Logical Pluralism, take 2

This post supersedes the previous one on logical pluralism; it's the post I would've written, had I bothered to do a bit of research before posting. I apologize in advance for how long it is...

Beall and Restall say we should be pluralists about logical consequence, because there are multiple acceptable ways of spelling out the notion of case in the standard criterion of validity:
(V) C is a logical consequence of P1 ... Pn iff in every case where all of P1...Pn are true, C is true also.
Different specifications of case, say B&R, yield different consequence relations.

My impulse is to address this issue 'from above'; that is, in general, when is any sort of pluralism an acceptable stance? Well, one case (though probably not the only one) in which it can be acceptable is when ambiguity is present. E.g., there are multiple, equally acceptable ways of spelling out 'Aaron is at the bank'. And Beall and Restall, on the 3rd page of "Defending Logical Pluralism," state that they think logical consequence is ambiguous; presumably the ambiguity is traced back to the word 'case' in (V) (what else in (V) could it be?). So if 'case' really is ambiguous, and (V) captures the core notion of consequence correctly, then we should be logical pluralists.

But I'm not sure 'case' is ambiguous -- prima facie, it doesn't feel like 'bank,' or 'duck'. It might just be 'general in sense' or 'lack specificity': for example, 'sibling' is general in sense between 'brother' and 'sister', the same goes for 'parent' and 'mother' and 'father.' And nobody wants to be a 'sibling-pluralist' or 'parent-pluralist.' (Note: 'thing', which seems to me closer to 'case', does not seem ambiguous either.)

Linguists, fortunately, have devised a test to distinguish ambiguous expressions from ones that are general in sense: the 'conjunction reduction' test. Suppose my friend Pat is making a monetary deposit, and my friend Tracy is sitting next to a river. Then the two sentences
'Pat is at the bank' and
'Tracy is at the bank'
are both true. However, these truths cannot be expressed in a 'conjunction reduced' sentence
'Pat and Tracy are at the bank,'
for this can only mean that both of Pat and Tracy are next to a body of water, or both are at a financial institution. No 'crossed reading' is possible. The impossibility of crossed readings in the reduced sentence suffices for the ambiguity of 'bank' (Jay D. Atlas, Philosophy without Ambiguity, OUP 1989, p.40). From the little I've seen, almost all linguists and linguistically-inclined philosophers accept this as an ambiguity test. And most also accept: If crossed readings are possible, then there is no ambiguity. E.g., 'Pat and Tracy are parents' can be read as saying that Pat and Tracy are both fathers, both mothers, or one of each -- and the possibility of that 'one of each' reading is what makes 'parent' non-ambiguous, but rather just general in sense.

With a test for ambiguity in hand, we can see if 'case' and 'consequence' are ambiguous (so pluralism is the right stance) or merely general in sense (in which case pluralism is not obviously the right stance). Let C0 be a construction (of the sort used in semantics for intuitionistic logic), and let S0 be a situation (of the sort used in semantics for relevance logic). (Cf. previous pluralism post if that doesn't make sense.) Then, I think, Beall & Restall would say
'C0 is a case'
is true, and so is
'S0 is a case'.
(If they don't, then 'case' is not ambiguous, but something else.)
Now the question is: Are 'crossed readings' possible for
'C0 and S0 are cases'?
That is: can that last sentence express that C0 is a construction and S0 a situation, or can it only be read as saying that C0 and S0 are both constructions or both situations? I lean towards the possibility of crossed readings, but I'm just not sure.

We might try the conjunction-reduction test with logical consequence as well:
not-not-A (relevantly) implies A [but not intuitionistically]
not-not-A (intuitionistically) implies 'If B, then B' [but not relevantly]
The conjunction reduced sentence is:
not-not-A implies A and 'If B, then B'
Here, a crossed reading DOES seem impossible, to me at least. So that makes it look like 'implies' is ambiguous. (Note: I'm not certain I've formed the correct conjunction reduction sentence.)

Now I'm stuck: it looks like 'implies' is ambiguous, while 'case' is not. If this is right, then we could deny (V) -- but what could we put in its place?

Final speculative thought: 'implies' is like 'before' in relativistic physics. If two events are spacelike related (= no signal can be passed between them), then 'e1 is before e2' is neither true nor false until a frame of reference is specified. In one frame of reference e1 will be before e2, and in another frame, e2 will be before e1. But once the frame (and a simultaneity convention) is chosen, then it becomes fully determinate which precedes the other. BUT no one frame is the 'right' one; each frame 'has equal rights'.

The analogy: picking a particular frame of reference is like picking situations over constructions or models etc. As there is no one right frame, so too there is no one right truth-maker. So 'A implies B' is then NOT like 'Aaron is at the bank', for I can use that second sentence meaningfully (= truth-valued-ly) without supplying some further information -- unlike 'e1 is before e2' for spacelike-related events. But once one picks a frame, all instances of 'e1 is before e2' become truth-valued; analogously, once one picks a specification of 'case', every instance of 'Does C follow from {P1, ... Pn}?' has a determinate answer.

In sum: ambiguity may not be the best way of thinking about the different consequence relations; rather, perhaps we should see how well we can make out this analogy with relativistic physics.


Kenny said...

I suspect you're right about moving away from ambiguity and towards the analogy with relativity. "Case" here really seems like some sort of technical term, rather than just an ordinary one. In fact, it's a term that seems to be deliberately chosen to be neutral on several issues here.

On another note, I saw Gillian Russell once give a talk where she points out another potential source of ambiguity in the account of consequence. The relata of the consequence relation might be taken to be propositions, sentences, utterances, or possibly something else (like Kaplanian characters or contents or whatever). This allows for one notion of logic on which "I am here now" is logically valid, and another on which the proposition expressed by the sentence is not valid.

Unknown said...

Thanks for an interesting post. I'm actually trying to write something about this myself at the moment, and I've been convinced that JC and Greg's indeterminacy talk is missing the target. They use vagueness as one of their analogies in the book (I think indeterminacy - or perhaps ambiguity - is better since logical consequence is not sorities-prone), but I fail to make sense of *what* it is they think is indeterminate.

Certainly, even if there is some ambiguity in locutions such as 'implies' or 'if ... then', it won't follow that logical consequence is indeterminate. For example, in some of the logics which are candidates for their pluralism, implication and logical consequence won't be related by a deduction theorem.

I think Graham Priest has put it well elsewhere: Logical consequence is an encoding of our practice or of our inferential rules. It is therefore not straightforward what such an ambiguity could amount to.

Unknown said...

Regarding the test: I'm not sure how convinced I am by the conjunction reduction sentence for 'implies'. At least I'm not getting the intuition that the crossreading is impossible.

Here is a similar case study (case in the non-technical sense here):

Let's look at 'follows from'. (This is the expression used by JC and Greg in 'Defending Logical Pluralism'.)

(REL) A follows from A & -A.

(CL) B follows from A & -A.

Both (REL) and (CL) are presumable true from the pluralist perspective, the former referring to relevant logic, the latter to classical logic. What is the target sentence? I would have thought it was the following:

(AMB?) A and B follow from A & -A.

Is this cross-reading permissible? Not really sure what my intuition is here. Help appreciated.

Greg Frost-Arnold said...

Kenny --

So it seems like the next question to ask is whether whatever makes going relativistic in physics OK is also present in the logic case. Einstein had to make an argument for relativity; could a roughly analogous one be constructed for B&R?

Also, since you've got John MacFarlane out there with you, do you have a sense of when he thinks going relativistic is legitimate/ warranted? (I think I asked him roughly that question when he came out to visit, but I don't recall him saying more than "whenever going relative solves the relevant puzzles".)

Ole --

1. I definitely don't have strong intuitions about the impossibility of crossed readings for 'implies' or 'follows from' (the 2 feel the same to me). I actually said as much in an earlier draft of the post, but because it ran so long, I cut it out. So does 'follows from', for you, feel more like 'parent'(='mother' or 'father') than 'bank'? Strangely enough, B&R (in the book) are willing to treat 'logical truth' as having this disjunctive reading; i.e., p is a logical truth (simpliciter) iff p is a logical truth in some logic. I keep reading the bit about why we shouldn't say the same thing about 'follows from', but I still don't get it.

2. And as I think I said, I think the source of ambiguity/ indeterminacy/ for B&R is in the word 'case' -- and then that filters up to 'follows from' via (V).

3. If you're writing something on this, (i) you may want to check out the comments on my previous post -- people made some good comments (and I blather some more); and (ii) I'd be very interested to see what you come up with... or you could at least post something about it!

Unknown said...

1. I'm honestly not sure what I think about it. It does seem awkward to go on from (CL) and (REL) to assert (AMB). It seems to contain a confusion similar to that of the 'bank' case.

Perhaps one reason why the cases are different is that whereas the 'bank' case belongs to colloquial English, the 'follow from' expression tested here is a specialized one. (I doubt if there is a common use of 'follows from' which allows for the type of multiple senses JC and Greg have in mind.)

We should ask some linguists about the examples. Actually, indeterminacy in logic is an interesting topic for other reasons as well. I'm thinking about Ted Sider argument from vagueness for four-dimensionalism. One of his premises is that there is no vagueness in logic.

2. That seems more sensible to me too. Perhaps this is what the authors would have liked to say as well. But, as you also said in the earlier post, I'm not sure what sort of indeterminacy I would attribute to 'case'. I haven't checked this but I thought they also talked about 'case' as a variable somewhere.

3. I'll have to rethink some of my ideas in light of what's been said here, but when I've written a draft I'll post something about it.

Greg Frost-Arnold said...

Note to myself:
I've just found something VERY useful on the internet: Beall and Restall's book blog, where they discuss this very issue (http://pluralism.pitas.com).

They say there are 3 options for the pluralist (quoting):
1. The word "valid" is ambiguous.
2. The word "valid" means something like "truth preserved in all situations" and the scope of the "all" here is contextually determined in some way.
3. The word "valid" means something like "truth preserved in all situations" and the extension of "situations" is indeterminate by practice and can be made more precise in different ways.

I had not really thought of 2, though it may be what Richard Chappell meant in his comment on the earlier version of this post.

In any case, if I ever get around to writing something up about this, B&R's work blog looks like it will be EXTREMELY useful.