Wolfgang Schwarz

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The gamma rule must not be applied if the result of its previous application has not yet been replaced by the closure rule.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

Objects of Fiction

Here comes a positive theory of fictional characters. Disclaimer: Only read when you are very bored. I've started thinking and reading about this topic just a weak ago, so probably the following 1) doesn't make much sense, 2) fails for all kinds of well-known reasons, and 3) is not original at all. The main thesis certainly isn't original: it is simply that fictional characters are possibilia. Anyway, I begin with an account of truth in fiction, which largely derives from what Lewis says in "Truth in Fiction".

Do We Need Fictional Truth?

J from Blogosophy proposes that we use "in a manner of speaking" instead of "accoring to the fiction" as a prefix for fictional statements. This, J says, would also work for the problematic cases like "Sherlock Holmes consumed drugs that are illegal nowadays". I'm afraid I don't quite understand this operator. What are the truth conditions of "in a manner of speaking, p"?

Counterfactuals and Counterexamples

It is controversial whether indicative conditionals with false antecedents are generally true. As far as I know, which really is not very far at all, it is equally controversial whether counterfactual conditionals with necessarily false antecedents are generelly true. What's interesting is the different kinds of counterexamples that are brought forward against these views. For indicatives, the counterexamples are indicative conditionals with false antecedents that nevertheless appear to be false, e.g. "if I put diesel in my coffee, the coffee tastes fine." For counterfactuals however, the alleged counterexamples (brought forward e.g. by Field in §7.2 of Realism, Mathematics & Modality, Katz in §5 of "What mathematical knowledge could be", and Rosen in §1 of "Modal fictionalism fixed") are counterfactual conditionals with necessarily false antecedents that appear to be true, e.g. "if the axiom of choice were false, the cardinals wouldn't be linearly ordered". Isn't this quite puzzling? How can the fact that some instances are true be a problem for a theory that claims that all instances are true?

Parsimony and Ontological Dependence

This is part 2 of my comments on Fiction and Metaphysics.

Amie Thomasson argues that fictional objects are not as strange and special as one might have thought because they belong to the same basic ontological category as works of art, governments, chairs and other objects of everyday life. Doing without fictional entities, she says, would merely be "false parsimony" unless one can also do without other entities of this category.

I have three complaints.

Amie Thomasson's Fiction and Metaphysics

Brian has made so many puzzling remarks about fictional characters being real but abstract that I've decided to read Amie Thomasson's Fiction and Metaphysics. Here is my little review.

Thomasson's theory, in a nutshell, is that the Sherlock Holmes stories are not really about the adventures of a detective who lives at 221B Baker Street, but rather about the adventures of a ghostly, invisible character who lives at no place in particular and never does anything at all. We don't find this written in the Sherlock Holmes stories because, according to Thomasson's theory, Arthur Conan Doyle simply doesn't tell the truth about Holmes. In fact the only thing he gets right is his name: That ghostly character he is telling wildly false stories about is really called "Sherlock Holmes".

More About Analyticity

Here comes the promised reply to Sam's reply to my previous posting. In that posting, I first suggested that some sentence S (in a given language) is analytic iff you can't understand it unless you believe it. Then I said that, "put slightly differently", S is analytic iff it is impossible to believe that not-S.

As Sam notes, the first definition implies that even very complicated analytic truths have to be believed in order to be understood, which might be somewhat unintuitive. I'm not sure how bad this is for lack of a clear example. Sam uses "the sum of the digits of the first prime number greater than 1 million is even", but this is not analytic, so here I can perfectly well admit that you may understand it without either believing or disbelieving it. He also mentions infinitely long sentences, but I don't believe there are any of those in ordinary languages.