# Is "understanding" a proposition necessary, but not sufficient, for "believing" that same proposition? Further, where could one find arguments (discussion) for and/or against either position?

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...

# Are definitions falsifiable? It seems that if I find something of category X that does not fit category X's definition, then it isn't actually of category X, and thus doesn't prove anything. But on the other hand, if that is the case, it seems no definition cannot be falsified or otherwise demonstrated to be inadequate (unless it is inherently contradictory or so).

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 was reading some questions on this site regarding vagueness and the Sorites conundrum and I'm not sure I understand the fascination with figuring out what does or doesn't qualify as a heap. Isn't the word heap useful precisely BECAUSE it doesn't have a strict quantitative requirement? We choose to use the word "heap" and not a different word (like grams, or tons, or twenty-seven, etc.) because it offers us flexibility. I'm not sure exactly why this "puzzle" has received so much attention. The fact that there hasn't been an accepted solution makes perfect sense to me because there is nothing to solve. It seems like trying to apply precision to a word intentionally designed to be imprecise. It seems to me that if we figure out the exact point at which something becomes a heap then we will no longer be able to use the word as freely. Am I misunderstanding the problem? Thanks in advance!

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 recently asked a question about the sorites paradox, and I received the following response, which seems to me to have a logical fallacy in it. In other words, the answer below does not seem to "explain" the paradox as much as it "contains" the paradox.... Here is the reply: "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.'" It seems that point (c) above presupposes that either we have 100% heap or 0% heap; however if we can have a number of grains such...

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...

# I read about the sorites paradox, especially "what is a heap?" and was a bit puzzled about the reasoning. Isn't it fairly straightforward to say, "fiftenn grains is not a heap" and "fifteen thousand grains is a heap" and then say, "even if we cannot give a single precise number where "not a heap" ends and "is a heap" begins, we can narrow down the range within which it occurs, right? In other words, a sort of "bounded fuzziness" applies, where we know for sure what is a heap and what is not a heap (the "bounded" part) while we cannot say exactly where the transition occurs (the "fuzziness" part). It also reminds me of Alexander the Great's solution to the Gordian Knot problem, in a way. People are getting confused because they are using the wrong tools, not because of the nature of the problem itself. the argument seems reminiscent of the supposed paradox about achilles and the tortoise, you can calculate the exact time at which Achilles catches and passes it.

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...