Barriers to non-classical implication

I’ve put another paper on my university page. It’s called ‘Barriers to Non-classical Implication’, and builds on some results in another paper by Greg Restall and Gillian Russell. You might understand mine better if you read theirs first. Even if you’d still understand mine fine I recommend you read theirs anyway if you like that sort of thing. It’s a delight. When I gave a talk on the material in my paper in February, this was the abstract:

Intuitively there are some kinds of statements you can't infer from some other kinds. You probably can't get general information from particular information, and maybe you can't get normative information from descriptive either. Gillian Russell and Greg Restall have shown how to make these barrier theses precise and immune to counterexamples, if the barriers don't divide the statements into two exhaustive groups. I explain why this happens, and extend their results to show how to make the divisions exhaustive by changing the consequence relation.

That’s pretty much still what it’s about. It's still a work in progress so please don't cite it without asking. If it sounds like your idea of fun, click here. If not, here’s another link to that romantic short story I posted ages ago.

New webpage and paper

I’ve now got a new webpage on my university’s site. It isn’t very fancy, but it does have some information about some papers I’ve been writing. There’s also a link to one of them, with more to follow soon.

All quiet on the Curry front

Sorry if you've been looking for my posts about Curry's paradox from a while back. I've taken them down because I'm writing a paper about that sort of thing, and taking them down seemed a good idea.

A solution to the lawn mowing puzzle

This one's got references and a slightly different style, because it's a paper. Given the length and the subject matter, I thought it'd be more at home on a blog than in a journal, so here it is:

Jeremy Gwiazda (2010) presents a puzzle: can someone mow eight lawns with one mower and never have exactly seven left? Clearly something strange would have to happen but if supertasks are possible then it can be done.

Supertasks are not new. Russell (1936, p.144) gives one example, Hawthorne and Weatherson (2004) give another and there are many more. The term appears to come from J. F. Thomson (1954). Supertasks involve doing an infinite number of tasks in a finite stretch of time. For example, if you want to clap your hands an infinite number of times in an hour then you can clap your hands at 2pm, 2.30, 2.45, 2.52 and thirty seconds and so on, halving the gap between claps each time. You will have clapped your hands infinitely many times by 3pm, but at any time before 3pm you will have only clapped a finite number of times. Of course, no human can clap their hands that quickly. Supertasks are strange, but not logically incoherent.

To solve the lawn mowing puzzle you must schedule the mowing of the first two lawns as two supertasks which finish at the same time, although no part of one supertask is simultaneous with a part of the other. They must finish at the same time because if one finished first then there would be exactly seven lawns left until the second one finished. They cannot have parts at the same time because you only have one mower. This is possible because a supertask need not have a last part. Here is how to do it.

Start at 2pm. Mow half of lawn one in fifteen minutes, and half of lawn two in fifteen minutes, then half of what remains of lawn one in seven and a half minutes, then half of what remains of lawn two in seven and a half minutes, then half of what remains of lawn one in three minutes and forty-five seconds, and so on. This will get both lawns mown by three o’clock, except for the very edges. To solve this problem you can start at 1.58 instead and mow the very edges before the supertasks begin. Then at three o’clock you have a well-earned rest before mowing the other six.

References

-Gwiazda, Jeremy 2010: ‘The lawn mowing puzzle’, Philosophia, vol. 38, no. 3, p.629

-Hawthorne, John and Weatherson, Brian 2004: ‘Chopping up gunk’, The Monist 87, pp.339-50

-Russell, Bertrand 1936: ‘The limits of empiricism’, Proceedings of the Aristotelian Society, New Series, vol. 36 (1935-1936), pp.131-150

-Thomson, J. F. 1954: ‘Tasks and super-tasks’, Analysis vol. 15, no. 1 (Oct. 1954), pp.1-13