PARADOXES AND STATES OF AFFAIRS
LauLuna
proposition according to a particular linguistic code L iff X depicts a
well defined state of affairs according to L.
Most logicians think that not all sentences express propositions, for
example the Liar-like sentences do not.
Nevertheless, the meta-paradox with the Strengthened Liar seems to show
that a same sentence can sometimes express a proposition while it
cannot do it in other occasions. Let me remind you of the well known
argument on this. Consider the sentence-token:
(1) (1) expresses no true proposition
No classical truth-value can be consistently assigned to (1); so we
have to conclude that (1) expresses no proposition and in particular no
true one, so that we are entitled to state the following
sentence-token:
(2) (1) expresses no true proposition
Different tokens of one and the same sentence seem to possess different
logical values. The usual tokenist approach does not go much farther
this way. And a consequence is that one still remains wondering why (1)
is unable to assert the state of affairs that (1) expresses no true
proposition, precisely what (2) is entitled to assert.
The question can be reformulated this way: why is the state of affairs
that (1) expresses no true proposition available for (2) to state but
not for (1)?
Intuitively one sees that (2) avoids the circularity that affects (1):
once (1) and (2) are distinguished as different logical objects, (1)
but not (2) is (or 'tries to be') self-referential. If we examine
the mental events resulting on the assertion of (2), as depicted above,
we see that when we are prepared to assert (2) we are able to make a
reference to (1) and its truth-value that we wouldn't be able while
uttering (1), i. e. before we've had the chance of performing an
assessment of (1). So, it appears that the thought undelying (2) could
never underlie (1).
The phenomenological approach (in Hussserl's sense) I want to propose
accounts, I think, for this all. The mental act A that accomplishes the
assertion of a proposition p is an 'intentional act', that is, a
mental act referring to some object called the 'intentional object'
of A. This object is precisely p.
It seems to me that it is a general or 'eidetic' feature of
intentional acts that no act can be its own intentional object or a
part of it; no thought can think itself, so to say. I will call this
the 'principle of no self-reference' (PNS) A thought underlying (1)
would have to break PNS and, consequently, would be phenomenologically
impossible. This is why there can be no thought behind (1) and this
explains the difference between it and (2).
Now it seems to be a sensible claim that whatever cannot be an
intentional object for an act of thinking cannot be an available state
of affairs for it to assert. This would in turn explain why the state
of affairs that (1) expresses no true proposition is available for the
one who asserts (2) but not for whoever utters (1).
This proposal also fits cases of indirect self-reference. You may
remember the example proposed by Daryl McCullough on the thread
'Halting Problem for Humans' at
http://groups.google.com/group/sci.logic/browse_frm/thread/25164bcecd12ec30
.
Peter and Daryl are trying to correctly answer respectively the
questions 'Will Daryl answer "yes"?', 'Will Peter answer
"no"?'. Each knows what the other is trying to do but no
interaction between them is allowed. Under these conditions there is no
possibility of both working out the correct answer.
Daryl McCullough defended that the questions were well defined, i. e.
that they proposed well defined states of affairs, for Peter will
either answer "no" or he will not do so and correspondingly for
Daryl and "yes". I think he was right but I think too that the
(eventual) state of affairs 'Peter will answer "no"' is not
available for Daryl and the (eventual) state of affairs 'Daryl will
answer "yes"' is not available for Peter. That is why they get
trapped in vicious circularity when trying to answer correctly. Daryl
McCullough's case is a version of the Card Paradox and an example of
indirect self-reference. Peter, for instance, will confront the
situation 'Daryl answers "yes" iff I answer "no"', so Peter
must take into account how his own calculation will behave in order to
perform it. Out of PNS, Peter's reasoning is not an available state
of affairs for Peter while he is still accomplishing it.
Here I found a difference between humans and Turing machines, namely
that a behavior of a Turing machine is always in principle an available
state of affairs for every human while this is not the case for human
mental behaviors.
The phenomenological approach to the state-of-affairs problem in
paradoxes does not apply to all of them, it is only meant for paradoxes
where direct or indirect self-reference is involved. Abo pointed out on
the thread 'Incompatibility of Computationalism and Classical
Logic' at
http://groups.google.com/group/sci.logic/browse_frm/thread/866ad885e012fd13/aca261aa6e66f2e4?hl=en#aca261aa6e66f2e4
that this approach does not apply to the paradoxes of the kind of the
Infinite Liars introduced by Yablo; for example, everyone in an
infinite queue says 'everyone behind me says an untruth' or 'the
next token is not true'.
However, there is no reason to require that the solution to all
sentential paradoxes be exactly the same. I think that the reason why
'everyone behind me says an untruth' is not an available state of
affairs for each queuer can perfectly be quite different from the
reason applicable to the usual Liar, without this disproving the said
above.
Regards
Laureano
|-|erc
> Nevertheless, the meta-paradox with the Strengthened Liar seems to show
> that a same sentence can sometimes express a proposition while it
> cannot do it in other occasions. Let me remind you of the well known
> argument on this. Consider the sentence-token:
>
> (1) (1) expresses no true proposition
>
> No classical truth-value can be consistently assigned to (1); so we
> have to conclude that (1) expresses no proposition and in particular no
> true one, so that we are entitled to state the following
> sentence-token:
>
> (2) (1) expresses no true proposition
>
> Different tokens of one and the same sentence seem to possess different
> logical values. The usual tokenist approach does not go much farther
> this way. And a consequence is that one still remains wondering why (1)
> is unable to assert the state of affairs that (1) expresses no true
> proposition, precisely what (2) is entitled to assert.
>
(g) (g) has no proof
Godels proof followers believe (g) is true, much like how (2) is derived.
Herc
Rupert
That's completely different. (g) is defined to be a certain
arithmetical proposition, which, as it happens, is equivalent to the
unprovability of (g). Since it is an arithmetical proposition, there is
no doubt that it is either true or false. No paradox arises because
truth and provability are different concepts. A paradox would arise if
truth in a given language L were definable in that very language L,
Tarski used this line of thought to prove that that is not the case.
> Herc
|-|erc
there are arithmetical propositions that are equivalent to their own negation.
no paradox arises because you accept truth and provability are different concepts.
Herc
aatu.kos...@xortec.fi
> there are arithmetical propositions that are equivalent to their own negation.
Could you give an example?
> no paradox arises because you accept truth and provability are different concepts.
If there's an arithmetical proposition equivalent to its negation then
arithmetic is totally bonkers.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Rupert
Nonsense.
Rupert
It seems to me we need to give an account of the nature of the
connection between the sentence token and the mental events that take
place when asserting it. I am reminded of Wittgenstein's
thought-experiment: Say "It's warm in here" and mean "It's cold in
here". Presumably the sort of mental event that "should" accompany the
assertion of the sentence is in some way determined by the form of the
sentence. Then we still seem to have a problem since it looks as though
the sentences have identical forms.
aatu.kos...@xortec.fi
> It seems to me we need to give an account of the nature of the
> connection between the sentence token and the mental events that take
> place when asserting it. I am reminded of Wittgenstein's
> thought-experiment: Say "It's warm in here" and mean "It's cold in
> here".
But the whole point of such thought-experiments was to show that
meaning something by something does not consist in some mental act.
Rupert
aatu.kos...@xortec.fi wrote:
> Rupert wrote:
> > It seems to me we need to give an account of the nature of the
> > connection between the sentence token and the mental events that take
> > place when asserting it. I am reminded of Wittgenstein's
> > thought-experiment: Say "It's warm in here" and mean "It's cold in
> > here".
>
> But the whole point of such thought-experiments was to show that
> meaning something by something does not consist in some mental act.
>
True. I recently bought a book called "A Spectical Guide to Meaning and
Rules: Defending Kripke's Wittgenstein." It defends the "meaning
scepticism" attributed by Kripke to Wittgenstein in "Wittgenstein on
Rules and Private Language". I remember reading a discussion
Wittgenstein had with Turing in which he was rather dismissive of the
problem posed by the liar paradox. But it seems to me it is worthwhile
to try and get clearer about the puzzle posed by the strengthened liar.
I'm not sure how you would do that under an assumption of meaning
scepticism.
aatu.kos...@xortec.fi
> I remember reading a discussion Wittgenstein had with Turing in which
> he was rather dismissive of the problem posed by the liar paradox.
It's a famous encounter indeed, described, if I recall correctly, in
Cora Diamond's _Wittgenstein's Lectures on the Foundations of
Mathematics, Cambridge, 1939_.
> But it seems to me it is worthwhile to try and get clearer about the
> puzzle posed by the strengthened liar.
Possibly, and equally possibly it might result in pretty much nothing.
> I'm not sure how you would do that under an assumption of meaning
> scepticism.
I'm not sure how meaning scepticism is relevant, and in any case I'm
not very fond of Kripke's reading of Wittgenstein. Exegetical accuracy
aside, often the best way to read Wittgenstein is to glean whatever
nuggets of insight, whatever thought-provoking questions, strike ones
fancy, and reflect on those, instead of trying to form some coherent
and systematic theory that underlines what he says, relating it all in
a painstaking fashion to this and that character in the history of
philosophy, of to the thought of this or that contemporary figure.
Take, for example, Wittgenstein's comments on the liar, and forget all
the stuff about propositions, meaning, mental states and what not. As
Wittgenstein, quite correctly, notes when faced with the liar we never
reason like: "This statement is false. Hence it is true. Hence it is
false. A contradiction. I'm a wet noodle of bad repute". Rather, the
only purpose the liar serves seems to be to make ones head swim in a
pleasant confusion (and, obviously, give rise to endless philosophical
to and fro).
So, what are we to make of this? Consider the (strengthened) liar
(1) This sentence is not true.
or the "Gödel sentence of Rupert"
(2) Rupert can never know this sentence.
or any of such conundrums people like to reflect on, sometimes reaching
mystical levels of awareness in the process, their eyes gleaming as
their minds expand. These sentences involve notions we in ordinary
circumstances have no problem using, reasoning about them in all sorts
of complex ways, and by means of them effectively communicating stuff
to others. Indeed, it is only in extraordinary circumstances that we
have any difficulty in using these concepts. No-one has any problem
understanding "What Nixon just said about Watergate was not true!" or
accepting "Yes, he studied under Shelah" as an answer to "Does he
really know all this stuff about PCF theory he's going on about?". Or,
if they do, we can give everyday explanations, explications, gentle
nudges to the right direction, hit them on the head until they come to
their senses, and so forth. Not so with the likes of (1) and (2).
The problem with (1) (and (2)) is that we simply have no idea what
truth (or falsity) is supposed to mean in such context. If we want to
be vaguely Wittgensteinian - it's never out of fashion - we could say
that "language has gone on holiday" or that "the wheels are spinning
free". We are using a concept we have a pretty good grasp of in many
contexts in a way that is not connected to its usual usage. Ok, then!
Let's give a meaning to "true" that can be applied in such a context,
as done in the myriards of theories of truth dealing with the liar and
related puzzles. Say, we stipulate that truth is to mean "grounded
truth" as in Kripke's theory of truth. We get then a new reading of (1)
(1') This sentence is not groundedly true.
The paradox goes away, since (1') is indeed not grounded and hence
certainly not groundedly true. But we have then established (1'), and
hence, by any ordinary understanding of truth, that (1') is true. The
same can be repeated for *any* proposed explication of truth that is
applicable in case of sentences like (1). What does this tell us?
The lesson I wish to draw from the observation in the preceding
paragraph is that truth is one of the many concepts that are
"indefinitely extensible", that is, such that for any specific
explication of them we can always find cases which fall under the
informally understood concept, but which are not captured by the
explication. As we all know, another familiar indefinitely extesnible
concept is mathematical provability; whatever formal theory we
recognize as correct, and possibly offer as an explication of
mathematical provability, will fail to capture mathematical truths that
are recognizable as true on basis of correctness of the formal theory.
Indeed, for any property of sentences, a "modality" if you wish, that
satisfies certain (modally definable) conditions will be inexhaustible
in this sense. This might be an interesting observation in itself, but
the real fun begins once we are able to get some real mathematics out
of it. This can be done, and is significant in context of the study of
reflection in proof theory, where we are able to relate the
inexhaustibility of the notion of truth with the inexhaustibility of
statements acceptable as correct on some basis. But that's a whole
another ballgame, on which I might or might not comment in the future.
LauLuna
aatu.kos...@xortec.fi wrote:
> The lesson I wish to draw from the observation in the preceding
> paragraph is that truth is one of the many concepts that are
> "indefinitely extensible", that is, such that for any specific
> explication of them we can always find cases which fall under the
> informally understood concept, but which are not captured by the
> explication. As we all know, another familiar indefinitely extesnible
> concept is mathematical provability; whatever formal theory we
> recognize as correct, and possibly offer as an explication of
> mathematical provability, will fail to capture mathematical truths that
> are recognizable as true on basis of correctness of the formal theory.
> Indeed, for any property of sentences, a "modality" if you wish, that
> satisfies certain (modally definable) conditions will be inexhaustible
> in this sense. This might be an interesting observation in itself, but
> the real fun begins once we are able to get some real mathematics out
> of it. This can be done, and is significant in context of the study of
> reflection in proof theory, where we are able to relate the
> inexhaustibility of the notion of truth with the inexhaustibility of
> statements acceptable as correct on some basis. But that's a whole
> another ballgame, on which I might or might not comment in the future.
I agree with the spirit of this paragraph, and it would be interesting
to have more comments on the topic. I tend to think that paradoxes of
truth hint at essentially the same kind of inexhaustibility we find in
formal provability. This topic promises both mathematical and
philosophical insights.
Regards
LauLuna
Rupert wrote:
> It seems to me we need to give an account of the nature of the
> connection between the sentence token and the mental events that take
> place when asserting it. I am reminded of Wittgenstein's
> thought-experiment: Say "It's warm in here" and mean "It's cold in
> here". Presumably the sort of mental event that "should" accompany the
> assertion of the sentence is in some way determined by the form of the
> sentence. Then we still seem to have a problem since it looks as though
> the sentences have identical forms.
Constant correlation between the (context-independent) sentences we use
to express thoughts and the thoughts expressed by those sentences is
the usual case. What the Strengthened Liar shows is that it cannot
always be the case.
Regards
|-|erc
> > > > (g) (g) has no proof
> > > >
> > > > Godels proof followers believe (g) is true, much like how (2) is derived.
> > > >
> > >
> > > That's completely different. (g) is defined to be a certain
> > > arithmetical proposition, which, as it happens, is equivalent to the
> > > unprovability of (g). Since it is an arithmetical proposition, there is
> > > no doubt that it is either true or false. No paradox arises because
> > > truth and provability are different concepts. A paradox would arise if
> > > truth in a given language L were definable in that very language L,
> > > Tarski used this line of thought to prove that that is not the case.
> > >
> >
> > there are arithmetical propositions that are equivalent to their own negation.
>
> Nonsense.
syntactically its trivial but the system will become inconsistent.
Herc
Rupert
In the inconsistent theory, there are arithmetic propositions
equivalent to their own negation. In any consistent theory, there are
not.
huangx...@yahoo.com
What about a theory which consistently predicts it's own inconsistency
? Is it consistent because of it's ability to predict correctly, or
simply inconsistent because that's what it is ?
Can a theory be both ?
Consistency is just like spacetime continuity. It is and it is'nt. Both
simultaneously.
That's why there are paradoxes.
Rupert
PA+~Con(PA) is consistent but proves its own inconsistency. It's
consistent but not 1-consistent.
> Can a theory be both ?
>
No.