nouns and adjectives. Surely there are rich people and toxic chemicals, but the sorites paradox seems to show that there can't be. It's as ubiquitous as it is hard to solve.

I supplied the response you found unsatisfying, so thanks for not pretending you were satisfied by it!

You're right that my response did assume that there's only a "yes" or "no" answer to such questions as "Can N grains (for some particular N) make a heap?", "Can N grains definitely make a heap?", and so on. I also claimed that my assumption was an element of common sense.

As I understand it, your counter-proposal is that N grains can be enough to make, for instance, "an 85% heap" (or maybe "85% of a heap") but not "a 100% heap" (or maybe "100% of a heap"). But what's an 85% heap? What's 85% of a heap, except a smaller heap? More plausibly, maybe you're proposing that the statement "N grains can make a heap" is only 85% true rather than 100% true. Proposals of this sort are well-known in the literature on the sorites paradox, usually under the heading of "many-valued logics" (see section 3.4 of the SEP article "Sorites Paradox" that I linked to in my previous reply).

These many-valued approaches face serious problems. For instance:

(1) What's the smallest number N that's enough grains to make a 100% heap (or 100% of a heap, or that makes it 100% true that N grains can make a heap)? I take it there's no non-arbitrary answer, not simply that we can't know or say what the answer is.

(2) The logical systems that go with many-valued approaches produce implausible results: suppose that some N makes the statement "N grains can make a heap" only 50% true; then the clearly false conjunction "N grains can make a heap and N+1 grains can't make a heap" ends up being roughly 50% true, rather than 0% true.

Further problems are mentioned in section 3.4 of the SEP article and discussed thoroughly in Keefe (2000), Chapters 4 and 5, cited in that article. Indeed, I recommend Keefe (2000) as an expert guide to the issues (even though I think she's far too optimistic about the ability of supervaluationism to solve the paradox).

So my advice is to read the SEP articles "Sorites Paradox" and "Vagueness" and then dive into some of the works they cite. There's no substitute for engaging with the writings of those who've thought carefully and systematically for years about this paradox. They've proposed many solutions, including the one you propose yourself. If you discover a many-valued approach that overcomes the objections arrayed against it, please let us know. I'll be very pleasantly surprised!

The sorites paradox -- the paradox of the heap and similar paradoxes exploiting more important concepts than heap -- is a terrific topic. It's great to see people thinking about it.

You wrote, "we cannot say exactly where the transition occurs." Some philosophers would respond, "It can't occur exactly anywhere, because heap (or bald or tall or rich ...) isn't a concept that allows exact status-transitions. To say that there's an exact point of status-transition, even a point we can't know or say, is to misunderstand what vague concepts are."

Some philosophers would also object to your suggestion that the fuzziness can be "bounded," if by that you mean "sharply bounded." They'd say that any boundary around the fuzzy cases must itself be a fuzzy boundary: like the boundary between heap and non-heap, the boundary between definitely a heap and not definitely a heap isn't precise to within a single grain. (This phenomenon is usually called "higher-order vagueness.") In that case, the fuzziness that brought into doubt the existence of heaps also brings into doubt the existence of fuzzy boundaries themselves.

I don't think there's any way to cut the Gordian Knot here, if by that you mean finding a commonsense solution that cuts through the paradox. Why? Because the paradox itself results from commitments of common sense: (a) some number of grains is clearly too few to make a heap (maybe 15, as you say); (b) some number of grains is clearly enough to make a heap (maybe 15,000); and yet (c) one grain never makes the difference between any two different statuses (heap vs. non-heap, definitely a heap vs. not definitely a heap, etc.). Given commonsense logic, (a)-(c) can't all be true, but which one should we reject? Most philosophers who try to solve the paradox attack (c), but I certainly haven't seen a refutation of (c) that I'd call "commonsense."

I wish I knew the answer, or even knew of an answer that comes close to being satisfying. I sometimes worry that we human beings are smart enough to have discovered the sorites paradox but constitutionally too dumb to solve it. I'd love to be shown that my pessimism is unwarranted!

Recommended reading:
SEP, "Sorites Paradox"
SEP, "Vagueness"