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previous application has not yet been replaced by the closure rule.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:
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".
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"?
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?
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.
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".
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.
Here at Humboldt University, there's a reading group about analytic
philosophy (Sam already mentioned it). The flyer
advertising this group describes analytic philosophy as a sort of new and
fascinating kind of philosophy characterised by its perspicuity and
ignorance of philosophical tradition. The funny thing is that the
organisers of the reading group decided that we'll be discussing David
Wiggins' Sameness and Substance Renewed. I don't want to know
how much Hegel one has to read to find Wiggins perspicuous (and
ignorant of philosophical tradition).
Some expression can't be properly understood unless one believes certain
things: In some sense you don't understand "irrational number" unless you
believe that no natural number is irrational; You don't understand "grandmother"
unless you believe that grandmothers are female; Maybe you don't understand
"cat" unless you believe that cats are animals.
This is all quite vague because "understanding" and "believing" are vague.
I now want to suggest that a sentence is analytic iff you can't understand
it unless you believe it. Analyticity is also vague, so the vagueness of
the explicans is fine for this purpose.
I've made some slides about logic programming (PS) for my presentation next week in the logic seminar.