Wednesday, October 8, 2008

Classical Logic and Inconsistent but Non-Trivial Fiction

This is a short excerpt from my paper on "What You Don't Need Paraconsistency or Pluralism For." The basic claim of the paper, as mentioned in the last post, was that while it is certainly true that if dialetheism is right, classical logic is wrong, since rules like Disjunctive Syllogism won't be universally truth-preserving, none of the traditional (non-dialetheist) motivations for paraconsistency give the classical monist (i.e. the logical monist who thinks that classical logic is the correct logic) any reason to abandon that position in favor of either logical pluralism or paraconsistent monism, since they can happily accept the relevant claims and accommodate them in their classical monist framework. Here's the bit about fiction:

"Start with inconsistent fiction. That it is possible for the writers of fiction to create inconsistencies, whether on purpose (as in Graham Priest’s whimsical short story ‘Sylvan’s Box,’ in which Priest himself, appearing as a character, finds that the late Richard Routley owned a box that both contained and did not contain a statuette, at the same time and in the same sense) or, as is vastly more common, by accident, is clear enough. To take a classic example, in Arthur Conan Doyle’s Sherlock Holmes stories, Watson’s war wound is always referred to in the singular, strongly suggesting that he only has one of them, but sometimes it is on his left shoulder (and thus, presumably not on his right shoulder) and sometimes it is on his right shoulder (and thus not on his left shoulder). Moreover, in both the Conan Doyle case and the Priest case, not everything is true in the worlds of the stories. It is not true in the world of ‘Sylvan’s Box’ that Graham Priest agrees with everything in this paper, and it is not true in the world of Conan Doyle’s Sherlock Holmes stories that Professor Moriarty’s ultimate aim was to raise up Cthulu and Azathoth to destroy humanity.

"How can the classical monist make sense of this? Easily, and in pretty much the same way that a conscientious paraconsistentist would have to make sense of it. Regardless of what your view is on fictional characters--whether they have no ontological status at all, or are abstract objects, or are Meinongian objects that fail to have the property of existing but have all sorts of other properties, like ‘being referred to in philosophy papers’--statements like, for example, “Sherlock Holmes lives on 221B Baker Street” are, strictly speaking, false. Go to 221B Baker Street in London, and you will not find Holmes or Watson there, but a bank, one with a special employee in charge of responding to correspondence written to Holmes. To the extent that we regard such statements as true--and this is true regardless of our stance on the ontological status of fictional characters--it is because we read them as containing implicit fiction operators. Sherlock Holmes, if he exists as an abstract object, has, for example, the property of being written about by Arthur Conan Doyle, but he most certainly does not have the property of actually living on 221B Baker Street. He only has this property in fiction.

"Once we realize this, however, the apparent conflict between classical monism and our obvious ability to non-trivially reason about fictional inconsistencies disappears. If Dr. Watson had the property of both being written about by Arthur Conan Doyle and not being written about by Conan Doyle, in precisely the same sense, that would be a real contradiction symbolized as (α ∧ ¬α), that would indeed entail any and every β in classical logic. If, however, Dr. Watson has the property of having in his war wound on his left shoulder and having it on his right (i.e. not on his left) shoulder in fiction, then acknowledging that that’s true commits us not to (α ∧ ¬α), but F(α ∧ ¬α), which is a very different thing.

"When we extend classical logic, to include such operators I see no reason why we should mimic the rules about what goes on outside of them when we make the rules for what goes on within their scope. A classical monist can happily admit that there are cases where F(α ∨ β) and F(¬α) are both true, but F(β) is not. More radically still, we can allow inferences within the scope of the fiction operators to be not only ‘paraconsistent’ but non-adjunctive. That is to say, F(α) and F(¬α) need not always entail F(α ∧ ¬α). This preserves the intuitive difference between the ‘Sylvan’s Box’ case, in which it is true in the world of the story that the box is both empty and non-empty, and the Sherlock Holmes case, in which both ‘Watson’s war wound is on his left shoulder rather than his right shoulder’ and ‘Watson’s war wound is on his right shoulder rather than his left shoulder’ are both true, but ‘Watson’s war wound is on both of his shoulders’ does not seem to be true. Notice that even here, we have not revised classical logic one whit, as evidenced by the fact that even if F(α) and F(¬α) do not jointly entail F(α ∧ ¬α), they continue to jointly entail (F(α) ∧ F(¬α))."


Aidan said...

Watson's war wound moves between his shoulder (Scarlet) and his leg (Sign of Four).

It's worth thinking about the role of the narrators in these stories. Some writers on the Holmes stories have suggested (somewhat implausibly to my mind) that Watson, rather than Conan Doyle, has trouble keeping things straight between the first two stories. On this interpretation, the reader is invited to infer that the event was sufficiently traumatic to Watson that he has trouble bringing it back to consciousness. Likewise, Daniel Nolan points out (in PQ) that we get a consistent reading of 'Sylvan's Box' if we take Priest as narrator to be deluded or confused. Both the (relevant) Holmes stories and Priest's story have a first-person narrative, which makes the suggestion of an unreliable narrator particularly salient. Things are trickier with third-person narratives, but it would be worth thinking about whether the same issues come up - my guess, fwiw, is that they do.

The point is there are issues to be resolved before we even draw any conclusions about the logic of truth in the fiction, let alone take the further step which you challenge here.

Ben said...

RIght, well, those are moves that you can make to dodge the issue of fictional inconsistency in any given case, but they all smack of desperation to me, and in any case would certainly beg the question against the paraconsistentist by taking for granted the background assumption that fiction must be consistent, and allowing that assumption to trump everything that seems to be the case about a given work of fiction. (Moreover, the dodge raises more questions than it solves. What, after all, could make things true about the world of the fiction other than the actual words on paper, particularly when, as in the Slyvan's Box case, the intention of the author was clearly *not* to portray his fictional avatar as an unreliable narrator?) That said, I don't think I need to respond or resolve any of these issues for the purpose at hand. My point is that even if (non-trivial) fictional inconsistency is possible, as it certainly seems to be, we don't need to step outside of classical logic to accommodate this. If fictional inconsistency is impossible, better yet. I think the idea that it is impossible is an extremely implausible one, but if it is, that's all to the good as far as the purpose at hand, which is defending classical monism against the claim that we need to switch logics to accommodate inconsistent fiction.

Rafal Urbaniak said...

Okay, say we introduce such an operator F, so that `F(A)' reads (say) `According to the story, A'. Here's a question: can we even specify a reasonable minimial consequence operation that would "work within the scope of F"? Call the set of sentences that literally belong to a story `S'. For any story S, its operator F, and any sentence A, we (probably) want the following to hold:

If A belongs to S, then F(A).

(Although even this isn't obvious: sometimes one might insist that a sentence contained in a story shouldn't be taken for granted in the story, because it resulted from an obvious mistake on the part of the author.)

But is there anything else that we may reasonably require to hold for any story and its operator? Let T(F) be the set of all those A for which it is the case that F(A). Why can't T(F) be just any arbitrary set A containing S and contained in the set of classical consequences of S (or even: an arbitrary superset of S)? If yes, it seems there is not much formal work to do. If no, there's lots of further problems:

1. S won't determine F(A) - how do you "extract" a consequence operation from a story? (arguably, different stories might involve different "internal" consequence operations, and since the authors usually don't specify it, how do you tell?).

2. If you can't clearly specify the consequence operation, how do you tell lack of acceptance from rejection? Suppose I'm writing a story and I want to reject conjunction elimination. I say:

P and Q.

and in some other context in the same story I say:

It is not the case that P.

Now, how do you decide whether you should rather (a) say that I accepted both P (by accepting `P and Q') and not-P, or (b) insist that I rejected conjunction elimination, without accepting any straightforward contradiction? Even saying that at some other place I explicitly used conjunction elimination (say I did) won't help, because I can accept only specific applications of it, for instance...

3. There also might be sentences that aren't in S, but are tacitly taken for granted in the story. So, for some A, A is neither in S nor in the set of classical consequences, and yet it should be the case that F(A) (for instance, if from the context it is clear that the world described abides by the same laws of physics, and yet the author doesn't say that explicitly).

I don't know, the idea that you could approach this mess with a decent formal tool so that its usage is philosophically justified seems quite hopeless...