The set of ‘heaps’ is an interesting example of a set.
Do you suppose, given our sense of pleasant dissonance with respect to trying to identify the smallest number of elements in a heap, that one of the necessary qualities of a ‘perfect’ (most typical) heap is that a person doesn’t know how many elements there are?
Or, rather … thinking about it further—being a ‘heap’ refers to how it appears to have been formed, with items randomly piling on top of one another. And at first, it is difficult to imagine ‘heaping’ of just one thing. Yet I suppose a person could heap themselves on the floor, and then given that context, it suddenly makes sense for there to be a heap of even one thing. Say, a pair of pants thrown carelessly on a chair.
So just a few moments were needed to trace the correct conditions for being a heap, and then it makes sense. The set of ‘heaps’ is defined based on the verb (items belong to the set based on the believability that a configuration may have been formed in such a way) and not on the number of elements.
Hm, that makes sense. For a similar fuzzy-quantity-border problem to what I was thinking of originally, though, consider these other nouns:
Crowd
Bunch
Drop (of liquid)
Bundle
Shard
And so on. And that’s not even getting into adjectives where the fuzzy-border property is even more pronounced (tall, short, big, small, heavy, light).
Some of these words I have ‘concepts’ for, others I don’t. If I don’t have a concept for the word, it seems to be understood just at the association stage—I can only come up with a list of contexts where I would use the word.
If a set is defined by necessary and sufficient conditions, then having an association means that you have some sufficient conditions but you don’t know the necessary conditions. That is, you can identify a region of ideas that are in the set but you don’t know where to draw any boundaries.
At this sufficient but unknown-necessary-conditions stage, you don’t really understand what the word means (you don’t have a well-defined concept) because you need to be able to delineate something that is not in a set in order to really understand anything about what a set is. I agree that not knowing the exact boundary may not be important—but it is important that there is some known interior and some known exterior. My conclusion is that any concept that is understood at any elementary level must have both sufficient and necessary conditions.
Regarding those words which are more associations than concepts, on NPR today the following sentence stuck out as relevant:
‘headwaters is kind of a political word to a lot of people around here; nobody really knows what that means or wants to agree that is where the mine is but—’
Hearing this statement sort of cemented a hypothesis I was considering (in a draft reply to this comment this morning) that many words don’t have real concepts behind them because their main use is to signal something. For example, I would rarely use the word ‘crowd’ when thinking to myself. I would only use the word ‘crowd’ hyperbolically, to indicate to someone that they should imagine more people than they would otherwise expect. In other words, the word augments a concept of how many and the meaning depends on a particular context of myself and a listener.
I think all words have some sort of concept behind them (for example, ‘headwaters’) but since they’re not literally used to convey that concept very often, a speaker of that language can be confused about or forget the concept behind the word.
In an internal dialogue, I would only use the word ‘crowd’ if there was literally jostling of elbows; I think that is the “original” concept for me. What does ‘crowd’ mean to you?
Hm, I see where you’re coming from, and I agree with you so far as it goes when mapping between sets and concepts. But to switch back to talking about words instead of concepts, I am not sure why you (seem to be) going with a set-based approach. Thinking of words as fuzzy regions seems more useful to me when trying to analyze how words are used, and when trying to figure out how words ought to be used.
As shown in the OP, people tend to use words not as strict sets but as regions in thing-space with more and less typical members (i.e. a bluebird is a more typical member of “birds” than a penguin or an ostrich). Though some situations will demand strict borders, in general the fuzzy region approach seems more beneficial, because it allows for quick low-level inductive reasoning. If you know that a bird is typical in bird aspects A and B (say, it has feathers, and it has a large wingspan), then you can more confidently predict that it’s typical in bird aspect C (it can fly). It’s less brittle to use probabilities in situations like these than strict boolean sufficient-or-necessary values.
Though some situations will demand strict borders, in general the fuzzy region approach seems more beneficial, because it allows for quick low-level inductive reasoning
You’ve convinced me on the importance of fuzzy sets. I’m sure quick low-level inductive reasoning is of vital importance, but even more immediately it seems that flexibility in meaning is useful for communication. I don’t have to hunt for the exact right word, many approximate words will do, and I can describe new concepts by stretching an old one.
However, when you write:
It’s less brittle to use probabilities in situations like these than strict boolean sufficient-or-necessary values,
I don’t see how sufficient-or-necessary defined sets don’t allow fuzziness. For example, saying that a bird must have feathers (a necessary condition) and that a bluebird is a bird (sufficient condition) then there is still plenty of gray area for penguins and ostriches.
I think of sufficient conditions as being associations. We learn that penguins, ostriches and doves are all birds so those are sufficient conditions. A necessary conditions puts an edge on your set—staplers aren’t birds because they don’t have feathers.
On the other hand, a stapler that looks like a bird could be thought of as a bird to oneself. If anything has enough of the characteristics of a bird you could consider it one. For example a small plastic object with no feathers whatsoever is a ‘birdie’ because it moves like a bird.
So now I’m leaning towards agreeing. We just have associations, things are like other things if the characteristics overlap, and there aren’t any necessary conditions for deciding if something is ‘like’ something else.
[...]it seems that flexibility in meaning is useful for communication. I don’t have to hunt for the exact right word, many approximate words will do, and I can describe new concepts by stretching an old one.
Yes, strongly agreed. This idea makes me want to think of adjectives as tugging a concept-defining region of thing-space in a new direction.
That works especially well in languages like Lojban, where adjectives and nouns are not distinguished from each other. For example there is “blanu” which means “x is blue” and there is “zdani’ which means “x is a house”, and you can say “blanu zdani” (blueish-house) just as easily as “zdani blanu” (houseish-blue).
[...]saying that a bird must have feathers (a necessary condition)
That actually is a good example of a brittle requirement, in ways that are even more directly problematic than shuttlecocks and bird-shaped staplers. What about a plucked chicken? What about a duck that, due to a genetic disease, never had feathers? What about (NOTE: This example isn’t valid re: real evolutionary history) a member of an intermediary species between pterodactyl and modern birds?
Not that it particularly affects your point, but pterodactyls are not genetic precursors to birds (they split off before the clade Dinosauria,) and feathers predate the first true dinosaurs capable of flight.
That actually is a good example of a brittle requirement,
Yeah, good point. I’m entirely convinced. Even for an apparently straight-forward category like ‘bird’, there’s not a single necessary condition you can point to. Even if there are some examples of categories with necessary conditions (I don’t know), this is evidence that the necessary conditions aren’t an intrinsic part of the way we structure a concept.
The set of ‘heaps’ is an interesting example of a set.
Do you suppose, given our sense of pleasant dissonance with respect to trying to identify the smallest number of elements in a heap, that one of the necessary qualities of a ‘perfect’ (most typical) heap is that a person doesn’t know how many elements there are?
Or, rather … thinking about it further—being a ‘heap’ refers to how it appears to have been formed, with items randomly piling on top of one another. And at first, it is difficult to imagine ‘heaping’ of just one thing. Yet I suppose a person could heap themselves on the floor, and then given that context, it suddenly makes sense for there to be a heap of even one thing. Say, a pair of pants thrown carelessly on a chair.
So just a few moments were needed to trace the correct conditions for being a heap, and then it makes sense. The set of ‘heaps’ is defined based on the verb (items belong to the set based on the believability that a configuration may have been formed in such a way) and not on the number of elements.
Hm, that makes sense. For a similar fuzzy-quantity-border problem to what I was thinking of originally, though, consider these other nouns:
Crowd
Bunch
Drop (of liquid)
Bundle
Shard
And so on. And that’s not even getting into adjectives where the fuzzy-border property is even more pronounced (tall, short, big, small, heavy, light).
Some of these words I have ‘concepts’ for, others I don’t. If I don’t have a concept for the word, it seems to be understood just at the association stage—I can only come up with a list of contexts where I would use the word.
If a set is defined by necessary and sufficient conditions, then having an association means that you have some sufficient conditions but you don’t know the necessary conditions. That is, you can identify a region of ideas that are in the set but you don’t know where to draw any boundaries.
At this sufficient but unknown-necessary-conditions stage, you don’t really understand what the word means (you don’t have a well-defined concept) because you need to be able to delineate something that is not in a set in order to really understand anything about what a set is. I agree that not knowing the exact boundary may not be important—but it is important that there is some known interior and some known exterior. My conclusion is that any concept that is understood at any elementary level must have both sufficient and necessary conditions.
Regarding those words which are more associations than concepts, on NPR today the following sentence stuck out as relevant:
At 1:18 Daysha Eaton says:
Hearing this statement sort of cemented a hypothesis I was considering (in a draft reply to this comment this morning) that many words don’t have real concepts behind them because their main use is to signal something. For example, I would rarely use the word ‘crowd’ when thinking to myself. I would only use the word ‘crowd’ hyperbolically, to indicate to someone that they should imagine more people than they would otherwise expect. In other words, the word augments a concept of how many and the meaning depends on a particular context of myself and a listener.
I think all words have some sort of concept behind them (for example, ‘headwaters’) but since they’re not literally used to convey that concept very often, a speaker of that language can be confused about or forget the concept behind the word.
In an internal dialogue, I would only use the word ‘crowd’ if there was literally jostling of elbows; I think that is the “original” concept for me. What does ‘crowd’ mean to you?
Hm, I see where you’re coming from, and I agree with you so far as it goes when mapping between sets and concepts. But to switch back to talking about words instead of concepts, I am not sure why you (seem to be) going with a set-based approach. Thinking of words as fuzzy regions seems more useful to me when trying to analyze how words are used, and when trying to figure out how words ought to be used.
As shown in the OP, people tend to use words not as strict sets but as regions in thing-space with more and less typical members (i.e. a bluebird is a more typical member of “birds” than a penguin or an ostrich). Though some situations will demand strict borders, in general the fuzzy region approach seems more beneficial, because it allows for quick low-level inductive reasoning. If you know that a bird is typical in bird aspects A and B (say, it has feathers, and it has a large wingspan), then you can more confidently predict that it’s typical in bird aspect C (it can fly). It’s less brittle to use probabilities in situations like these than strict boolean sufficient-or-necessary values.
You’ve convinced me on the importance of fuzzy sets. I’m sure quick low-level inductive reasoning is of vital importance, but even more immediately it seems that flexibility in meaning is useful for communication. I don’t have to hunt for the exact right word, many approximate words will do, and I can describe new concepts by stretching an old one.
However, when you write:
I don’t see how sufficient-or-necessary defined sets don’t allow fuzziness. For example, saying that a bird must have feathers (a necessary condition) and that a bluebird is a bird (sufficient condition) then there is still plenty of gray area for penguins and ostriches.
I think of sufficient conditions as being associations. We learn that penguins, ostriches and doves are all birds so those are sufficient conditions. A necessary conditions puts an edge on your set—staplers aren’t birds because they don’t have feathers.
On the other hand, a stapler that looks like a bird could be thought of as a bird to oneself. If anything has enough of the characteristics of a bird you could consider it one. For example a small plastic object with no feathers whatsoever is a ‘birdie’ because it moves like a bird.
So now I’m leaning towards agreeing. We just have associations, things are like other things if the characteristics overlap, and there aren’t any necessary conditions for deciding if something is ‘like’ something else.
Yes, strongly agreed. This idea makes me want to think of adjectives as tugging a concept-defining region of thing-space in a new direction.
That works especially well in languages like Lojban, where adjectives and nouns are not distinguished from each other. For example there is “blanu” which means “x is blue” and there is “zdani’ which means “x is a house”, and you can say “blanu zdani” (blueish-house) just as easily as “zdani blanu” (houseish-blue).
That actually is a good example of a brittle requirement, in ways that are even more directly problematic than shuttlecocks and bird-shaped staplers. What about a plucked chicken? What about a duck that, due to a genetic disease, never had feathers? What about (NOTE: This example isn’t valid re: real evolutionary history) a member of an intermediary species between pterodactyl and modern birds?
Not that it particularly affects your point, but pterodactyls are not genetic precursors to birds (they split off before the clade Dinosauria,) and feathers predate the first true dinosaurs capable of flight.
Whoops, didn’t know that, thanks.
Yeah, good point. I’m entirely convinced. Even for an apparently straight-forward category like ‘bird’, there’s not a single necessary condition you can point to. Even if there are some examples of categories with necessary conditions (I don’t know), this is evidence that the necessary conditions aren’t an intrinsic part of the way we structure a concept.