You need to be a little more careful about such absolute statements. The definition of factorial(.) as
factorial(n) = n*factorial(n-1)
factorial(0) = 1
references itself and is valid.
You need to be a little more careful about such absolute statements. The definition of factorial(.) as
factorial(n) = n*factorial(n-1)
factorial(0) = 1
references itself and is valid.
Recursive definitions are possible, but they must still be founded on a base level that does not reference itself. Each other level can then be defined in a way that is not self-referential.
Q: What's the most important leg of a three-legged stool? A: The one that isn't there. - traditional joke-riddle
A specific neurological lesion can sometimes damage or impair specific neurological functions without touching others. In the condition famously known as "Ondine's Curse", for example, automatic control of breathing is destroyed while conscious control remains, so that without modern medical intervention nerve-damaged patients can survive only as long as they can remain awake. Such conditions are nevertheless unusual exceptions to the more general principle that complex, recently-developed, and 'meta'-functions (those that monitor and control others) are first to be impaired and lost when the nervous system is stressed, damaged, or altered.
If a fuzzy definition becomes a massive problem, then that definition clearly wasn't in existence merely to simplify speech.
Regarding mammals, is there a use for the term that requires its inclusion of dolphins? Does the existence of sweat glands usefully separate mammals from other animals? After all, mammals in general share a variety of properties: most give live birth, most have hair, most are warm-blooded, etc.--but we admit to the category of mammals many animals that fail one or more of these criteria.
A well-defined but useless category (I am not arguing that "mammal" is such a category, as there may well be a biological use for it) may be pedagogically interesting but otherwise may merely confuse our understanding of thingspace.
".--but we admit to the category of mammals many animals that fail one or more of these criteria."
No, we don't. Dolphins have all of the required attributes to be considered mammals. If they didn't, we couldn't call them mammals any longer.
Which weighs more: a pound of feathers, or a pound of gold?
[...rationality discussion...]
I must be missing something. I thought the point of this riddle was the difference between avoirdupois and Troy weight -- a simple matter of (rather esoteric) factual knowledge not contained in the structure of the question.
That is an absolutely charming interpretation, and one that makes a lot of sense. However, in my experience, it's not how the riddle is commonly used.
That would be a great way to show off your knowledge of jeweler's weights, though.
Questions about a property rarely contain their own answers in a trivial way
I think this fact may explain most of the reaction; the answer is slipped into the question with a couple words that don't interrupt the flow. The questions is pointless, but we don't expect questions to be pointless, so we don't really hear it right or interpret it correctly, and we answer the question that would actually make sense.
There's more to it, of course. Ask the question with substances that don't produce strong associations regarding "weight" (really, density), and people tend not to get it wrong no matter how much time pressure is involved.
Which weighs more: a pound of feathers, or a pound of gold?
Close consideration of this riddle - and the conditions under which people tend to get it wrong - is helpful in understanding the limits of human rationality. It is a specific example which leads us to general principles of rationality failure.
At what point do we say that the problem lies in the definition of a category? Since ordinary people have no especial use for the category "bird," it's unsurprising that they haven't nailed down characteristics that would allow such a use.
Categories that we need--that must reliably possess some characteristic(s) such that they are useful--tend to have strict necessary and sufficient conditions for inclusion. Categories that we use purely to simplify speech can get away with fuzzier definitions.
Is the dolphin really a fish? That depends: is that thing over there really a blegg?
The biological category of 'mammal' is quite well-defined, thank you.
And fuzzy definitions are fine until you're dealing with a case that lies in the penumbra, at which time it becomes a massive problem.
This looks sincere to me, and given that it's sincere, people really ought to be allowed more chance than this to recover from their mistakes.
I say that depends entirely on the nature of the mistake. Gross negligence should not be forgiven, although the proper response is not necessarily retributive.
This could be criticized as trying to find an argument that fits the conclusion, but I think this is uncharitable.
I think it's exactly right. All reasons are rationalizations.
Not in the way that 'rationalization' is used in natural language. That refers to a non-rational statement that is used in place of rationality in order to satisfy the desire to present an argument as rational without having to go through the trouble of actually constructing and adopting a rational position.
The biggest functional difference: when a reason is abolished, the behavior goes away. When a rationalization is abolished, the behavior remains.
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While alcohol does in fact reduce doubt and insecurity, I wouldn't jump to: any crude global brain impairment is likely to first impair self-doubt/monitoring.
Otherwise, nice pep talk.
Alcohol is an just example. It's well-known that crude global brain impairment reduces self-monitoring first.