Archive for the ‘logic’ Category
I’ve not posted for a while: I’ve been busy doing things other than procrastinating! OK that’s a lie, but the procrastination hasn’t taken the form of blog posts for a while. My website looks much the same as ever, but lots has changed under the hood, as it were. It now validates as XHTML and the columns are the same height and extend to accommodate as much text as needed. Hoorah. I also have an essay I’m rather proud of. (Actual work, shock horror!) It is probably over long and not all that much of it can be adapted to fit into my literature review, but I’m still happy with the (almost) finished product. On an unrelated note, here are some poems that appealed to the geek in me.
Here is the halting problem proven in poem form.
A poem composed entirely of punctuation. (I may have linked to this before.)
I remember a maths lecturer at Warwick starting a lesson by telling us:
Integral t-squared dt
from 1 to the cube root of 3
times the cosine
of three pi over 9
equals log of the cube root of ‘e’.
More maths limericks here.
A history of Western philosophy in limerick form you say? Well why not?
And while it’s not a poem, it’s certainly the same ballpark.
And finally my own contribution thanks to getting bored during measure theory lectures. I give you a haiku about basic measure spaces:
A finite union
of disjoint rectangles is
I have tons more of these on some scrap of paper in my old notes folder. I also wrote a limerick about Rene Magritte once… (I rhymed “Rene Magritte” with “ceci n’est pas une pipe”)
When I went for an interview at Oxford for my undergrad degree one of the things we had to do was a kind of logic test. It had things like “All men wear hats, some men wear ties” and you were asked what it was possible to conclude from that. Presumably they were looking for something like “Some men wear ties and hats”. One of the questions was “Most men wear ties, most men wear hats”. I said that it wasn’t possible to conclude anything from this. But it was then explained to me that if hat-wearers and tie-wearers are both in a majority, then there must be some who are both: the proper conclusion is that “Some men wear ties and hats”.
“All men” is standing in for the universal quantifier (x) and “Some men” stands for the existential quantifier (Ex). But “Most men” is awkward to put in similar terms without resorting to some sort of logic enriched with predicates relating to numerical proportions. But that sort of “statistical logic” is too complex. The fact that some predicate applies not to all but to a majority of some domain is an easily understood fact that does not need a whole logic of proportions behind it. Let’s make “Most of” a new kind of quantifier: (Mx). What properties does (Mx) have?
This is the property my interviewer at Oxford was exploiting. If most x’s are P’s and most x’s are Q’s, then some x’s must be both.
These two properties simply say that if all men wear hats then most men wear hats and if most men wear hats then some men wear hats.
Do we really need a third quantifier here? Is there some way to express “most of” in terms of universal and existential quantifiers? I’m not sure there is. Here’s a first stab. When you say “(Mx)Px” what you are really saying is:
But that’s no good, because that expression still contains an “Mx”. We are kind of going beyond standard predicate logic, but only a little bit. Perhaps Mx shouldn’t be a quantifier, but a property of predicates? But I don’t think that is a particularly satisfying suggestion. A property of predicates would be a third-order entity, and that seems extravagant given the modest goal of formalising the easily understood concept of a majority.
Soon I’ll post again and discuss the “dual” of (Mx): “Only a few men wear hats” “Hat-wearers are in a minority”.
A note on the symbolism: I wanted to use and but I couldn’t think of a way to “invert” the letter “M” in a similar fashion. It shares the vertical symmetry of “A” but when reflected around a horizontal axis as is, you get a “W”, which is no good if the idea is to come up with a new symbol. So I reverted to old-style quantifiers. Suggestions regarding how to “symbolise” majority are most welcome.