Jim:
Back in September 2002, you asked me for something; and I didn't give
it to you then, because I didn't know at the time that I had it to
give. I hope this post is an example of better late than never.
<quote>
[ ... ]
Burns:
In the search for new-and-improved versions of equality, it
seems to me that a case can be made for weakening the axiom
extensionality in some domains. A typical example of the use
of intensional propositions:
(1) The little boy believes the man in the fake beard is
Santa Claus.
(2) The man in the fake beard is his father.
Therefore,
(3) The little boy believes his father is Santa Claus.
[FALSE]
Clearly, we can't get away with using substitution in all
circumstances.
[GD]
That's just the invalid reference (Bb(m=s) & (m=f))-> Bb(f=s)
(B=believes; b=boy; m="man in fake beard"; s="Santa Claus")
which reflects the fact that B is not a truth-functional
operator. The inference would be just as invalid using
different predicates.
[Burns]
I agree that what I wrote above is invalid. I intended it to be
invalid, I even marked it so. I don't understand why what
you've written is invalid: (Bb(m=s) & (m=f))-> Bb(f=s)
If (m=f), why can't we substitute f for m in Bb(m=s), giving
us Bb(f=s)?
"B is not a truth-functional operator" I take to mean
"B is not a function of {persons}x{true, false}". I agree with
that but don't see what that explains.
I think the inference would be valid if we restrict ourselves
to _extensional_ predicates, eg,
(1) The man in the fake beard is seated on a throne in a
department store.
(2) The man in the fake beard is the little boy's father.
Therefore,
(3) The little boy's father is seated on a throne in a
department store.
I think the inference above would be just as invalid using
different _intensional_ predicates: "The boy doubts that ...",
"The boy wishes that ... ". I don't claim to have a theory
explaining or even describing intensional predicates.
That's what I was asking for.
<quote>
Now that I've thought about it, and know to answer your points, I'd
like to reply to them in turn:
[Burns]
"I agree that what I wrote above is invalid. I intended it to be
invalid, I even marked it so. I don't understand why what
you've written is invalid: (Bb(m=s) & (m=f))-> Bb(f=s)
If (m=f), why can't we substitute f for m in Bb(m=s), giving
us Bb(f=s)? "
Because beliefs are not closed under entailment; if I believe that
Paul Martin is the Liberal leader in Canada, and the Liberal leader is
in fact the Prime Minister, it doesn't follow that I think Paul Martin
is PM (it could be that I'm confused about which party is in power,
instead).
Beliefs are closed under believed entailment - if I believe Martin is
the Liberal leader, and I believe that the Liberal leader is PM, I
believe that Martin is PM - and under believe entailment only.
[Burns]
""B is not a truth-functional operator" I take to mean
"B is not a function of {persons}x{true, false}". I agree with
that but don't see what that explains."
What I meant was that B is a non-truth-functional operator; and by
that I mean, not *completely* truth-functional (in the way ~ is; the
truth value of ~p can always be computed from the truth value of p,
and vice versa). "It is believed that p" does not imply p, nor does p
imply "It is believed that p."
[Burns]
"I think the inference would be valid if we restrict ourselves
to _extensional_ predicates, eg,
(1) The man in the fake beard is seated on a throne in a
department store.
(2) The man in the fake beard is the little boy's father.
Therefore,
(3) The little boy's father is seated on a throne in a
department store."
Yes. It would also be valid if we restricted ourselves by using only
intensional predicates; which, I've suddenly realized, is what we do
by combining extensional predicates with non-truth-functional
operators. For example, we can adopt a box-operator [], defined as
"It is believed that.
That allows us to distinguish between the valid
extensional-predicate-only arguments -
1. Ba -> Ta
[B=df."_has a fake beard"; T=df."_is seated on a throne in a
department store
2. Ba <-> Fa
[F=df."_is the little boy's father"]
3. Fa -> Ta
QED
- valid intensional-predicate-only arguments -
1. []Ba -> Ta
2. []Ba <-> Fa
3. []Fa -> Ta
QED
- and invalid arguments that mix intensional and extensional
predicates:
1. []Ba -> Ta
2. Ba <-> Fa
3. []Fa -> Ta
BZZT!
Well, the theory that I'd like to give you to try on is:
Intensional predicates are correctly formalized or symbolized in
predicate logic using extensional predicates modified by
non-truth-functional operators (as defined above). What truth
functions there are between those operators and the PC operators must
be modelled axiomatically, and different operators would require
different axioms - different relations would be axiomatic between D
"It is doubted that..." and W "It is wished that...", eg.
IOW, intensional predicates are correctly formalized or symbolized in
predicate logic by using one of the various axiomatic systems of modal
logic.
.
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