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Sunday, September 23, 2007

Relevant Consequence

Since my talk here at ANU last Thursday I have been thinking a lot about how to define 'relevant consequence'. There are various reasons why one might want a notion of relevant consequence (in addition to the standand notion of necessary consequence). For example, various attitudes are closed under some consequence relation. But they are not closed under necessary consequence. For example, 'I believe that snow is white' does not entail 'I believe that Goldbach's conjecture is true'. There is a lot of literature on how to define 'relevance'. One notion which I believe is due to (or at least inspired by) Graham Priest goes like this. q is a relevant consequence p iff q is a consequence of p, and q does not introduce any new non-logical constants (of course, it may be that new non-logical constants are introduced in the derivation of q from p, e.g. if existential elimination is applied). I like this way of defining 'relevant consequence'. But it is too strict for my purposes. I want 'I own a car' to be a relevant consequence of 'I own a Porsche', but 'car' is a new non-logical constant. So, I was thinking the following might do as a definition of 'relevant consequence'. q is a relevant consequence of p iff q is a consequence of p, and if q introduces a new non-logical constant P2, then P2 is a minimal predicate, and for some minimal predicate P1 in P, necessarily, all P1 are P2. Given this notion, 'I own a car' is a relevant consequence of 'I own a Porsche', because it is true that necessarily, all Porsches are cars. I can't think of any obvious counterexamples to this way of defining 'relevant consequence', though I am sure there are some.

5 comments:

Mike said...

Hi Brit,

q is a relevant consequence of p iff q is a consequence of p, and if q introduces a new non-logical constant P2, then P2 is a minimal predicate, and for some minimal predicate P1 in P, necessarily, all P1 are P2. Given this notion, 'I own a car' is a relevant consequence of 'I own a Porsche', because it is true that necessarily, all Porsches are cars.

Let P2 be the property of not being a squared circle. Any P1 entails that, since necessarily x is a P (say, a Porsche) only if x is not a squared circle. But not being a squared circle hardly seems relevant to owning a Porsche. So 'I am not a squared circle' does not seem to be a relevant consequence of 'I own a Porsche'.

Brit Brogaard said...

Thank you, Mike! The "minimal condition" was supposed to rule out this sort of case. So, "is a Porsche" and "is not a squared circle" are not minimal predicates. Perhaps "predicate" is misleading. I should probably say "constituent" or something like that.

Perhaps the following re-formulation is more in line with what I had in mind:

q is a relevant consequence of p iff q is a consequence of p, and if q introduces a new non-logical constant P2, then P2 is a minimal constituent, and for some minimal constituent P1 in P, necessarily, all P1 are P2.

Mike said...

Thanks Brit. Maybe you could say something more about what counts as a minimal predicate (min. consituent). In the initial example, if I am reading it right, your P2 is the new predicate 'is a car' and P1 is the predicate 'is a Porsche'; but I gather that neither is minimal predicate. So, maybe some necessary conditions on 'minimal predicate'?

Garen said...

Hello Brit,

Sorry for posting this as a would-be off topic comment, but I couldn’t find your e-mail.

I think you would be interested in joining www.hypertope.com , which is this new Academic Networking Site for researchers and scholars that we’ve launched 2 weeks ago.

Normally, joining is free, but by invitation only… but for a brief promotional period only we’ve “opened the gates”, so that the initial members can join in, and invite their own academic colleagues, and so on…

you can join here www.hypertope.com/user/register (it will take you just a couple of minutes. Don’t forget to post a link to this blog when you create your page on hypertope)

Hypertope has loads of handy tools for research networking. I guess rather than just describing them, it’s better to just create a user account and give it a test ride.

It would also be nice if you could post a blog item about it and also include it in your blogroll, so that the word spreads.

all the best

Garen

Brit Brogaard said...

Hi Mike, yes that's a good point. I would like to say that a minimal constituent is just a common noun (like 'car' and 'Porsche'). But that won't do. For it is arguable that 'entity-that-is-not-a-square-circle' is a common noun (or at least a common noun phrase). Perhaps *non-composite common noun* will do. Then we also avoid the undesirable consequence that 'I own a small car' is relevant to 'I own a small stretch limo'. But perhaps the 'non-composite' condition is too strict. It seems to rule out that 'I own a car' is relevant to 'I own a stretch limo'. But it's the best I can think of right now.

Garen: Sorry, I don't have the time or the inclination to advertise anything unless it is a philosophy conference or not-for-profit organization or the like.