Interestingly, something like the converse of your question was asked and answered at Question 4669, linked here . The earlier question was " Is 'understanding' a proposition necessary, but not sufficient, for 'believing' that same proposition?" Whether it's psychologically possible for someone to believe a given proposition without understanding the proposition will depend on whether it's even conceptually possible, i.e., whether it could even make sense to describe someone that way. When asking whether something is conceptually possible, philosophers often consult their linguistic intuitions. So you might ask whether you would sincerely assert something of the form "So-and-so believes that p but doesn't understand it." I myself wouldn't. Now maybe that shows only that such statements are unassertible rather than conceptually false, but I think it's conceptually confused to describe someone as believing a proposition without understanding it. If it is, then the answer to your second...

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. PLEASE NOTE : (3) above was taken from Professor Heck's original...

Let's focus on the phrase "something of category X that does not fit category X's definition." One on interpretation, we can't possibly find something of that description: if it doesn't fit category X's definition, then it's not something of category X, as you say. But that interpretation assumes that I've already got a correct definition of category X, a definition that's neither too broad nor too narrow. What if my definition of 'chair' is 'item of furniture with four legs' and you show me a bean-bag chair or an IKEA Poang chair? Haven't you shown me an item of category X that doesn't fit my definition of category X? Haven't you falsified my definition of 'chair', at least as a definition of the word in ordinary use, by showing that it's too narrow? (It's also too broad, as I realize when you show me a four-legged table.)

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...

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...