Wolfgang Schwarz

Fictionalism's Ontological Commitment

Fictionalism about a certain discourse is the view that statements belonging to this discourse are to be interpreted like statements in fictional discourse.

Now as Brian has observed, on the common account of fictional discourse, "Fictional(Fa)" implies "(Ex)Fictional(Fx)" (even though it normally doesn't imply "(Ex)Fx"). So one might think that on the common account, fictionalism can't do with fewer entities than realism, even though it can do with different entities. However, the common account is not committed to "Fictional(a != b)" implying "(Ex)(Ey)(x != y)". After all, it usually allows for "(Ex)(Fictional(Fx) and Fictional (not-Fx))", so why not allow for "(Ex)(Fictional(x != b) and Fictional(x = b))"? So maybe one could endorse fictionalism about mathematics and the common account of fictional discourse without being committed to an infinity of entities by claiming that all the "numbers" talked about in mathematics are in fact identical.

Realism, Reductionism, Determinatism

Allan Hazlett, in his Against Fictionalism, says that if numbers exist then "there is a fact of the matter about which sets numbers are", even if we can't find it out. I don't think realism and reductionsism about numbers implies this kind of determinatism. (Sorry, "determinism" was already taken.)

The point is general. I believe in psychological states, and I believe that psychological states are really just neurophysiological states. But I don't believe that it is possible to isolate a single brain state that realises the pain role, or the believes-that-the-meeting-starts-at-noon role. The problem is that folk psychology is probably far too unspecific to have a unique realisation. (This is not the problem of multiple realisability, or not quite. Multiple realisability is usually taken to be the problem that pain is or might be differently realised in different individuals. It would be interesting to know more about the relation between the two problems.)

Similarly, I believe in mountains, and I believe that mountains are really just mereological sums of rocks, stones, sand, etc. But I don't believe that it is possible to isolate a single sum of rocks etc. that is (determinately) Mt. Everest.

Vague Objects, Strange Properties

Assume some sentence "Fa" is neither determinately true nor determinately false. This might be due to the fact that

1) It is somewhat indeterminate exactly which object "a" denotes.

or

2) It is somewhat indeterminate exactly which property or condition "F" expresses.

If neither (1) nor (2), then "a" determinately denotes a certain object A and "F" determinately expresses a certain condition F. So whence the indeterminacy? Maybe

In Defense of the Impossibility Hypothesis

My wrists still don't feel quite fine, in particular after writing such a long piece as this one. So this will probably be the last entry for another couple of days.

I want to defend the Impossibility Hypothesis about Imaginative Resistance. The hypothesis is that when we find ourselves unwilling to accept that some statement explicitly made in a fiction is true in the fiction, this is (always) because what the statement says is in some sense conceptually impossible.

Visitors

I would like to know what is being discussed in some non-public forum on ubbforums.ubi.com, from where quite a lot of visitors have arrived within the past weeks. (From a brief glance at these forums, I don't even have a clue what they might be about.) So if you are one of these visitors, would you drop me a line and tell me?

For some reason, the two Gods puzzle also seems to have attracted some attention recently. Today, a clever reader called Ian Stern even found a new solution which, though not truthfunctional, is much more natural than my original solution: 'Would God A say that "qwer" means "yes"?'

Counterfactuals and Games of Make-Believe

In chapter 10 of The Varieties of Reference, Gareth Evans endorses a counterfactual analysis of truth in games of make-believe: When children play the mud pie game, an utterance of "Harry placed the pie in the oven" is true (in the game) iff (roughly) it would be true given that these globs of mud were pies and this metal object were an oven.

He then notices that this is a problem for the possible worlds analysis of counterfactuals because the relevant counterfactuals seem to have impossible antecedents: "there simply are no possible worlds in which these mud pats are pies" (p.355).

Semantic Properties and Bearerless Names

Linguistic expressions have all kinds of properties. In other words, they can be alike in all kinds of ways. For example, two sentences (of a particular language) can be alike in that

1. they have the same truth value
2. they attribute the same property to the same object
3. they are necessarily equivalent
4. they are a priori equivalent
5. they are such that noone who understands them could regard one as false and the other as true
6. they are cognitively processed in the same way in all speakers of the language
7. they invoke the same mental images in all speakers
8. they invoke the same mental images in some particular speaker
9. they have the same use in the community
10. they are verified by the same observations
11. they are constructed in the same way out of constituents that are alike in one way or another

and so on. All these properties are, I believe, worth investigating into, and all of them might be called "semantic".

Fiction and History

Not much blogging these days because for some reason my wrist hurts, and I think it's better to let it rest for a while. So here are just a couple of brief remarks, typed with my left hand, about some parallels between fictional and and historical characters.

We might distinguish two modes of speaking about historical characters:

1. Past: Immanuel Kant is a philosopher; he lives at Königsberg; etc.

2. Present: Immanuel Kant does not exist; he does not live at Königsberg; etc.

Operator Overload

I wanted to blog something on how to treat discourse about fiction in the framework of a general multi-dimensional semantics, but this turned out to work so well that the entry is growing rather long, and I won't finish it today. In the meantime, here is a nice example of multiple context-shifting operators, from this BBC story (via Asa Dotzler):

In this place, for a few hours each day, just after noon in the summer, there could be liquid water on the surface of Mars.

Detecting Satisfiability with Free-Variable Tableaux

Recently I suggested the following restriction on free-variable tableaux:

The gamma rule must not be applied if the result of its previous application has not yet been replaced by the closure rule.

I think I've now found a proof that the restriction preserves completeness:

Let (GAMMA) be a gamma-node that has been expanded with a variable y even though the free variable x introduced by the previous expansion is still on the tree. I'll show that before the elimination of x, every branch that can be closed by some unifier U can also be closed by a unifier U' that does not contain y in any way (that is, y is neither in the domain nor in the range of the unification, nor does it occur as an argument of anything in the range of the unification.) Hence the expansion with y is completely useless before the elimination of x.

Before the y-expansion, no branch at any stage contains y. After the y-expansion, every open subbranch of (GAMMA) contains the formula created by the y-expansion, let's call it F(y). Among these branches select the one that first gets closed by some unifier U containing y in any way. Now I'll show that, whenever at some stage a formula G(y) containing y occurs on this branch, we can extend the branch by adding the same formula with every occurrence of y replaced by x.

At the stage immediately after the y-expansion, the only formula containing y is F(y). And because (GAMMA) has previously also been expanded with x, and x has not yet been eliminated, the branch also contains F(x). Next, assume that G(y) occurs at some later stage of the branch. Then it has been introduced either by application of an ordinary alpha-delta rule or by the closure rule. If it has been introduced by application of an alpha-delta rule then we can just as well derive G(x) from the corresponding ancestor with x instead of y (which exists by induction hypothesis). Now for the closure rule. Assume first this application of closure does not close the branch (but rather some other branch). Then by assumption, the applied unifier does not contain y in any way, Moreover, it does not replace x by anything else. So in particular, it will not introduce any new occurrences of y in any formula, and it will not replace any occurrences of x and y. Hence if G(y) is the result of applying this unifier to some formula G'(y), then G(x) is the result of applying the unifier to the corresponding formula G'(x).

Assume finally that the branch is now closed by application of some unifier U that contains y in any way. Let C1, -C2 be the unified complementory pair. (At least one of C1, C2 contains y, otherwise U wouldn't be minimal.) Then as we've just shown, the branch also contains the pair C1(y/x), -C2(y/x). Let U' be like U except that every occurrence of y (in its domain or range or in an argument of anything in its range) is replaced by x. Clearly, if U unifies C1 and C2, then U' unifies C1(y/x), C2(y/x).

In the other posting I also mentioned that this restriction can't detect the satisfiability of x((Fx y Fy) Gx), which the Herbrand restriction on standard tableaux can. (A simpler example is x((Fx Fa) Ga).) These cases can be dealt with by simply incorporating the Herbrand restriction into the Free Variable system: