Logical pluralism and brother-in-law pluralism

In my philosophy of logic class this week, we discussed JC Beall and Greg Restall's version of logical pluralism. Our text was their 2000 Australasian Journal of Philosophy article, available on Restall's website here. I've been flipping through their fantastic book-length treatment (OUP, 2006) as well.

Here's their basic idea. The basic, accepted notion of logical consequence is adequately captured in the following:

(V) Consequence C is a logical consequence of premises P1, ... Pn = In every case in which P1, ... Pn is true, C is also true.

Beall and Restall further hold that the notion of case admits of a number of "precisifications" (2006, 88), that is, it can be 'spelled out' or 'fleshed out' in more than one way. [Note: I can't find the quotation now, but I think Beall and Restall said that 'case' is neither ambiguous nor vague (in the sense of having borderline examples). [CORRECTION (3/2/08): In their "Defending Logical Pluralism," Beall and Restall explicitly say that they think the concept of deductive consequence is ambiguous (p.3). This more or less vitiates the main point of this post. I say 'more or less' because there is a test accepted by linguists for distinguishing ambiguity from lack of specificity, and it's not clear that B&R's concept of 'case' passes the test; see my comment #6 in the comment thread.] Different spellings-out of 'case' give rise to different consequence relations (and thus different logics); as examples of cases, they give:
(i) Classical Tarskian models, (ii) possible worlds, (iii) constructions (which yield intuitionistic logic), and (iv) situations (which yield relevant logic).
Finally, because there are multiple ways of spelling out 'case', there is not one correct notion of consequence, since different consequence relations correspond to different ways of specifying the content of (V).

So if someone asks: "Does an arbitrary sentence p follow from a contradiction 'q and not-q'?", the pluralist answer is "Yes and No -- yes, it follows classically (when we take Tarskian models as cases), but no, it does not follow relevantly (when situations are the cases)." Similarly, the pluralist answers the question "Is 'p or not-p' a logical truth?" with "Yes and No -- yes, it is a classical logical truth (since it is true in all Tarskian models), but no, it is not an intuitionistic logical truth (since it is not true in all constructions)".

I find Beall and Restall's position attractive. But while thinking about it, I wondered about when, in general, pluralism is the right (or at least a reasonable) position to take. B&R's claim is the fact that 'case' can be precisified in more than one way -- the meaning of 'case' is somehow underspecified or indeterminate -- to justify being pluralists about 'case' and thereby via (V) about consequence. However, I wonder whether, if this rationale were accepted across the board, pluralism would be almost everywhere, and the appropriate answer to many, many questions would be "Yes and no".

Here's an example of what I mean. The meaning of the phrase 'my brother-in-law' is not completely specific; it is indeterminate between the brother of my spouse and the male spouse of my sibling. However, nobody is a "brother-in-law pluralist": When someone asks me "Is Leon your brother-in-law?", I shouldn't reply "Yes and No -- yes, he is the brother of my spouse, but no, he's not the male spouse of my sibling." And what holds for 'brother-in-law' holds for many, many other terms: lack of specificity is everywhere.

Hopefully the analogy is clear: 'case' and 'brother-in-law' can both be made (more) determinate in different ways. But if this underspecification in the notion of 'case' is all that is required to justify pluralism about consequence, then we should also be pluralists about 'brother-in-law', since there is underspecification there too.

How might someone sympathetic to logical pluralism (e.g. me) respond to this challenge? Well, we could find an example where pluralism seems like the right (or at least reasonable) attitude, and try to argue that 'case' is (more) like that example. For example, I think pluralism about the concept of 'thing' is reasonable: if someone holds out a deck of cards, and asks me "Are there 52 things here?", the right (or reasonable) answer should be "Yes and No -- yes, there are 52 cards, but no, there are far more than 52 molecules".

The question is then: What makes 'thing' different from 'brother-in-law'? And is 'case' (in Beall and Restall's use) more like 'thing' or 'brother-in-law'? The pluralist wants 'case' to be more like 'thing', but I haven't yet figured out how to draw a sharp line. Any ideas?


Kenny said...

The way I would want to go about these discussions is to figure out what uses we have for the relevant concept. I don't really understand whether there's a distinction here between pluralism and ambiguity (or perhaps better, homophony between different words). But in the case of "brother-in-law", I think we normally don't have very essential uses for the term, and so it's ok that it sometimes refers to one and sometimes the other. Whereas with logical consequence, I've worried (since reading that paper) that the pluralist position is missing something. If logic is really about truth, and how things actually are, then presumably one of the notions will be the relevant one. Perhaps it'll be a different one for inferential purposes than for truth purposes, but I just don't see why someone would be interested in the intuitionist or relevance notions if they're realists and correspondence theorists about truth and the like.

Greg Frost-Arnold said...

Hi Kenny --

Thanks for the rich comment. A few things occur to me in response:

(1) I'm very sympathetic to your basic idea of appealing to what the concept is FOR as a constraint on the problem-area. That said, I don't have a clear and distinct idea of what you mean by "very essential uses" of a term, and why 'brother-in-law' lacks one.

(2) Also, though I asserted in the post that 'brother-in-law'-like cases abound, I did not really justify that at all. So here's another extended example: 'sibling' is indeterminate between 'brother' and 'sister'. 'Sister' is indeterminate between 'older sister' and 'younger sister.' 'Parent' is indeterminate between 'mother' and 'father.' And I think it's clear that 'parent,' 'sister,' and 'sibling' are not ambiguous (more specifically, not homophonic) words. Examples of non-maximally-specific words could be multiplied. Will none of these have 'very essential uses'?

(3) I'm very sympathetic to your worry about logical pluralism missing something: "If logic is really about truth, and how things actually are, then presumably one of the notions will be the relevant one." I'm not sure how B&R would respond, but here's two tries:

(i) When we say C is a logical consequence of P1...Pn, that can't just be saying that the actual world is such that either one of P1...Pn is false or C is true (since then "Grass is purple" entails "The sky is green"). So we can't just be talking about truth (in our world). In my intro classes I say "A valid argument is one whose conclusion is true whenever all the premises are" (= B&R's (V)). What exactly does that 'whenever' quantify over? It's not specified, and (more importantly) it's not clear to me that being a "realist and correspondence theorist about truth and the like" forces you to pick one specification over the other (e.g., does the realist correspondence theorist go with possible worlds or Tarskian models as the domain of quantification for that 'whenever'?).

Try (ii): Logic is not about truth simpliciter, it is about (preservation of) true sentences. Sentences are 'made' true by something or another (at least in most projects of philosophical semantics). Is the realist correspondence theorist committed to the idea that there is only one kind of thing that can make sentences true? Or is there still some freedom in specifying what makes sentences true?

Those responses are a bit groping, I realize. As I said, I think you're placing pressure on B&R in the right spot.

Richard Y Chappell said...

Regarding (2), we should distinguish non-maximal-specificity from indeterminacy. 'Parent' is not "indeterminate between 'mother' and 'father'", it determinately includes both! So we may hope to distinguish pluralism on these grounds. It is not that there is some overarching concept of a 'case' which is the disjunction of any number of more specific implementations. Rather, there are competing conceptions of a 'case', each of which implicitly excludes the others, though objectively speaking no one approach is privileged.

Greg Frost-Arnold said...

Hi Richard -

Thanks, that strikes me as a good way to think about the difference between 'brother-in-law' and 'case'. Do you think various conceptions of 'thing' compete (instead of being disjunctive) also?

I was also hesitant about using the term "indeterminate", but I recently read a famous paper by the linguists Zwicky and Sadock, "Ambiguity Tests and How to Fail Them," where they say some linguists have called (what they call) 'lack of specificity' by the name of 'indeterminacy' inter alia. I'm not a linguist, so I was deferring to their jargon.

I do wonder whether the different notions of case really do compete/ 'implicitly exclude' one another in the way you're envisaging, for B&R. For some cases are proper subsets of others. E.g., one way of spelling out 'situation' is as a part of a world -- but not necessarily a proper part, so that worlds are situations too.

Now that I think about this, it strikes me as wrong to conclude that this precludes implicit exclusion/ competition: if someone asks me "Given any two distinct numbers, is there a third distinct number between them?", I have to say "Yes and No -- If by 'number' you mean 'real number', then yes; but if by 'number' you mean 'integer' then no." But the integers are obviously a proper part of the reals.

So can we figure out a test that sorts cases of lack of specificity into (implicitly) competing vs. disjunctive ones? (And then, of course, run that test on 'case' in B&R's usage.)

Greg Frost-Arnold said...

Notes mostly to self:

~ If 'case' (and therefore 'consequence') are ambiguous (as B&R say in "Defending Logical Pluralism" -- see correction in main post), then is 'thing' also ambiguous, and not merely very general in sense? For 'thing' seems to me much closer to 'case' than 'bank' or 'duck' are.

~ 'sibling' = 'brother or sister'. But B&R cannot think of 'case' as 'situation or possible world or Tarskian model or...'; if they did, there would be one notion of logical consequence, viz. C is a consequence of P iff there is at least one type of case in which C is true in every case in which P is true. So the question is then: is 'case' really disjunctive/ unspecific (like 'sibling') or not?

~ Jay David Atlas's 1989 Philosophy without Ambiguity argues (to a first approximation) that many supposed cases of ambiguity are really just general in sense. So if I ever think about this more, I'm going to have to dig into that book.

~ I asked my students today how they would respond to "Is pi a number?" All said yes, nobody was really tempted by the 'one the one hand yes (it's a real number), on the other hand no (it's not a natural number).'

~ B&R give (in the book, not the article) a definition of case as (basically) something that makes a sentence true. But does this definition really generate an ambiguous notion? I.e., Might this yield a univocal notion of case? [Student Tommy Lane pushed in this direction]

Greg Frost-Arnold said...

A couple more notes to myself:

- It looks like a common figurative way of putting the difference between lack of specificity and ambiguity is that in ambiguity, we SELECT a sense to give an utterance meaning, whereas in underspecification, we SUPPLY a sense. Using that metaphor, 'case' (and 'thing') don't look that ambiguous.

- One of the most common ambiguity tests is the "conjunction reduction test." Consider the two sentences
Aaron is at the bank
Beth is at the bank

Consider a circumstance in which both are true, because Aaron is next to the river, and Beth is making a monetary deposit.
Now, the 'reduced' sentence
Aaron and Beth are at the bank
cannot express this state of affairs: it can only say that both A and B are at the money depository or at the edge of a body of water. The linguists describe this as 'a cross interpretation' being impossible. This is evidence of ambiguity (instead of generality of sense). Why? Consider 'parent,' which is unspecified between 'mother' and 'father'.
Aaron is a parent
Beth is a parent

Consider a case where both are true, because Aaron is a father and Beth a mother. But unlike the above case involving 'bank', cross-interpretation is possible in the reduced sentence:
Aaron and Beth are parents.
That is, 'parent' means 'mother or father'.

Now how does this test fare when it comes to the pluralists' 'case'?

Let's suppose that S1 is some situation (of the sort used in semantics for relevant logic), and C1 is a construction (of the sort used in semantics for intuitionistic logic).

I think Beall and Restall would say
S1 is a case
C1 is a case
are both true (at least in the way 'Aaron is at the bank' is true when he is at the edge of a body of water but not at a monetary depository; if B&R say these two are not true, then 'case' is not even ambiguous). The 'conjunction reduction' would then be
S1 and C1 are cases.
Now, the question for the ambiguity test is: does this admit of crossed interpretations? (= Is this sentence true in the specified circumstance?) If Yes, then 'case' is not ambiguous; if No, it is ambiguous. The problem is I just don't know whether the answer is Yes or No -- or what evidence for one side or the other would look like.

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