![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
Happy Easter, everybody! Today, I'm going to talk about two of my favourite logical fallacies. This post got a bit long, so I've split it into two: the second part, about Sliding Definition Ploy, is here.
My favourite, for the name if nothing else, is the No True Scotsman Fallacy1, which canonically goes like this:
Hamish, unwilling to concede that his initial statement was wrong, has extended his definition of Scottishness to include "does not put sugar on his porridge". Angus, at least according to Hamish, has betrayed his nation and his ancestors by his choice of breakfast condiment, and thus does not deserve to be considered a true Scotsman. And yet an impartial observer, working to the generally-accepted definition of Scotsmanhood, would certainly say that Angus is a Scot, and a true one at that.
The NTS fallacy comes up a lot in arguments about evolution, where it commonly takes the form:
NTS, used appropriately, can be a useful tool in mathematics: if you find a counterexample to some desirable theorem ("all Scotsmen suffer the symptoms of salt overconsumption", say), it may well be best to go back and refine your original definitions so that the counterexample disappears, then claim that what you really meant all along were the porridge-salting Scotsmen, you were just misled by the adjacent notion of "Scotsman". The trick is to do this explicitly, making it clear that the new definition ("porridge-salting Scotsman") is different from the old one ("Scotsman"), and excludes the pathological cases (Angus MacAlasdair). I did encounter the bad form in a conversation with a fellow mathematician the other day, though:
By the way, porridge really is a lot tastier with salt on it. In fact, I'm eating some salted porridge now. Mmmm...
1 That's its name. Really!
My favourite, for the name if nothing else, is the No True Scotsman Fallacy1, which canonically goes like this:
Hamish: No true Scotsman puts sugar on his porridge.[We're assuming that Angus MacAlasdair's Scottish nationality is above reproach, apart from his unnatural and perverted breakfast habits.]
Jock: But my cousin Angus MacAlasdair from Glen Coe puts sugar on his porridge.
[Beat.]
Hamish: No true Scotsman puts sugar on his porridge.
Hamish, unwilling to concede that his initial statement was wrong, has extended his definition of Scottishness to include "does not put sugar on his porridge". Angus, at least according to Hamish, has betrayed his nation and his ancestors by his choice of breakfast condiment, and thus does not deserve to be considered a true Scotsman. And yet an impartial observer, working to the generally-accepted definition of Scotsmanhood, would certainly say that Angus is a Scot, and a true one at that.
The NTS fallacy comes up a lot in arguments about evolution, where it commonly takes the form:
Fundie: A belief in evolution is incompatible with my Christian faith, so I must reject it.
Member of the reality-based community: Actually, no. Many, many Christians have no problem accepting evolution. In fact, evolution is accepted by nearly every major denomination of Christianity, and the denominations that accept it account for nearly all the Christians in the world.
Fundie: In that case, they can't really be Christians, can they?
MRBC: *headdesk*
NTS, used appropriately, can be a useful tool in mathematics: if you find a counterexample to some desirable theorem ("all Scotsmen suffer the symptoms of salt overconsumption", say), it may well be best to go back and refine your original definitions so that the counterexample disappears, then claim that what you really meant all along were the porridge-salting Scotsmen, you were just misled by the adjacent notion of "Scotsman". The trick is to do this explicitly, making it clear that the new definition ("porridge-salting Scotsman") is different from the old one ("Scotsman"), and excludes the pathological cases (Angus MacAlasdair). I did encounter the bad form in a conversation with a fellow mathematician the other day, though:
Colleague: Are there any categories without pushouts?
Me: Yeah, sure. In fact, it's quite unusual for a category to have pushouts.
Colleague: Can you give me an example of a category without pushouts?
[There follows a few minutes in which we try to find a "naturally-occurring" counterexample, which proves tricky to think about. Eventually, I give up and write down the first thing I'd thought of. In terms of our island metaphor, it's a shard of rock sticking out of the ocean at low tide, with a solitary seagull perched precariously on top of it, but it does provide a counterexample.]
Colleague: No, that's too categorical! A real category!
[You ask a category theorist, and then complain that the answer he gives you is too categorical? OK, whatever...]
By the way, porridge really is a lot tastier with salt on it. In fact, I'm eating some salted porridge now. Mmmm...
1 That's its name. Really!
Tags:
no subject
(@int-e) Syzygy-: so partial orders are not REAL categories?
no subject
Of course - any poset without suprema provides a counterexample! I suspect that even they might not be "real" enough for her, but I'll pass that on :-)
no subject
no subject
Of course - any poset without suprema provides a counterexample!"
I have no idea what this means (and its probably not worth explaning to me), but the phraseology is just beautiful - a perfect example of the classical joke of the intelligentsia whereby one is shown to be a fool for not noting something superficial within one context, which the context remains opaque to an outsider.
no subject
A poset is a Partially Ordered Set, ie a set X with a relation <= defined on it which behaves as expected:
- x <= y and y <= z implies x <= z (transitivity)
- x <= x for all x (reflexivity)
- if x <= y and y <= x, then x = y (antisymmetry)
The only departure from the normal behaviour of <= is that not all pairs of elements of X have to be comparable: ie, given x and y in X, it doesn't have to be the case that either x <= y or y <= x. This is why it's a partially ordered set :-) Partially ordered sets are important examples of categories - in fact, they're exactly the categories with at most one arrow between any two objects (put an arrow a -> b if a <= b, and none otherwise).
A "supremum" of a subset Y of X is a least upper bound for Y, ie an element s of X such that y <= s for all y in Y (it's an upper bound) and for all upper bounds b of Y, s <= b (it's the least such upper bound). Not all posets have suprema for all their subsets: for instance, the rational numbers Q have no supremum for the subset {x in Q : x^2 < 2}. The set R of real numbers, however, does have a supremum of every subset: this is one way of stating the completeness of the real numbers, and it allows calculus to work properly.
no subject
The set R of real numbers has a supremum of every subset that is bounded above. There is no supremum for the integers, for instance.
That is all.
no subject
no subject
If you've never tried porridge with salt, give it a go (or preferably two or three - it might take a bit of getting used to). It's worth it, I promise!