Excluded Middle
Marko Amnell
Nothing but the Truth" today I was struck by the surprising role
of _tertium non datur_ in the problem of defining truth inside
a logic. It seems that if one drops the Law of Excluded Middle
as a principle in the construction of logic, it becomes _possible_
to build a logic in which the set of all true sentences can be
defined. Gabriel Sandu claims to show this rigorously in a new
paper "IF first-order logic and truth-definition".
What I wish to ask you all is your general opinion about the
_metaphysical_ status of the Law of Excluded Middle. Reading
Hintikka's article, I was struck by the dissimilarity of the
role of this principle in the problem of Truth, when compared
with its role in the metaphysical problem of Reality or
Existence.
In _The Logical Basis of Metaphysics_ Michael Dummett attempts to
show that if one accepts _tertium non datur_ then the realist
approach to physial objects and the existence of mathematical
objects is supported. (This is an over-simplification, but bear
with me, I have a point) This suggested to me that if the Law
of Excluded Middle supported Realism, then it would also be
on the side of the existence of a set of all truths.
But Hintikka shows that this is manifestly false. Klaus Grue's
work in CS also supports the same conclusion. What we see is
that if we maintain Excluded Middle, we are led to realism in
metaphysics and philosophy of mathematics (what one might call
`global realism') but when it comes to examining the problem
of the definition of truth in logic, and the metaphysical
problem of whether there is or is not a set of class of any other
kind of collection of all truths _in the real world_ (and distinguished
from within a formal system) maintaining Excluded Middle has
a conclusion that seems antithetical to the aims or beliefs of the
realist.
I find this to be a deep result. As Von Wright would put it, the
status of this one principle is a clue about more profound things
in philosophy at large. I am quite surprised at the starkness of
the contrast in the consequences of dropping or maintaining
Excluded Middle in these two groups of philosophical problems:
those surrouding truth and the whole truth, and those surrounding
existence and the nature of the being of various kinds of
objects -- sets, chairs, persons.
I'm probably rambling a bit here. But, in any case, what do you
think about the Law of Excluded Middle? Is it valid in all cases?
Why? How would one justify dropping it? How does one decide
in cases like this which philosophical principle to maintain and
which to drop?
--
Marko Amnell
amn...@klaava.helsinki.fi
Graduate Student in Philosophy
Marko Amnell
believe that there is a set of all truths is simple. He believes
that there is such a thing as the Whole Truth about the world.
This is an objective, mind-independent entity, and the aim of
science is to approach this object, to have our theories match
the structure of this metaphysical object. This is a gross
caricature of the various realist positions, but it serves my
purpose.
So, if Dummett is right that Excluded Middle is necessary for
metaphysical realism, and my caricature above holds in broad
outlines, and one cannot define a set of all truths in a logic
without dropping Excluded Middle (such as in Hintikka's IF system,
which he claims matches the deep structure of natural language and
hence the way we think) then what follows is a very interesting dilemma.
If one holds Excluded Middle, one can maintain Popper-style
realism about the world (and Platonism about mathematical
reality) but this position is itself undermined by sticking
to Excluded Middle, for the only way that one could really
define a set of all truths explicitly is to drop the Law of
Excluded Middle.
On the other horn of dilemma we have the metalogical fact that
if one drops Excluded Middle as a basic principle in the
construction of logic, one can define a set of all truths,
eg. Hintikka and Sandu's IF system, or Grue's Map Theory.
But, if one drops Excluded Middle, and Dummett's meta-philosophical
analysis is right, then one cannot at the same time be a
realist about physical objects or mathematical reality.
One is led to eg. Berkeleyan idealism and intuitionism.
What is one to make of this, if my analysis is indeed correct?
Well, one cannot help being struct by the _irony_ of the
dilemma (and by the fact that a dilemma is only a dilemma if
Excluded Middle holds!!). Apart from this, it all looks pretty
confusing. Are we to say that there is a set of all truths in
a metaphysical sense, but since we are idealists and intuitionists,
there is no ultimate or objective reality, but only physical
objects constructed from sense data, and mathematical theorems
and axioms built from intuitions? Or, are we to say that there
is an ultimate mathematical and metaphysical reality, but that
there is no set or class or other collection of all true
propositions correctly ascribed of it?
What would these two alternatives mean? Neither seem to be very
appealing. The first is simply a surrendering of the realist
ground and an embracing of Idealism. The second is even more
problematical. If there is an ultimate reality, why is there not
a collection of all propositions that are true of it? Is this
because the very act of making an assertion can be repeated
indefinitely, and hence one could always `step back' from any
completed totality and assert something of the whole, which
would create a truth not included in the original totality?
This is essentially Peter Unger's argument against a Whole
Truth about the world.
Unger's third way is even more unappealing. There is an objective
reality but we cannot know it because our way of thinking, the
logic of our language, prevents us from defining a set of all
truth about the world. But, if we drop Excluded Middle and try
to build a logic that avoids this kind of radical skepticism,
then we run up against Dummett's claims that Excluded Middle
is necessary to have realism at all. Popperian philosphy
of science might survive, since there is no directly appeal
to Excluded Middle in theories of verisimilitude or notions
of truthlikeness.
One possibility does suggest itself. One could say that there is
a set of all truths about the world and Unger is right that our
limited understanding cannot grasp it, but it exists nonetheless.
No realist logic that maintains Excluded Middle can define its
own truth-set since a realist logic that could do this would
grasp the Whole Truth about the world in its totality. But, no
logic can do this as Tarski showed. If one relaxes the implicit
critaria that Tarski's Truth Theorem imposes on the features of
the logic, then one can try to define the truth-set within a
logic (one can do this in other logics in stronger systems).
Now, since such a logic drops Exluced Middle in order to define
Truth, and since dropping Excluded Middle leads to anti-realism,
then one could say that Hintikka's or Grue's systems no longer
talk about the real world (or ultimate, noumenal reality)
but only about the world of experience (the phenomenal world,
the world described by our best scientific theories, and the
world as it appears to our minds) not the objective reality
itself. This is an explanation of the apparently contradictory
status of Excluded Middle in the problems or reality and truth.
But, is it a satisfactory explanation? Unger would say that if
our current categorial framework does not match the real world,
but only somehow approximates it, then we have absolutely no
guarantee that the way we think the world is is even close to
the way it really is (in-itself, as Kant would say). This
position sounds crazy, but it does point out the fact that no
notion of verisimilitude has even really been shown to be more
than a nice picture (and few are still trying to carry out
this application of intensional logic). Nevertheless, one can
argue on pragmatist grounds that the relative or approximate
truth of our world-view can be verified by experience.
One can say that if our world-view was totally false, and reality
were completely different (as Unger claims is possible) then
how could we successfully interact with the world? The fact that
we are able to avoid walking into walls, and that we can build
a rocket and fly to the moon, these feats show that the way we
see the world is close in some sense to the way things really are.
We could be wrong here and there, just as the Ptolemaic astronomy
was shown to be incorrect, but again, the fact that we could
_see_ that it was incorrect defeats the radical sceptic. We
have a grasp on reality, but the picture of the world is always
incomplete, only an approximation of the objective and complete
reality, which may indeed be too complex for a human being to
ever grasp in all its detail.
Torkel Franzen
FI (Marko Amnell) writes:
>What we see is
>that if we maintain Excluded Middle, we are led to realism in
>metaphysics and philosophy of mathematics (what one might call
>`global realism')
This sounds to me like a typical piece of Dummett babbling. What
is "Excluded Middle" supposed to mean here?
Torkel Franzen
(Marko Amnell) writes:
>A bit of elaboration. The reason why a realist might want to
>believe that there is a set of all truths is simple. He believes
>that there is such a thing as the Whole Truth about the world.
Associating this particular piece of babbling with "realism" is
quite arbitrary.
Marko Amnell
(Torkel Franzen) writes:
>In article <1992Nov5.2...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
>
> >I think we already covered all this two months ago. I don't see
> >why realism about scientific theories (which is what I have in
> >mind here) can't include a belief that science approaches some
> >sort of ultimate truth about reality.
>
> Realism about scientific theories "can include" all sorts of things,
>for example the belief that the universe is a banana being eaten by
>the Composite Principle of Aloofness. But why associate these ideas
>with realism?
Peirce's realism also includes this idea, the notion of the "ideal end
of inquiry". If one considers realism to be the thesis that there
is an objective, mind-independent reality, then it seems natural to
me that this metaphysical goal is in some sense the totality of
all true propositions about the world. Allow me to quote Peirce's
definition of reality in "The Fixation of Belief".
Such is the method of science. Its fundamental hypothesis, restated
in more familiar language, is this: There are Real things, whose
characters are entirely independent of our opinions about them;
those Reals affect our senses according to regular laws, and, though
our sensations are as different as are our relations to the objects,
yet, by taking advantage of the laws of perception, we can ascertain
by reasoning how things really and truly are; and any man, if he have
sufficient experience and he reason enough about it, will be led to
the one True conclusion. The new conception involved here is that
of Reality.
Note that Peirce calls the end of inquiry "the one True conclusion".
This is what I have in mind when I speak of the totality of all truths
as a metaphysical entity and bring up Popperian notions of verisimilitude.
Peirce's definition of reality is inseparable from his conception of
scientific method.
The anti-realism of Putnam, eg. was developed in response to the ideas
that Dummett first expressed in the lectures that have now appeared
as _The Logical Basis of Metaphysics_. Putnam notes in the Preface
to _Realism with a Human Face_ that his "inner realism" does not
include the Peircean idea of "the ideal end of inquiry". This is
why he calls it "inner realism" and I call my realism "metaphysical
realism" following the introduction of this term into the realism
debate by Putnam.
I think I already mentioned K"orner. His essay _Metaphysics_ seemed
to me to be a good example of an effort to do metaphysics in as
unprejudiced a way as possible. The book is an essay in meta-metaphysics.
K"orner is aiming to examine the general features of all metaphysical
theories (this is also Dummett's aim). K"orner emphasizes the Kantian
distinction between noumena and phenomena. Peirce also read Kant from
a very early age. He later repudiated much of his Kantianism, but
it seems to me there is still a lot of Kant in his definition of
reality. The "one True conclusion" sounds a lot like the things-in-
themselves to me. Science is directed to this end, but will it ever
reach it?
Also, Unger's radical skepticism seems to me to be very Kantian. Like
K"orner, he speaks of our _concepts_ and how they order reality. His
theses all arise from this Kantian conception of reality. Much of my
own thinking about philosophy is a re-examination of the Kant's
Copernican Revolution. Was it a mistake? If not, how do we reconcile
the change in the intuition of time with Einstein's work? How can we
reconcile Kant's constructivism with our belief in an objective reality?
If we begin with Peirce's definition of reality, then you see why I am
so interested in the problem of the existence of Truth. If there is
no totality of all true propositions (if, say, Tarski's result about
formal logics with the usual configuration of quantifiers -- Hintikka
changes the way quantifiers work, and it is this that enables him to
avoid Tarski) then there is no Truth as the ideal end of inquiry,
and the whole Peircean metaphysics crumbles. We are left with some
kind of radical constructivism (that reality is a purely social
construct) or with some kind of relativism (a variety of relatvism
that avoids most of the old knock-down arguments can be found in
Margolis' book _The Truth about Relativism_)
It is because all these metaphysical theories are living options to
me, and because I more or less follow Peirce's definition of reality
and his linkage of it to Truth, that I am interested in the problem
of whether a truth predicate can be defined internally in a formal
system that does the things we want it to do (like arithmatic).
Note that this problem is not the same problem as the metaphysical
problem of whether there is a totality of all truths or not (what
is usually called simply Truth). But, there is a linkage here,
whose strength depends on how far we follow, eg. Hintikka or
Penrose that human thought can be modelled by a formal system.
Marko Amnell
(Torkel Franzen) writes:
I think we already covered all this two months ago. I don't see
why realism about scientific theories (which is what I have in
mind here) can't include a belief that science approaches some
sort of ultimate truth about reality. But call it Amnellian
babble-realism (ABR) if you like.
Marko Amnell
(Torkel Franzen) writes:
>In article <1992Nov5.2...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
>
> >By the Law of Excluded Middle I mean any philosophical principle that
> >says that sentences of some language (formal system, natural language)
> >can only have the truth values "true" or "false".
>
> What does this mean? Mind, I am as fanatical a realist as ever walked
>the surface of the earth, but this makes it no easier for me to understand
>this talk about truth values. What does it mean for a sentence to have
>a truth value, and what are the truth values "true" and "false"?
Frege's conception of the truth values was of them as pure Platonic
objects, and of propositions as functions with "true" and "false"
as values. I don't know how many people can believe this today.
Our age has been permanently changed by the influence of psychology.
But, how is one to define truth without reference to objective
truth values? One way is to follow Horwich's minimalism in his recent
book _Truth_ and assert that the meaning of truth is exhausted by
saying that a sentence (or statement, proposition, I don't want to
get into the truth-bearer debate here) is true just in case the
state of affairs it describes obtains in the world. This is the
dis-quotational theory of truth. It is purposefully empty of any
explanatory content.
To me, the notion of truth must include some kind of correspondance
with reality. If we define reality as the Truth about the world
(following Peirce) then this idea of correspondance is illuminated
somewhat. A sentence I assert is true if it corresponds to one of
the infinitely many true timeless propositions that make up the
ultimate objective reality. These include mathematical propositions
(maybe a proof of Fermat's Last Theorem, maybe a proof of the
consistency of NF, or maybe not) and propositions that would be
expressed in physical theories. To over-simplify again, a TOE
in physics would be some sub-set of the Truth in a Peircean sense.
I'm using these ideas in a purposefully loose way here, just to give you
a rough idea of how I'm trying to link up the definition of truth with
the metaphysical problem of reality.
So, what are the truth values? I don't know. They are some way that
we have of expressing that a certain state of affairs obtains in the
real world or it does not. But, of every possible state of affairs,
is it the case that it obtains or does not? This is the question of
the status of the Law of Excluded Middle. We have this ability of
ascribing truth or falsity in our language. But we don't understand
what it is. I don't believe the minimalists. I think that the idea
of Truth is linked to the idea of Reality, as Peirce suggests.
Gary Merrill
In article <1992Nov5.2...@klaava.Helsinki.FI>, amn...@klaava.Helsinki.FI (Marko Amnell) writes:
|> In article <TORKEL.92...@echnaton.sics.se> tor...@sics.se
|> (Torkel Franzen) writes:
|>
|>
|> predicate" in their systems by allowing sentences to have more
|> than two truth-values. Grue explicitly adds a third truth-value
|> called "indefinite" and in Hintikka's IF system some sentences
|> are said to have "no truth-value".
This type of approach is pretty old now. I recall a book edited by
(I think) Robert Martin entitled something like "The Liar Paradox"
that was published in about 1971 or so. (I believe that it was
a collection of papers presented at some symposium on the liar.)
A couple of papers in there took a many-valued approach to the
liar paradox. You should, of course, be aware (as you may be)
that the pardox can be reconstructed in many multi-valued
systems. In addition, these approaches are not without their
own oddities. I have the vague recollection (the only kind I
have any more) of analyzing and criticizing approaches of this
sort in my dissertation. If I can remember this weekend, I'll
wipe the mold off it and see if there is anything useful in it.
--
Gary H. Merrill [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC 27513 / (919) 677-8000
sas...@theseus.unx.sas.com ... !mcnc!sas!sasghm
Marko Amnell
>What is "Excluded Middle" supposed to mean here?
Both Hintikka and Grue are able to define what they call a "truth
predicate" in their systems by allowing sentences to have more
than two truth-values. Grue explicitly adds a third truth-value
called "indefinite" and in Hintikka's IF system some sentences
are said to have "no truth-value".
By the Law of Excluded Middle I mean any philosophical principle that
says that sentences of some language (formal system, natural language)
can only have the truth values "true" or "false". I am interested in
the consequences that these logicians see in either asserting or
denying this principle.
There is the further question of under what conditions a sentence
has a truth value in a logic. In natural deduction set theories
like Gilmore's, some sentences fail to have a truth value because
they're not allowed by the rules of deduction.
Torkel Franzen
FI (Marko Amnell) writes:
>If one considers realism to be the thesis that there
>is an objective, mind-independent reality, then it seems natural to
>me that this metaphysical goal is in some sense the totality of
>all true propositions about the world.
Well, to me it seems natural to couple the thesis that there is an objective
mind-independent reality with a rejection of the idea of a "totality of
true propositions about the world" as confused and unjustifiable. Eating a
banana, I have no doubt about its mind-independent reality. The assertion
that there is a "totality of true propositions" about the banana, on the
other hand, is not so much doubtful as senseless.
Torkel Franzen
FI (Marko Amnell) writes:
>By the Law of Excluded Middle I mean any philosophical principle that
>says that sentences of some language (formal system, natural language)
>can only have the truth values "true" or "false".
What does this mean? Mind, I am as fanatical a realist as ever walked
Torkel Franzen
FI (Marko Amnell) writes:
>But, in what way is this phrase a tautology? That is what I'm asking.
It's a tautology in the ordinary sense of being a logical truth in
propositional logic.
>Is it just an empty locution (as you seem to think) or is it a law
>of thought (as Aristotle thought)? I don't really have an opinion on
>the status of such tautologies. I'm curious, and as long as my
>curiosity lasts, I'll stick with philosophy.
I'm not asking for any opinion on the status of tautologies. I am
just asking if you can explain what you yourself are saying. In
ordinary discourse, a logically equivalent formulation of a statement
conveys the same information. In your discourse this is not the case,
since apparently "the conditions obtain or do not obtain" cannot be
replaced by e.g. the logically equivalent "if the conditions obtain then
they obtain". Hence you are making some special use of the tautology.
I am asking you if it is at all possible to explain what you mean by
this special use.
Randall Holmes
"states of affairs"; there are objects in reality and there are
sentences. Now the terms "fact" and "state of affairs" may be used to
refer to these, I suppose; a "fact" might be a true sentence (or class
of true sentences equivalent in some way) (while a "proposition"
would be a sentence or class of sentences without the commitment to
truth). A "state of affairs" might be a chunk of 4-space (if it is a
state of affairs in the physical world).
--
The opinions expressed | --Sincerely,
above are not the "official" | M. Randall Holmes
opinions of any person | Math. Dept., Boise State Univ.
or institution. | hol...@opal.idbsu.edu
Torkel Franzen
FI (Marko Amnell) writes:
>But, how do you avoid this "mystification", Torkel? How do you describe
>what you are really "getting at" without introducing some kind of
>metalogical machinery? Dummett's terminology happens to be at hand, so
>I use it. Show me a better analytic machinery, and I will adopt it
>instead.
What "analytic machinery" are you using now? I don't see any analytic
machinery in statements such as "every possible state of affairs
obtains or does not obtain", but only a tautology used as a rhetorical
device. By this I don't mean that such formulations are to be rejected.
They are often convenient, but they don't really take us anywhere.
To amplify this, suppose you try to explain to me what you mean by
the statement "every possible state of affairs obtains or does not
obtain". Clearly you are using the tautology here in some special
way.
Marko Amnell
(Torkel Franzen) writes:
> I'm not asking for any opinion on the status of tautologies. I am
>just asking if you can explain what you yourself are saying. In
>ordinary discourse, a logically equivalent formulation of a statement
>conveys the same information. In your discourse this is not the case,
>since apparently "the conditions obtain or do not obtain" cannot be
>replaced by e.g. the logically equivalent "if the conditions obtain then
>they obtain". Hence you are making some special use of the tautology.
>I am asking you if it is at all possible to explain what you mean by
>this special use.
OK. What I have in mind is the relativist argument that there is no
such thing as "the way things really" are. Margolis makes this attack
on realism. If there is no objective, mind-independent reality, then
it makes no sense to ask of a certain proposition, is it a fact or is
it not. This is the point where the Law of Excluded Middle comes in.
It says that every possible proposition is true or false. It follows
that in every instance, it _does_ make sense to ask whether a certain
proposition describes some fact in the real world, or does not.
The picture is very simple now. If there is a collection of all true
propositions, then what the Law of Excluded Middle does is say that
every possible proposition either belongs to this collection or it
does not. If it does, it is true, if it does not, it is false.
While if there is no collection of all truths, the relativist can say
that it makes no sense to ask whether a certain proposition is
really true ie. does it belong to the collection of all truth?
All we say that it is "true from us" or "true for them".
So, what I'm setting up here is a parallel between the realism/relativism
issue and the Truth exists/Truth does not exist issue. It would be very
nice if we could get these two issues to stand or fall together, if we
could just build a logic in which we can rigorously show that there is
a collection of all truths, and if this achievement then meant that we
had refuted the skeptic or relativist. You see what I mean? It is an
appealing picture. Maybe it's not a good picture. But there it is.
What I'm trying to do in a sense is to reduce metaphysics to logic.
To reduce the problem of existence to the problem of whether there is
a collection of all truths about the world or not. There are a vast
number of different arguments for the conclusion that one can never
have a collection of all truths in a logic. Tarski's Theorem says that
no nice logic can pick out all its true sentences. If we change the
principles of the construction of logic, the truth-set can be defined.
But, given the correctness of this metaphilosophical picture, and
assuming that Hintikka's IF logic, eg. can actually define its own
truth-set, and assuming that he is right in claiming that IF
corresponds in some way to the logic of our actual every-day language,
then we must ask ourselves: Is IF an acceptable logic? One of the
things that happen in this logic is that the Law of Excluded Middle
Fails. This follows in a natural way from the altered nature of the
quantifiers in IF (the scope rules are changed).
But, he has achieved his results by violating the Law of Excluded Middle.
Someone who holds that this tautology ought to apply to all logics can
reject his results. But how does he justify this rejection? How does
anyone justify the rules of logic, rules of deduction? We set up a
formal system and hope that it does some things for us. It is another
mattter entirely to say that building such a logic shows something
about the real world. But this is what Hintikka suggests.
To suggest this is to call for some kind of non-circular justification
of logical laws. Can this be done? Dummmett tries to do just that.
I can't give his analysis off the top of my head, although I've read
his book twice. What it boils down to in his opinion is what basic
form our meaning-theory is to take, and in particular, what is the
meaning of the logical constants. In classical logic their meaning
differs from that in intuitionism or quantum logic. And this
difference in meaning results in different logical laws.
So, the question of the validity of Hintikka's system would,
according to Dummett, turn on the meaning of the symbols involved,
the logical constants and in particular the quantifiers. There is
an underhanded trick in the way Hintikka introduces branchin
quantifiers in his paper. He merely states that it is a prejudice of
logicians that quantifiers should have the scope rules they have.
But this is absurd. The quantifiers have the scope rules they have
because the resulting logic is well-understood. To change them is
to change the whole ball-game. IF is so different from classical
logic that it is fair to ask is Hintikka talking about truth in
the classical sense at all.
We are faced with the question of the applicability of formal logic
to philosophical problems. Assume certain axioms or principles and
certain results follow. Assume others and things look totally
different. But what does any of this formal manipulation have to
do with reality and truth in the real world any more. One can make
the same kind of criticim about epistemic logic. If the agent knows
the consequences of all his beliefs, what does this formal system
have to say about the actual beliefs of actual people? If the
nature of the `beliefs' is so different, why even call it belief
any more.
But, it is just too easy to say that formal logic has _nothing_
to do with real thinking. I thought this for a long time, while
I studied mathematics and when I started to study philosophy.
Prof. Rota of MIT argues in an article "The Pernicious Influence
of Mathematics on Philosophy" that all of 20th century Anglo-Saxon
philosophy is the result of one big mistake. The mistake of
assuming that formal logic describes how people really think.
But, can it be that simple? Have we learned nothing about thought
by studying formal logic? If not, why the stange similarity
between implication in logic and the way we infer about facts in
the real world?
If mathematical logic is just another branch of mathematics and
has nothing to say about real thinking, a lot of philosophers
have been kidding themselves in a dumb way. If we deny this,
how are we to show that formal logic has something so say?
What I have been describing, my Master's thesis topic, is one
exploration of the extent to which a question in logic, can
the collection of all truths be defined has bearing on a
problems in philosophy. I really have no idea how valid all this
is.
Gary Merrill
In article <TORKEL.92...@bast.sics.se>, tor...@sics.se (Torkel Franzen) writes:
|> In article <1992Nov6.1...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
|> FI (Marko Amnell) writes:
|>
|> >It says that every possible proposition is true or false.
|>
|> "the conditions obtain or do not obtain"? But this is just what I am
|> asking about. What do you intend to assert here? How is this statement
|> different from "if the conditions obtain, then they obtain" or any other
|> statement that is usually regarded as a logical truth? Apparently you
|> attach some peculiar significance to this particular tautology, at least
|> when it is uttered in certain contexts, but your present remarks don't
|> explain what this signficance is.
I think the confusion results from Marko's failure to express what he
really means (or ought to mean?). Marko keeps putting things in ways
that encourage Torkel to see a trivial tautology. And when he responds
to Torkel's questions, he just ends up rephrasing his claims in
fundamentally the same way.
Certainly Marko wants to deny that a certain sentence (or sentential
form) *is* a tautology. That sentence is
Syn: P v ~P
This is the syntactic version of Excluded Middle. The semantic
version is
Sem: For every sentence x, either x is true or x is false.
And Syn expresses Sem (in the object language) via the interpretation
of '~'.
To begin, Marko wants to deny Sem and replace it with something like
Sem': For every sentence x, either x is true (T) or x is false (F)
or x is indeterminate (?).
Given the three truth-values we now have, we need to provide a new
semantics for our sentential operators. One *may* provide such
a semantics that *still* makes Syn come out to be tautological (true
under every truth-value assignment. But there are *other* semantics
that render Syn a non-tautology. For example, (a common one)
P | ~P P | Q | P v Q
__|___ __|___|_______
T | F T | T | T
F | T T | F | T
? | ? T | ? | T
F | T | T
F | F | F
F | ? | ?
? | T | ?
? | F | ?
? | ? | ?
Of course, (P -> P) remains tautologous under the (common) semantics
that the conditional is false only when its antecedent is true and
its consequent false. This certainly removes the equivalence of
"Either the conditions obtain or do not obtain" and "If the conditions
obtain, then they obtain" since if "the conditions obtain" has the
third value (?), the first of these is indeterminate while the second
is true.
As to what the metaphysical significance of all this may be, I will
not even speculate.
Randall Holmes
>Nothing but the Truth" today I was struck by the surprising role
>of _tertium non datur_ in the problem of defining truth inside
>a logic. It seems that if one drops the Law of Excluded Middle
>as a principle in the construction of logic, it becomes _possible_
>to build a logic in which the set of all true sentences can be
>defined. Gabriel Sandu claims to show this rigorously in a new
>paper "IF first-order logic and truth-definition".
How about the sentence "The truth value of this sentence is not
"true""? However many other truth values there may be, the truth
value of this sentence will be "true" precisely if its truth value is
anything other than "true". The thing which this paradox shows cannot
be defined is the function which takes a sentence to its truth value;
of course, the set of true sentences gives us this function if there
are two truth values and does not if there are more. But the paradox
has not actually been evaded.
>
>What I wish to ask you all is your general opinion about the
>_metaphysical_ status of the Law of Excluded Middle. Reading
>Hintikka's article, I was struck by the dissimilarity of the
>role of this principle in the problem of Truth, when compared
>with its role in the metaphysical problem of Reality or
>Existence.
>
>In _The Logical Basis of Metaphysics_ Michael Dummett attempts to
>show that if one accepts _tertium non datur_ then the realist
>approach to physial objects and the existence of mathematical
>objects is supported. (This is an over-simplification, but bear
>with me, I have a point) This suggested to me that if the Law
>of Excluded Middle supported Realism, then it would also be
>on the side of the existence of a set of all truths.
Not necessarily. Not all collections are sets, as Russell's paradox
showed us; similarly, not all properties can be reified.
>
>But Hintikka shows that this is manifestly false. Klaus Grue's
>work in CS also supports the same conclusion. What we see is
>that if we maintain Excluded Middle, we are led to realism in
>metaphysics and philosophy of mathematics (what one might call
>`global realism') but when it comes to examining the problem
>of the definition of truth in logic, and the metaphysical
>problem of whether there is or is not a set of class of any other
>kind of collection of all truths _in the real world_ (and distinguished
>from within a formal system) maintaining Excluded Middle has
>a conclusion that seems antithetical to the aims or beliefs of the
>realist.
But observe that you cannot in any case have the function which sends
a sentence to its truth value, which is just as compelling an object
to the realist, no matter how many truth values there are.
>
>I find this to be a deep result. As Von Wright would put it, the
>status of this one principle is a clue about more profound things
>in philosophy at large. I am quite surprised at the starkness of
>the contrast in the consequences of dropping or maintaining
>Excluded Middle in these two groups of philosophical problems:
>those surrouding truth and the whole truth, and those surrounding
>existence and the nature of the being of various kinds of
>objects -- sets, chairs, persons.
Again, it is NOT profound; what happens when you add more truth values
is that the set of all truths becomes an innocuous object, but there
is still an interesting object which cannot be defined for the same
reasons that truth could not be defined originally. The profound
result remains the non-existence of Truth (or of the full truth-value
assignment scheme).
>
>I'm probably rambling a bit here. But, in any case, what do you
>think about the Law of Excluded Middle? Is it valid in all cases?
>Why? How would one justify dropping it? How does one decide
>in cases like this which philosophical principle to maintain and
>which to drop?
I think that Excluded Middle is true and that the existence of Truth
cannot be supported (as opposed to the existence of truths). The
class of all classes that are not members of themselves is a pretty
appealing object, too.
>
>--
>Marko Amnell
>amn...@klaava.helsinki.fi
>Graduate Student in Philosophy
For Mikhail's benefit (since he has been waiting for me to say
something about truth), I think that truth is a property of sentences
in a language (and so Truth[L] is a set of sentences of the language
L). The result I have in mind when I say that Truth does not exist is
the following: let L be the language of first-order logic with
equality, membership, and whatever other predicates you favor, and let
L be adequate to talk about arithmetic and "enough" set theory and
contain a constant for every object in the universe (names for
everything). Whatever the "set of all truths" should be, it should
enable us to construct Truth[L], the set of all true sentences of L.
But Truth[L] can be neither a set nor even a definable proper class,
by an application of Tarski's paradox. This works in NFU, in ZFC, and
in any other reasonable theory. If you switch to multi-valued logic,
I invite you to consider the function which takes sentences to their
truth values, rather than the set of truths (which can then exist).
Now Truth[L] can exist for languages L which do not have names for all
objects (they will not, in particular, have names for Truth[L], if it
happens to be a set). It can exist for languages L which do not
purport to provide all predicates (Truth[L] would be such a
predicate). Truth can be saved by limiting the scope of reference.
But then one is no longer able to talk about all "truths".
Just as Russell's paradox does not show that non-self-membership does
not occur, this argument does not show that there are no truths. But
there is no abstract object which captures Truth as a whole. I
continue to maintain that statements are either true or false, if they
are well-defined. "Not well-defined" is not a truth value; it calls
upon one to withdraw the question.
Notice that the language L with names for all objects in the universe
_can_ exist as a definable proper class (in ZFC) or as a full-fledged
set (in NFU); this object can be thought of as the collection of all
"propositions". The collection of all propositions appears not to be
paradoxical, but the set of all true propositions, or, more generally,
the function which assigns truth-values to propositions, is
paradoxical.
Truth lurks in the Absolute Infinite.
Randall Holmes
order to search for truths. This is fortunate, since it seems clear
that Truth (as an abstract object) does not exist. Of course, the
iterative hierarchy of the usual set theory does not exist as an
abstract object either, and remains useful; I think that the status of
Truth is similar.
Burt Voorhees
>What I wish to ask you all is your general opinion about the
>_metaphysical_ status of the Law of Excluded Middle. Reading
>Hintikka's article, I was struck by the dissimilarity of the
>role of this principle in the problem of Truth, when compared
>with its role in the metaphysical problem of Reality or
>Existence.
>In _The Logical Basis of Metaphysics_ Michael Dummett attempts to
>show that if one accepts _tertium non datur_ then the realist
>approach to physial objects and the existence of mathematical
>objects is supported. (This is an over-simplification, but bear
>with me, I have a point) This suggested to me that if the Law
>of Excluded Middle supported Realism, then it would also be
>on the side of the existence of a set of all truths.
>But Hintikka shows that this is manifestly false.
You might also enjoy reading Heidegger's The Metaphysical Foundation
of Logic.
The law of excluded middle is one of the three "laws of thought" that
give the basic axioms of Aristotelean logic. Other two, of course,
being the law of identity and the law of contradiction. Taken together
these three laws provide the basis for definition of equivalence
classes of things. In other words, we need them to assign generic
names. Thus, if we accept these laws, and in particular excluded
middle, as having metaphysical validity then we are assuming that
the objects which we identify through application of these laws
are really existant.
The catch with this is that the identity established via these laws
is static. Nothing can change (as Parmenides showed) since change
violates the law of contradiction. And excluded middle, unless we
view change as an instantaneous jump. Thus we cannot take these laws
as describing reality. Rather, they are descriptive of the way that
our mind parcels up sensation and assigns names.
Some of the attempts at a quantum logic do away with excluded middle,
as do the intuitionist (and constructivist) mathematicians. (This is
why they do not accept proof by contradiction.)
A book that might interest you is: Horn, R.E. (ed.) Trialectics: Toward
a Pratical logic of Unity. Can be ordered from Lexington Institute,
80 Marrett Road, Lexington, MA 02173, USA
Randall Holmes
>In article <1992Nov5.2...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
>
> >I think we already covered all this two months ago. I don't see
> >why realism about scientific theories (which is what I have in
> >mind here) can't include a belief that science approaches some
> >sort of ultimate truth about reality.
>
>for example the belief that the universe is a banana being eaten by
>the Composite Principle of Aloofness. But why associate these ideas
>with realism?
Realism is a technical term meaning "maintaining the reality of
universals". Amnell's proposed position is obviously relevant.
Truth is on the face of it a property which we might wish to maintain
corresponds to an abstract object.
Marko Amnell
(Torkel Franzen) writes:
>In article <1992Nov6.0...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
>
> >But, of every possible state of affairs,
> >is it the case that it obtains or does not? This is the question of
> >the status of the Law of Excluded Middle.
>
> This again is a piece of Dummett-style mystification. Let me be blunt:
>there is no "law of excluded middle" in this sense. There is only a very
>natural tendency to use tautologies in expressing metaphysical convictions.
>Saying "every possible state of affairs obtains or does not" is at best
>only a way of suggesting what one is getting at, and it's a poor way of
>doing this.
But, how do you avoid this "mystification", Torkel? How do you describe
what you are really "getting at" without introducing some kind of
metalogical machinery? Dummett's terminology happens to be at hand, so
I use it. Show me a better analytic machinery, and I will adopt it
instead.
--
Randall Holmes
I don't think that "Reality is the Truth about the world". Reality is
the world itself; Truth is what can be said about the world. Of
course, what can be said about the world is part of the world, and
when we start considering what can be said about that aspect of the
world, we get into trouble. Thus Tarski's paradox, etc.
Marko Amnell
>In article <1992Nov6.1...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
>
> >This is the point where the Law of Excluded Middle comes in.
> >It says that every possible proposition is true or false.
>
> That is, that it is true or is not true? As in your earlier formula,
>"the conditions obtain or do not obtain"? But this is just what I am
>asking about. What do you intend to assert here? How is this statement
>different from "if the conditions obtain, then they obtain" or any other
>statement that is usually regarded as a logical truth?
I see your point. You must be saying that a tautology is vacous, and
hence it is no use trying to use it as an explanation, since it is
futile to rephrase it. But, I am wondering where this tautology
comes from, and to ask this is to ask for a psychological explanation
of how we think.
Apparently you
>attach some peculiar significance to this particular tautology, at least
>when it is uttered in certain contexts, but your present remarks don't
>explain what this signficance is.
This is fair. I'm not sure my own thoughts are very clear about this.
Nevertheless, I remain interested in the apparently crucial role this
tautology plays in Hintikka's and Dummett's work. I don't think I
understand why it should crop up in all these places.
> >It follows
> >that in every instance, it _does_ make sense to ask whether a certain
> >proposition describes some fact in the real world, or does not.
>
> How does this follow, and why doesn't it follow from other
>tautologies? Let's take a concrete example. I say, "this painting is
>beautiful, or it is not beautiful". How do you deduce from this
>statement that the matter of the painting being beautiful is in some
>non-trivial sense a matter of fact?
The picture is of the world as a totality and each proposition has a
place in this collection. I can see that if this kind of Tractatus-
like ontology is to hold, any tautology would do the trick. But,
if this is the case, why do we ascribe special significance to the
law of logic? A trick of history? It occurs to me that the way we
actually think -- when I think of how I think myself -- is very
much the way Locke described it. I associate ideas almost randomly,
and when there is order, I don't have an explanation of where this
order comes from. All I can appeal to is a further idea. This is
very muddled.
Or perhaps you don't actually mean
>that anything follows from this statement, but rather that when you
>yourself make such statements you intend to express some metaphysical
>conviction - as people do when they utter a tautology of the form "if
>A then A" in order to express a fatalistic view of events?
The first part is probably true. The conviction is that there is a fact
of the matter about the things we would call facts, that it is legitimate
to ask if something is a fact or a fiction. This is so despite the
profusion of languages and alternative possible descriptions of the
same event. But, does the tautology express this conviction? How?
But, I don't think I have a fatalistic view of events, if you mean by
this some kind of determinism. I think that we are more free than
animals, which are more free than plants, but that to ask if we are
_really_ free is to misunderstand what we mean by freedom. This is a
very Wittgensteinian position about free will.
Marko Amnell
(Burt Voorhees) writes:
>You might also enjoy reading Heidegger's The Metaphysical Foundation
>of Logic.
Yes! I have this book, but have only skimmed it so far. I read the
interesting Translator's Introduction by Michael Heim to the English
Edition from Indiana University Press (1984).
>The law of excluded middle is one of the three "laws of thought" that
>give the basic axioms of Aristotelean logic. Other two, of course,
>being the law of identity and the law of contradiction. Taken together
>these three laws provide the basis for definition of equivalence
>classes of things. In other words, we need them to assign generic
>names. Thus, if we accept these laws, and in particular excluded
>middle, as having metaphysical validity then we are assuming that
>the objects which we identify through application of these laws
>are really existant.
>The catch with this is that the identity established via these laws
>is static. Nothing can change (as Parmenides showed) since change
>violates the law of contradiction. And excluded middle, unless we
>view change as an instantaneous jump. Thus we cannot take these laws
>as describing reality.
But, how do you know that the laws of logic do not describe reality?
Perhaps Plato was right and we are just too focused on the world of
shadoews to see the true world of Forms?
Rather, they are descriptive of the way that
>our mind parcels up sensation and assigns names.
>Some of the attempts at a quantum logic do away with excluded middle,
>as do the intuitionist (and constructivist) mathematicians. (This is
>why they do not accept proof by contradiction.)
I find your locution "how our mind parcels up sensation and assigns
names" interesting, but highly elliptical. Are you saying that we
order the flux of sensation by introducing a framework of concepts
and intuitions? If so, where does the structure of these psychological
constructs come from? If everything is really flux, where does the
order come from?
>A book that might interest you is: Horn, R.E. (ed.) Trialectics: Toward
>a Pratical logic of Unity. Can be ordered from Lexington Institute,
>80 Marrett Road, Lexington, MA 02173, USA
Thanks.
Marko Amnell
(Gary Merrill) writes:
>In article <1992Nov5.2...@klaava.Helsinki.FI>, amn...@klaava.Helsinki.FI (Marko Amnell) writes:
>|> In article <TORKEL.92...@echnaton.sics.se> tor...@sics.se
>|> (Torkel Franzen) writes:
>|>
>|> >What is "Excluded Middle" supposed to mean here?
>|>
>|> Both Hintikka and Grue are able to define what they call a "truth
>|> predicate" in their systems by allowing sentences to have more
>|> than two truth-values. Grue explicitly adds a third truth-value
>|> called "indefinite" and in Hintikka's IF system some sentences
>|> are said to have "no truth-value".
>
>This type of approach is pretty old now. I recall a book edited by
>(I think) Robert Martin entitled something like "The Liar Paradox"
>that was published in about 1971 or so. (I believe that it was
>a collection of papers presented at some symposium on the liar.)
>A couple of papers in there took a many-valued approach to the
>liar paradox. You should, of course, be aware (as you may be)
>that the pardox can be reconstructed in many multi-valued
>systems.
Yes, this is true, and Hintikka explicitly deals with the "Strengthened
Liar" and tries to show that his IF system avoids it too. Frankly, I
have my doubts about whether he pulls it off or not. In Grim's new
book it is shown that a Liar-type Paradox can be built in any logic
having any features, including an infinite number of truth values.
But, right now I see arguments and counter-arguments on both sides.
I have no idea who is right and who is confused.
Marko Amnell
OK, this is your view. I understand it. Allow me to add another quotation,
this time from the Hintikka paper I referred to.
One partial way of seeing the relevance of [a definition of truth in a
logic that avoids Tarski's Truth Theorem] is to recall the extremely
strong, albeit partly tacit tradition or trend in philosophy (not only
in the philosophy of language) that maintains that semantics is ineffable,
at least the semantics of our actual working language. This tradition
is represented by no lesser figures than Frege, Wittgenstein, the
Vienna Circle of 1930-32, Quine and Church, and similar views frequently
rear their ugly heads elsewhere in contemporary philosophical discussions.
And if semantics at large is inexpressible, then the key concept of all
semantics, the concept of truth, will _a fortiori_ be ineffable,
according to this tradition. Furthermore, we are not dealing here with
a sect of analytic philosophers, either. Martin Kusch has shown that
a similar view is one of the key ingredients of Heidegger's philosophy.
This entire multiple tradition comes to a screeching halt if truth turns
out to be definable after all in a nontrivial way.
Now, I'm not saying I underline everything Hintikka is saying here, nor
does Jaakko think that what I focus on in metaphysics is "central".
But, I cite this passage as further evidence of the importance of truth
definitions for philosophy at large.
Marko Amnell
> To amplify this, suppose you try to explain to me what you mean by
>the statement "every possible state of affairs obtains or does not
>obtain". Clearly you are using the tautology here in some special
>way.
But, in what way is this phrase a tautology? That is what I'm asking.
Is it just an empty locution (as you seem to think) or is it a law
of thought (as Aristotle thought)? I don't really have an opinion on
the status of such tautologies. I'm curious, and as long as my
curiosity lasts, I'll stick with philosophy. I really can't
answer your question beyond this.
Torkel Franzen
FI (Marko Amnell) writes:
>I think we already covered all this two months ago. I don't see
>why realism about scientific theories (which is what I have in
>mind here) can't include a belief that science approaches some
>sort of ultimate truth about reality.
Realism about scientific theories "can include" all sorts of things,
Torkel Franzen
FI (Marko Amnell) writes:
>But, of every possible state of affairs,
>is it the case that it obtains or does not? This is the question of
>the status of the Law of Excluded Middle.
This again is a piece of Dummett-style mystification. Let me be blunt:
Gary Merrill
|> own oddities. I have the vague recollection (the only kind I
|> have any more) of analyzing and criticizing approaches of this
|> sort in my dissertation. If I can remember this weekend, I'll
|> wipe the mold off it and see if there is anything useful in it.
Let me begin with a kind of "vote of confidence" for Torkel Franzen.
I confess that my feelings about the discussion and issues raised
here pretty much mirror his, and I think that his remarks and
criticisms (aside from the name-calling, which is somewhat pardonable)
should be taken seriously.
I did unearth my dissertation and scrape the mold off it (this is not
an exaggeration). If you are seriously into alternative (non-Tarski)
approaches to theories of truth (or truth predicates) you might
benefit from looking at a copy (if you could get one). The title
is _A Semantically Closed Theory of Truth_. In skimming it I was
reminded that the multi-valued logic approach to all this is even
older than I recalled: Chrysippus (I forget the dates). My own
approach bears certain similarities to that of Jean Buridan (14th
century; see _Sophisms on Meaning and Truth_).
An analysis of several more modern attempts (most based upon denial
of excluded middle; van Fraassen, Skyrms, Robert Martin) is performed
and weaknesses in these approaches are described. If you are inclined
toward such an approach, you need to beware of certain unsavory
features it may exhibit. For example, you will discover that the
deduction theorem frequently fails in systems of the sort under discussion.
And it often turns out that you cannot meaningfully say things that
you seemingly should be able to. For example, if you use a three-
valued approach in which each sentence is either true or false or
neither (sometimes "meaningless" or "neuter" is used here), you
may likely discover that the statement
Every sentence is either true or false or neuter.
is neither true nor false. You may discover that your truth
prediate is non-extensional. Other problems can arise as well.
My point is that there will be *some* price to pay for solving
the paradoxes, and that price may be regarded as too high.
While I'm at it, I should describe the approach I took. This
involved constructing a formal theory T* which embodies S5 as its
sentential logic and is otherwise a "free" logic (in that
denotationless terms are allowed). The language contains a
quotation operator and a concatenation operator (suitably
axiomatized). The quantifiers receive a substitution inter-
pretation. Given this, the expressive power of T* is somewhat
greater than that of the standard first-order predicate
calculus. A truth predicate for T* can be defined within T*
itself, and theorems are proved to show that this predicate
saatisfies criteria of adequacy for such predicates. The
theory is a classically two-valued semantically closed theory
capable of defining its own notion of truth.
A full axiomatization and semantics for T* is provided and
Tarsk's Convention T is valid relative to that semantics.
Several familiar semantic paradoxes (including the Liar and
the Grelling-Nelson pardoxes) are formulated in the language
of T* and it is shown that no contradictions are
generated within the theory. Soundness and completeness
proofs for T* are provided.
Late in the game Pollock discovered an error in the proof of
a fundamental lemma in the soundness proof. The problem was
quite complex and involved the key definition of "representing
function" in the semantics. It was clear to me how to avoid
the problem, but the truth predicate in the resulting theory
became non-extensional.
None of this was ever published. I felt that I had failed in
achieving some of my original goals -- which included having an
extensional truth predicate and having all the semantic
paradoxes fail in a uniform and principled way. As it turned
out (in order to repair the problem discovered by Pollock)
the truth predicate was not extensional. And while the pardoxes
could be formulated and seen to fail (i.e., they weren't
pardoxes -- didn't result in contradictions), different
paradoxes tended to fail for different reasons. While the
resulting theory certainly wasn't any goofier than what others
had done, it did not seem to me to offer any genuine advantages
either. You win some; you lose some.
Randall Holmes
But note that the multi-valued logics do not escape the problem. The
assignment of truth values remains ineffable; it is only because you
have more than two of them that you become able to get a set of all
truths -- but the set of all truths ceases to be the supremely
interesting object!
Torkel Franzen
FI (Marko Amnell) writes:
>This is the point where the Law of Excluded Middle comes in.
>It says that every possible proposition is true or false.
That is, that it is true or is not true? As in your earlier formula,
"the conditions obtain or do not obtain"? But this is just what I am
asking about. What do you intend to assert here? How is this statement
different from "if the conditions obtain, then they obtain" or any other
statement that is usually regarded as a logical truth? Apparently you
attach some peculiar significance to this particular tautology, at least
when it is uttered in certain contexts, but your present remarks don't
explain what this signficance is.
>It follows
>that in every instance, it _does_ make sense to ask whether a certain
>proposition describes some fact in the real world, or does not.
How does this follow, and why doesn't it follow from other
tautologies? Let's take a concrete example. I say, "this painting is
beautiful, or it is not beautiful". How do you deduce from this
statement that the matter of the painting being beautiful is in some
non-trivial sense a matter of fact? Or perhaps you don't actually mean
Marko Amnell
I was already familiar with Randall Holmes' basic position on Truth and
the "set of all propositions" from our correspondance, but I appreciate
seeing it layed out in some detail. Thanks. Gary Merrill's clarification
of the syntactic/semantic distinction is also greatly appreciated.
Torkel Franzen is probably right that my notion of "realism" doesn't
correspond to realism about universals. Although if one includes
mathematical realism, realism about universals would be included in
`global realism'. A Fregean notion of truth values also seems to
involve such realism about Truth as a Platonic universal. I can't think
of too much to add to any this at the moment. This is primarily the
result of my own confusion about the possible philosophical relevance
of all this, and the fact I haven't read Hintikka very thoroughly yet.
Torkel Franzen
FI (Marko Amnell) writes:
>But, I don't think I have a fatalistic view of events, if you mean by
>this some kind of determinism.
I was merely giving another example of a (natural) use of
tautologies to express metaphysical views. In this case, though, few
people take the view that the "law of the repeated antecedent" - i.e.
the tautology "if A then A" - stands or falls with philosophical
fatalism.
Michael Zeleny
hol...@garnet.idbsu.edu (Randall Holmes) writes:
>I don't think that "Reality is the Truth about the world". Reality is
>the world itself; Truth is what can be said about the world. Of
>course, what can be said about the world is part of the world, and
>when we start considering what can be said about that aspect of the
>world, we get into trouble. Thus Tarski's paradox, etc.
Let me see if I am getting this straight. You, Randall, are
presumably "part of the world" itself, and _ipso facto_ part of
Reality. Consequently, what I can say about you is Truth.
So we have that
Randall Holmes is a wood duck
is part of Truth, insofar as I can, and indeed do say it right now.
Did I get your proposal right, or is there anything more profound to it?
>--
>The opinions expressed | --Sincerely,
>above are not the "official" | M. Randall Holmes
>opinions of any person | Math. Dept., Boise State Univ.
>or institution. | hol...@opal.idbsu.edu
cordially,
mikhail zel...@husc.harvard.edu
" -- I shall speak bluntly, because life is short."
Michael Zeleny
>I am a realist myself, but I do not think that Truth is required in
>order to search for truths. This is fortunate, since it seems clear
>that Truth (as an abstract object) does not exist. Of course, the
>iterative hierarchy of the usual set theory does not exist as an
>abstract object either, and remains useful; I think that the status of
>Truth is similar.
Randall, it never ceases to amaze me just how intent you seem to be on
brashly recapitulating the egregious philosophical errors of yore.
What sort of bullshit neoKantianism have you been reading lately? was
it, by any chance, Vaihinger? Now, repeat after me: "Nothing can be
said truly about that, which does not exist." (Church) Your homework
assignment is to translate this thesis into a rigorous positive form.
Remember your pal Parmenides?
>--
>The opinions expressed | --Sincerely,
>above are not the "official" | M. Randall Holmes
>opinions of any person | Math. Dept., Boise State Univ.
>or institution. | hol...@opal.idbsu.edu
cordially,
Torkel Franzen
>Realism is a technical term meaning "maintaining the reality of
>universals".
This is not the sense of "realism" Amnell intended.
Marko Amnell
>In article <1992Nov5.2...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
> >If one considers realism to be the thesis that there
> >is an objective, mind-independent reality, then it seems natural to
> >me that this metaphysical goal is in some sense the totality of
> >all true propositions about the world.
The following paragraph contains some pretty strong statements. Let's
break it up.
> Well, to me it seems natural to couple the thesis that there is an objective
>mind-independent reality with a rejection of the idea of a "totality of
>true propositions about the world" as confused and unjustifiable.
To say that something is _confused_ is to assert something like that there
is no clear and distinct idea associated with the locution. Why is this
phrase confused? Can you not even _imagine_ what it would be like for
there to be a collection of all true propositions about the world?
Suppose eg. that there is an omniscient being, would it not know all
true propositions? Apparently so, and this means that you are saying
that the very notion of omniscience is confused. I disagree. I think
I can conceive what an omniscient being would be like. Just imagine
a succesion of larger and larger computers, with the limit as the
series approaches an infinitely large memory. What is confused about
this idea?
Then you say that the idea of a "set of all true propositions about
the world" is also _unjustifiable_. I take this to mean that you would
assert that there is no conceivable way that it could be verified
whether a certain collection of true propositions T was the total
Truth or not. How do you know that such a thing could not be checked?
Suppose there is a candidate collection T of propositions. To
justify that it is the whole truth we would need an infinite number
of operations. But, unless you are a mathematical finitist, there
is nothing unacceptable about the _idea_ of a justification procedure
that consists of an infinite number of steps.
Eating a
>banana, I have no doubt about its mind-independent reality. The assertion
>that there is a "totality of true propositions" about the banana, on the
>other hand, is not so much doubtful as senseless.
Now you add that the idea of a "totality of true propostions" even about
one object, eg. a banana, is _senseless_. To say it is a doubtful idea
would be to say that one could imagine such a totality existing for eg.
God but that it is questionable that such a totality could actually exist
in the real world. This would be akin to the old criticism of sense data
theorists that it is doubtful that one could ever translate the notion
of a physical object into a infinite number of observation statements.
But you don't say this. You say that the very idea of a totality of all
propositions about any single object is _senseless_. This is to say you
deny any meaning to it whatsoever. Do you really mean this? Do you
mean to say that the words "the collection of all propositions correctly
asserted of the banana" carries no meaning for you at all? If this is
indeed your position, I don't understand it. It seems absurd to deny
meaning to this perfectly ordinary phrase of English. Maybe you can
show me that such collections of propositions are not possible, but it
is absurd to deny that this phrase lacks any sense whatsoever.
Torkel Franzen
FI (Marko Amnell) writes:
>Suppose eg. that there is an omniscient being, would it not know all
>true propositions?
You tell me. But first you need to explain "all true propositions".
The idea that there is a totality of true propositions is by no means
self-explanatory. What, in your scheme of things, is the relation
between propositions and sentence in languages? How is the totality of
propositions delimited? Is "the totality of true propositions"
assumed to be a notion better defined than "the totality of true sentences
in all possible languages"?
>I take this to mean that you would
>assert that there is no conceivable way that it could be verified
>whether a certain collection of true propositions T was the total
>Truth or not.
Well, no, nothing of the sort. I simply doubt that you or anybody else can
give any content to this talk of "the totality of true propositions" that
justifies your apparently confident use of the term in posing philosophical
questions.
>But you don't say this. You say that the very idea of a totality of all
>propositions about any single object is _senseless_.
Nope. I say that it's not so much doubtful as senseless. In other words,
what is primarily lacking is any serious attempt on the part of those who
would speak of "the totality of true propositions about this banana" to
explain what they believe themselves to be referring to.
Randall Holmes
>In article <1992Nov6.1...@guinness.idbsu.edu>
>hol...@garnet.idbsu.edu (Randall Holmes) writes:
>
>>I don't think that "Reality is the Truth about the world". Reality is
>>the world itself; Truth is what can be said about the world. Of
>>course, what can be said about the world is part of the world, and
>>when we start considering what can be said about that aspect of the
>>world, we get into trouble. Thus Tarski's paradox, etc.
>
>Let me see if I am getting this straight. You, Randall, are
>presumably "part of the world" itself, and _ipso facto_ part of
>Reality. Consequently, what I can say about you is Truth.
>
>So we have that
>
> Randall Holmes is a wood duck
>
>is part of Truth, insofar as I can, and indeed do say it right now.
It would be an element of the set of true propositions in your
language, i.e., an element (not a part) of what I am terming
Truth[MZL] (where MZL is your language); but all of this depends on my
being a wood duck, which does not hold.
AHHH! I see that I did goof here! Of course, Truth is not merely
what can be said about the world, but what can be said _correctly_
about the world. Thank you!
Michael Zeleny
hol...@opal.idbsu.edu (Randall Holmes) writes:
>In article <1992Nov7.0...@husc3.harvard.edu>
>zel...@husc10.harvard.edu (Michael Zeleny) writes:
>>In article <1992Nov6.1...@guinness.idbsu.edu>
>>hol...@garnet.idbsu.edu (Randall Holmes) writes:
RH:
>>>I don't think that "Reality is the Truth about the world". Reality is
>>>the world itself; Truth is what can be said about the world. Of
>>>course, what can be said about the world is part of the world, and
>>>when we start considering what can be said about that aspect of the
>>>world, we get into trouble. Thus Tarski's paradox, etc.
MZ:
>>Let me see if I am getting this straight. You, Randall, are
>>presumably "part of the world" itself, and _ipso facto_ part of
>>Reality. Consequently, what I can say about you is Truth.
>>
>>So we have that
>>
>> Randall Holmes is a wood duck
>>
>>is part of Truth, insofar as I can, and indeed do say it right now.
RH:
>It would be an element of the set of true propositions in your
>language,
Am I hearing the term `proposition' from a self-professed extensionalist?
RH:
> i.e., an element (not a part) of what I am terming
>Truth[MZL] (where MZL is your language); but all of this depends on my
>being a wood duck, which does not hold.
What do you mean by "does not hold"?
RH:
>AHHH! I see that I did goof here! Of course, Truth is not merely
>what can be said about the world, but what can be said _correctly_
>about the world. Thank you!
Now all that remains is for you to explain the meaning of the adverb...
MZ:
>>Did I get your proposal right, or is there anything more profound to it?
Same question once again.
Marko Amnell
(Randall Holmes) writes:
>I am a realist myself, but I do not think that Truth is required in
>order to search for truths. This is fortunate, since it seems clear
>that Truth (as an abstract object) does not exist.
This is the point where the relativist jumps in and says that since
there is no ultimate standard of truth, but only individual truths
in such and such circumstances, there is no objetive reality. He
would seek to undermine your realism by questioning the validity
of your conviction that there is an objective standard of fact.
If there is no ultimate Truth, what can you appeal to besides
contingent cirumstance?
If it turns out that Truth cannot be defined in any logic, it could
still play the role of what Kant called a regulative idea of the
understanding. Our reason, said Kant, eschews the idea of God or
an immortal soul, but we can use these idea as `limits' as it were
to reason. In a similar way, the notion of Truth as the "ideal end
of inquiry" regulates our search for individual truths. It is almost
a question of faith. We have faith that science is taking us closer
to the Truth, although our reason eschews Truth as a completed totality
because it is paradoxical (supposing that it turns out this way).
Marko Amnell
Randall Holmes has made four points which seem of particular interest:
- If one sets up a logic with more than two truth values, and hence
manages to avoid Tarski's paradox, then a further paradox can be set
up by considering the function that sends sentences to their truth
values. In particular, a paradox is formed by the sentence "the truth
value of this sentence is not `true'". So, this sentence is true iff
it is anything but `true'. Is it just like Tarski's, then? Can you
elaborate a bit about this paradox? I will try to see if Hintikka's
IF system avoids it or not.
- Truth vs. reality. Certainly one can say that the truth about the
world--if it exists--is not the world itself. But, the interesting
theoretical object is the collection of truths, not the Lebenswelt of
experience. How would one characterize the world? In terms of the
contingent chracteristics of human experience? If so, would other
beings, with different sense organs, not inhabit different worlds?
This is why Peirce's definition of reality focuses on the Reals, not
on how they appear to us--or to other beings--in sensation.
- Truth and the iterative hierarchy of sets. The iterative hierarchy
is endless in the sense that in eg. ZF one builds the universe of set
in a well-founded way and never reaches V. The universe of eg. NF, does
exist, but is not charaterized by a nice picture like in ZF. How is
the iterative hierarchy like Truth? We don't have a picture of truths
being built in a systematic way. What we do have is an idea, Truth,
that corresponds to the "ideal end of inquiry". But there is no full
characterization of this totality. Hence to me Truth looks more like
the universe of NF than the iterative hierarchy (if we wish to follow
this conceit). I dunno how useful it is.
- Finally, objects, sentences, facts and states of affairs. Quine's
ontology is not the only possible ontology. It is legitimate to
have facts in one's ontology (eg. Reinhardt Grossmann's ontology).
The identification of states of affairs with chunks or hunks of
4-space is, again, not the only possibility, although it is natural.
States of affairs could be characterized by way of our senses. A
state of affairs might be eg. a collection of sense data by one
subject at a particular moment in his subjective sense of time. Note
that it is possible to build a very sophisticated notion of time
from this subjective basis. Saul Basri builds full General Relativity
in such a way in _A Deductive Theory of Space and Time_. He defines
the "class of all living human beings who have adequately functioning
sense organs, can communicate with each other, and do so honestly and
without bias". He defines the "objective universe" based on this
collection of observers. He then builds GR in a rigorous way.
Marko Amnell
>But first you need to explain "all true propositions".
>The idea that there is a totality of true propositions is by no means
>self-explanatory. What, in your scheme of things, is the relation
>between propositions and sentence in languages? How is the totality of
>propositions delimited? Is "the totality of true propositions"
>assumed to be a notion better defined than "the totality of true sentences
>in all possible languages"?
I think that when one makes a true statement (this is a psychological act)
what one does it to assert that a certain sentence in a language in which
one is fluent is true. Now, depending on the circumstances of the act
of assertion, the sentence assumes a certain meaning for speakers of the
language. This meaning is by no means always (or even never) totally
clear. Ambiguities remain (about indexicals eg.).
Also, one must remember that I am here talking about descriptive use of
language, as distinguished from other acts of speech with other types
of "illocutionary force" or however one wants to classify speech acts.
I am also limiting myself to spoken language. This is arbitrary, and
the relation between spoken assertions and assertions in written texts
is not trivial.
Now, if the statement is judged to be correctly asserted in totality of
the state of affairs which it describes (this judgement is made by those
who are fluent in the language, and is really an idealization of the whole
linguistic community in the sense that if one refers to "gold" then all
the expert knowledge about gold could affect the judgement) then we would
say that it was a true statement. Note the enormous tacit complexity
of such a simple descriptive speech act (and even now there are over-
simplifications).
If we wish to introduce the notion of propositions as distinct from both
sentences and statements, then what it added is this: The true statement
about the contingent state of affairs in its human context picks out one
of an infinity of timeless abstract objects called propositions. These
objects are timeless in the sense that they are not indexed by time.
All true statements will correspond to some proposition. In the case of
mathematical propositions, many instances of "2+2=4" uttered by various
people at various times in various circumstances, will correspond to
the one timeless arithmatical truth that 2+2 indeed equals 4.
Other propositions will be picked out in a similar way. Things are
simple if we assume a deterministic ontology--a reified universe of
4-space. The universe exists as one abstract object, and we are
certain chunks of this objects--what we experience as time is our
extension on the t-axis. If we allow for free will things get more
complicated. We have to introduce some kind of possible-worlds
structure and say that a person can pick out where his timelines
will stretch in this profusion of possible universes.
In any case, we have here a way of matching up true assertions by
people with abstract objects called propositions. The nature of
propositions is not clear. One can think of them as objects whose
nature is wholly exhausted by considerations of structure. A part
of this structure is truth value. Each proposition is either
true or false. What these truth values are is also not clear.
One can think of them as abstract objects that are the values of
propositions--considered as functions (as in the _Tractatus_).
This whole scheme up to now is a model. We set up a metaphysical
model of the world and hope that it has some explanatory value.
If it is a good model we gain some understanding of our place in
the scheme of things. To what extent one _believes_ that the
real world is like this metaphysical model is also unclear. There
are other possible models. What I have been describing is a
simplification of the various "metaphysical realist" positions of
analytic philosophers like Plantinga or Lewis. There are other
schemes in the analytic tradition and further schemes from
other contemporary schools and history (think of Heidegger, or
Leibniz's monadology, or Spinoza's Ethics). All of these
metaphysical theories are to me different models of the world.
And one of the primary tasks of philosophy is to survey these
possibilities and compare their merits. By understanding as many
different models as possible, we can begin we formulate desiderata
for what we would like from a metaphysical theory.
The metaphysical realist model satisfies some of the desiderata that
seem appealing. It posits an objective reality (as distinguished
from relativist or constructivist pictures). It has the idea of
an ultimate standard of truth (as distinguished from pragmatist or
"inner realist" pictures). Why we see these two desiderata to be
appealing is a value judgement. We value objectivity and certainty.
So here I agree with Putnam that the ultimate justification of one's
metaphysics is ethical. This is in agreement with my conviction of
the supreme importance of ethics in philosophy. Here I mean my own
`philosophy of life'. I have certain ethical values, and these
values lead me to desire a certain kind of explanation of experience.
It is pointless to ask where these values come from. Explanations
must end somewhere.
Now, in this metaphysical realist theory, the key object is the
totality of all propositions. There are also the truth values and
the physical universe as an abstract object with a certain mathematical
structure--it is a 4-space with some TOE imposing a structure on
it (also fundamental constants, initial conditions, and so on).
This is pure Platonism--and so all other objects, physical objects,
people, and so on are merely abstract objects fleshed out by the
way we experience ourselves and other 4-space slices. This is only
a picture--if it aids us in understanding it is good. This is also
a caricature of the metaphysical realist picture. A fuller picture
along these lines can be found in John Post's _The Faces of Existence:
An Essay in Nonreductive Metaphysics_. The main simplification here is
the replacement of a suitable supervenience relation by a relation
of reduction. A better picture would not say that there are only
abstract objects (including 4-space slices). It would say that all
other entities--people, physical objects, emotions, metaphors, gods,
and so on--supervene on the objective level of existence here
described as the totality of propositions, truth values and 4-space
with a certain mathematical structure.
A collection of all true propositions is a sub-collection of the
totality of all propositions. Tarski's paradox appears to say that
in a classical logic with two truth values, the elements of this
sub-collection cannot be picked out. There are also arguments
that the collection of all propositions cannot exist and would
be contradictory. These are not orthodox arguments (Grim presents
several in _The Incomplete Universe).
If one has this kind of metaphysical model as an ideal, then one
would like to find some way of chaning logic so that one could
pick out the sub-collection of all true propositions. I have this
as an ideal, but to be frank, I only _play_ with it as a model,
I have virtually no faith that the real world is like this. If
I am asked to describe my own world view, I can say this: For a
long time I was a nihilist and even today my hold on my own ethics
is tenous at best. I like to quote Valery to myself "God created
the world from the void, but the void shows through". I am
interested in finding a nice picture of the world, but emotionally
I am still most at home with the void. Even my moral values
supervene on this `existential' emptiness. All the meaning we find
in life is of our own making. There is only nothingness beyond
the social. But nothing can be asserted of the void--objectivity,
the world, these are all ideas we have brought.
Marko Amnell
I wrote:
>If one sets up a logic with more than two truth values, and hence
>manages to avoid Tarski's paradox, then a further paradox can be set
>up by considering the function that sends sentences to their truth
>values. In particular, a paradox is formed by the sentence "the truth
>value of this sentence is not `true'". So, this sentence is true iff
>it is anything but `true'. Is it just like Tarski's, then? Can you
>elaborate a bit about this paradox? I will try to see if Hintikka's
>IF system avoids it or not.
OK. Hintikka's logic avoids the Strengthened Liar Paradox, and hence
also Randall's modified Tarski Paradox, in the thoroughly uninteresting
way that these sentences cannot be formalized in the system IF. This
is because the sentence "This sentence is not true" contains contradictory
negation. In IF contradictory negation can only occur at the beginning
of a sentence, not inside it. Here's what Hintikka says:
Why cannot [a Liar-type] sentence be forthcoming with a weak negation
instead of a strong one in [IF extended to include weak negation]?
The answer lies in the fact that in the extended IF languages
contradictory negation can occur only sentence-initially. Therefore,
it cannot be used in constructing a sentence like [ eg. "The truth
value of this sentence is not `true'"]. The best such a sentence can
therefore do is to say ... ["This sentence is false"] and not [the
version with weak negation]. Furthermore, no matter how we manipulate
sentence-initial contradictory negations, we cannot produce a paradox.
So, it is the expressive limitations of IF which allows Hintikka to avoid
these kinds of paradoxes. (Incidentally, the deduction theorem does fail
in IF, as Merrill suggested).
Torkel Franzen
FI (Marko Amnell) writes:
>If we wish to introduce the notion of propositions as distinct from both
>sentences and statements, then what it added is this: The true statement
>about the contingent state of affairs in its human context picks out one
>of an infinity of timeless abstract objects called propositions.
That's all well and good, but it doesn't answer my question whether
"the totality of true propositions" is a notion better defined than
"the totality of true sentences in all possible languages". If it is
not, questions and speculations about the supposed totality are so
much hot air. If it is, this remains to be shown.
In metaphysics we can postulate all sorts of things, but we cannot
postulate that we are making good sense. You may postulate that there
is, corresponding to the sentence "this is a banana" uttered in a
certain context, a timeless abstract object, the proposition that this
is a banana. And so on. This postulate, however, is useless when we
try to make sense of the locution "the totality of true propositions",
in the absence of any explanation of which sentences, in which
languages, real or hypothetical, have corresponding propositions, and
how those sentences themselves are to be understood.
Wittgenstein, in the Tractatus, did not fall into this kind of loose
babbling about a "totality of true propositions" since he explicitly
postulated a particular formal ontology and a corresponding purely
truth-functional language. Of course, he paid a price for this: the
relation between that ontology and language and our ordinary ontology
and language was obscure in the extreme.
Gary Merrill
|> OK. Hintikka's logic avoids the Strengthened Liar Paradox, and hence
|> also Randall's modified Tarski Paradox, in the thoroughly uninteresting
|> way that these sentences cannot be formalized in the system IF. This
|> is because the sentence "This sentence is not true" contains contradictory
|> negation. In IF contradictory negation can only occur at the beginning
|> of a sentence, not inside it. Here's what Hintikka says:
[ quote omitted ]
|> So, it is the expressive limitations of IF which allows Hintikka to avoid
|> these kinds of paradoxes. (Incidentally, the deduction theorem does fail
|> in IF, as Merrill suggested).
This is symptomatic of all of these multi-valued approaches. You end
up with a formal system in which (at least some of) the pardoxes can
be escaped. But the manner of the escape is unsatisfying ("Oh, golly,
I can't seem to formulate the paradox!"). The result is that you
end up playing a kind of game with formal systems that has no real
value (aside from expanding your publications list) because the
systems you construct have ad hoc features designed purely to escape
a set of paradoxes and the resulting logic is grossly non-standard
and counterintuitive. When you contrast these approaches to that
of Tarski, it's real hard to make a case that you have accomplished
anything.
Randall Holmes
>Some further notes (I cancelled an earlier version of this):
>
>Randall Holmes has made four points which seem of particular interest:
>
>- If one sets up a logic with more than two truth values, and hence
>manages to avoid Tarski's paradox, then a further paradox can be set
>up by considering the function that sends sentences to their truth
>values. In particular, a paradox is formed by the sentence "the truth
>value of this sentence is not `true'". So, this sentence is true iff
>it is anything but `true'. Is it just like Tarski's, then? Can you
>elaborate a bit about this paradox? I will try to see if Hintikka's
>IF system avoids it or not.
I can't elaborate yet; I thought it up as I was writing. I think that
many systems of multi-valued logic would avoid this paradox -- but
this would reflect an inability to introspect on their system of truth
values just as unsatisfactory as the problem with two-valued logic.
>
>- Truth vs. reality. Certainly one can say that the truth about the
>world--if it exists--is not the world itself. But, the interesting
>theoretical object is the collection of truths, not the Lebenswelt of
>experience. How would one characterize the world? In terms of the
>contingent chracteristics of human experience? If so, would other
>beings, with different sense organs, not inhabit different worlds?
>This is why Peirce's definition of reality focuses on the Reals, not
>on how they appear to us--or to other beings--in sensation.
Here I should note an omission pointed out by M. Zeleny -- Truth
refers to what can be said about the world -- _correctly_. The
characterization of the world is rather thin and flavorless, I am
afraid -- it is simply whatever exists. It certainly has nothing to
do with _experience_ (specifically) -- our experience of the world is
more in the category of Truth than the category of Reality.
>
>- Truth and the iterative hierarchy of sets. The iterative hierarchy
>is endless in the sense that in eg. ZF one builds the universe of set
>in a well-founded way and never reaches V. The universe of eg. NF, does
>exist, but is not charaterized by a nice picture like in ZF. How is
>the iterative hierarchy like Truth? We don't have a picture of truths
>being built in a systematic way. What we do have is an idea, Truth,
>that corresponds to the "ideal end of inquiry". But there is no full
>characterization of this totality. Hence to me Truth looks more like
>the universe of NF than the iterative hierarchy (if we wish to follow
>this conceit). I dunno how useful it is.
The iterative hierarchy is the analogue to the "ideal end of inquiry",
not the universe of NF.
>
>- Finally, objects, sentences, facts and states of affairs. Quine's
>ontology is not the only possible ontology. It is legitimate to
>have facts in one's ontology (eg. Reinhardt Grossmann's ontology).
Actually, I do admit "facts", but I regard them as sentences or
equivalence classes of sentences, so I doubt that you would regard
them as facts. To me, truth is obviously an attribute of sentences.
>The identification of states of affairs with chunks or hunks of
>4-space is, again, not the only possibility, although it is natural.
I regard it as an expedient; I don't place any philosophical weight on
it.
>States of affairs could be characterized by way of our senses. A
>state of affairs might be eg. a collection of sense data by one
>subject at a particular moment in his subjective sense of time. Note
>that it is possible to build a very sophisticated notion of time
>from this subjective basis. Saul Basri builds full General Relativity
>in such a way in _A Deductive Theory of Space and Time_. He defines
>the "class of all living human beings who have adequately functioning
>sense organs, can communicate with each other, and do so honestly and
>without bias". He defines the "objective universe" based on this
>collection of observers. He then builds GR in a rigorous way.
Hmmm...
>
>--
>Marko Amnell
>amn...@klaava.helsinki.fi
>Graduate Student in Philosophy
>
>
>
>--
>Marko Amnell
>amn...@klaava.helsinki.fi
>Graduate Student in Philosophy
Randall Holmes
>(Randall Holmes) writes:
>
>>I am a realist myself, but I do not think that Truth is required in
>>order to search for truths. This is fortunate, since it seems clear
>>that Truth (as an abstract object) does not exist.
>
>This is the point where the relativist jumps in and says that since
>there is no ultimate standard of truth, but only individual truths
>in such and such circumstances, there is no objetive reality. He
>would seek to undermine your realism by questioning the validity
>of your conviction that there is an objective standard of fact.
>If there is no ultimate Truth, what can you appeal to besides
>contingent cirumstance?
Truth in particular languages is quite reliable; it can be defined in
a stronger system. The absolute Truth is analogous to Cantor's
Absolute Infinite. There is no ground for doubting particular truths.
>
>If it turns out that Truth cannot be defined in any logic, it could
>still play the role of what Kant called a regulative idea of the
>understanding. Our reason, said Kant, eschews the idea of God or
>an immortal soul, but we can use these idea as `limits' as it were
>to reason. In a similar way, the notion of Truth as the "ideal end
>of inquiry" regulates our search for individual truths. It is almost
>a question of faith. We have faith that science is taking us closer
>to the Truth, although our reason eschews Truth as a completed totality
>because it is paradoxical (supposing that it turns out this way).
It is possible to reason quite precisely about the iterative hierarchy
(the Absolute Infinite), and, similarly, it is possible to reason
about the Truth. But one can only refer to it indirectly.
>
>--
>Marko Amnell
>amn...@klaava.helsinki.fi
>Graduate Student in Philosophy
Randall Holmes
>In article <1992Nov8.1...@guinness.idbsu.edu>
>hol...@opal.idbsu.edu (Randall Holmes) writes:
>
>>In article <1992Nov7.0...@husc3.harvard.edu>
>>zel...@husc10.harvard.edu (Michael Zeleny) writes:
>
>>>In article <1992Nov6.1...@guinness.idbsu.edu>
>>>hol...@garnet.idbsu.edu (Randall Holmes) writes:
>
>RH:
>>>>I don't think that "Reality is the Truth about the world". Reality is
>>>>the world itself; Truth is what can be said about the world. Of
>>>>course, what can be said about the world is part of the world, and
>>>>when we start considering what can be said about that aspect of the
>>>>world, we get into trouble. Thus Tarski's paradox, etc.
>
>MZ:
>>>Let me see if I am getting this straight. You, Randall, are
>>>presumably "part of the world" itself, and _ipso facto_ part of
>>>Reality. Consequently, what I can say about you is Truth.
>>>
>>>So we have that
>>>
>>> Randall Holmes is a wood duck
>>>
>>>is part of Truth, insofar as I can, and indeed do say it right now.
>
>RH:
>>It would be an element of the set of true propositions in your
>>language,
>
>Am I hearing the term `proposition' from a self-professed extensionalist?
"Proposition" means "sentence".
>
>RH:
>> i.e., an element (not a part) of what I am terming
>>Truth[MZL] (where MZL is your language); but all of this depends on my
>>being a wood duck, which does not hold.
>
>What do you mean by "does not hold"?
>
>RH:
>>AHHH! I see that I did goof here! Of course, Truth is not merely
>>what can be said about the world, but what can be said _correctly_
>>about the world. Thank you!
>
>Now all that remains is for you to explain the meaning of the adverb...
>
This can be explained in the same sort of way that the iterative
hierarchy can be explained. "Not subsumed under a universal" does not
mean "incomprehensible". Truth in particular languages can be
explained, and a hierarchy of languages constructed which evades
Tarski's paradox by not being a completed totality.
Randall Holmes
should not be taken too seriously. Tarski's Paradox is worse than
some of the others; the sequence of "truths" in the hierarchy could
not even be definable (in the way in which the iterative hierarchy,
for instance, _can_ be defined). There could not be a sequence
Truth(a) (truth for language a) that could be described internally, or
we would be able to define Truth as "belonging to Truth(a) for some a"
and recover the paradox. But there are possible uses for such
hierarchies.
Michael Zeleny
hol...@garnet.idbsu.edu (Randall Holmes) writes:
MZ:
>>Am I hearing the term `proposition' from a self-professed extensionalist?
RH:
>"Proposition" means "sentence".
Certainly not to me, nor, I suspect, to the vast majority of
philosophers. If you mean "sentence", then write `sentence'.
RH:
>>> i.e., an element (not a part) of what I am terming
>>>Truth[MZL] (where MZL is your language); but all of this depends on my
>>>being a wood duck, which does not hold.
MZ:
>>What do you mean by "does not hold"?
I note a conspicuous absence of answer. Please be specific next time.
RH:
>>>AHHH! I see that I did goof here! Of course, Truth is not merely
>>>what can be said about the world, but what can be said _correctly_
>>>about the world. Thank you!
MZ:
>>Now all that remains is for you to explain the meaning of the adverb...
RH:
>This can be explained in the same sort of way that the iterative
>hierarchy can be explained. "Not subsumed under a universal" does not
>mean "incomprehensible". Truth in particular languages can be
>explained, and a hierarchy of languages constructed which evades
>Tarski's paradox by not being a completed totality.
Randall, I am willing to grant you the possibility that correctness
may be explained, say, by contextual definition, *only* if you follow
through by offering an adequate explanation thereof along these lines.
Analogously, impotence is not characterized by inability to imagine or
claim performance, but by a consistent failure to do so. I hereby
accuse your alleged extensionalist philosophy of a conspicuous lack of
erectile tissue. Feel free to prove me wrong.
MZ:
>>>>Did I get your proposal right, or is there anything more profound to it?
MZ:
>>Same question once again.
One more time, explicitly: so far, you have no proposal, just
equivocal periphrastic obfuscatology.
Gary Merrill
In article <1992Nov9.1...@husc3.harvard.edu>, zel...@husc10.harvard.edu (Michael Zeleny) writes:
|>
|> MZ:
|> >>Am I hearing the term `proposition' from a self-professed extensionalist?
|>
|> RH:
|> >"Proposition" means "sentence".
|>
|> Certainly not to me, nor, I suspect, to the vast majority of
|> philosophers. If you mean "sentence", then write `sentence'.
I agree with your sentiments, but I'm not at all sure that your
claim about the vast majority of philosophers. Since at least
the medievals, philosophers have used 'proposition' (or 'propositio'?)
when they meant 'sentence'. In the past when I have insisted on
the same distinction of usage I have been accused of pedantry --
something of which I am *never* guilty.
Frank Adams
>To say that something is _confused_ is to assert something like that there
>is no clear and distinct idea associated with the locution. Why is this
>phrase confused? Can you not even _imagine_ what it would be like for
>there to be a collection of all true propositions about the world?
>Suppose eg. that there is an omniscient being, would it not know all
>true propositions? Apparently so, and this means that you are saying
>that the very notion of omniscience is confused. I disagree. I think
>I can conceive what an omniscient being would be like. Just imagine
>a succesion of larger and larger computers, with the limit as the
>series approaches an infinitely large memory. What is confused about
>this idea?
But, unless you place the computer *outside* the world, the world
itself keeps getting larger and larger as the computer does. So the
computer never gets to omniscience.
It is not at all clear that it is sensible to talk about something
existing outside the world.
Marko Amnell
(Torkel Franzen) writes:
>In article <1992Nov8.1...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
>
> >If we wish to introduce the notion of propositions as distinct from both
> >sentences and statements, then what it added is this: The true statement
> >about the contingent state of affairs in its human context picks out one
> >of an infinity of timeless abstract objects called propositions.
>
> That's all well and good, but it doesn't answer my question whether
>"the totality of true propositions" is a notion better defined than
>"the totality of true sentences in all possible languages". If it is
>not, questions and speculations about the supposed totality are so
>much hot air. If it is, this remains to be shown.
The status of propositions is problematical in that it is not at all
clear in what sense a statement _expresses_ a proposition. This is
in addition, of course, to all the usual problems with abstract objects.
This suggests that one should try to stick to sentences in human
languages when it comes to formulating eg. truth-definitions--Quine eg.
takes this approach in his latest effort. But it is also true that
metaphysics looks aesthetically simpler with propositions rather than
messy sentences in natural language. This can be seen in eg. the
`crystalline clarity' of the Tractatus picture (as you note below).
> In metaphysics we can postulate all sorts of things, but we cannot
>postulate that we are making good sense. You may postulate that there
>is, corresponding to the sentence "this is a banana" uttered in a
>certain context, a timeless abstract object, the proposition that this
>is a banana. And so on. This postulate, however, is useless when we
>try to make sense of the locution "the totality of true propositions",
>in the absence of any explanation of which sentences, in which
>languages, real or hypothetical, have corresponding propositions, and
>how those sentences themselves are to be understood.
Would you be happier with "the totality of all possible sentences
(in all possible (or actual?) languages)" then? Maybe one could base
a metaphysics on such an ontology, following Quine.
> Wittgenstein, in the Tractatus, did not fall into this kind of loose
>babbling about a "totality of true propositions" since he explicitly
>postulated a particular formal ontology and a corresponding purely
>truth-functional language. Of course, he paid a price for this: the
>relation between that ontology and language and our ordinary ontology
>and language was obscure in the extreme.
--
Torkel Franzen
FI (Marko Amnell) writes:
>The status of propositions is problematical in that it is not at all
>clear in what sense a statement _expresses_ a proposition. This is
>in addition, of course, to all the usual problems with abstract objects.
But, as I said before, I am not objecting to associating abstract
objects with sentences. I am quite prepared to accept the description
"the proposition expressed by the statement ....".
>Would you be happier with "the totality of all possible sentences
>(in all possible (or actual?) languages)" then?
There is a considerable difference between "all possible languages"
and "all actual languages". After all, "Znorklozk frazk, znorkbul
frazk" is a sentence in any number of possible languages, but what
does it mean to say that it expresses a true proposition? And sticking
to actual languages, how do you propose to explain what "expresses a
true proposition" means applied to English sentences, in a way that
makes this locution any clearer than those sentences themselves? After
all, to speak of "the totality of true English sentences" is not to
make any obvious sense, unless one is able to provide an explication (based
on whatever considerations) of every possible English sentence.
Randall Holmes
I'm very much aware of the distinction; the reason why I feel free to
use "proposition" to mean "sentence" is that I do not recognize any
"propositions" in any sense intermediate between and independent of
truth values and sentences as existing. Thus, the word is available
(and _has_ been used in this sense, as Gary points out). When Marko
claimed that the set of all propositions was an inconsistent totality,
I indicated why I thought not using sentences in a suitable language
as "propositions" in NFU.
Marko Amnell
(Frank Adams) writes:
>>Can you not even _imagine_ what it would be like for
>>there to be a collection of all true propositions about the world?
>>Suppose eg. that there is an omniscient being, would it not know all
>>true propositions? Apparently so, and this means that you are saying
>>that the very notion of omniscience is confused. I disagree. I think
>>I can conceive what an omniscient being would be like. Just imagine
>>a succesion of larger and larger computers, with the limit as the
>>series approaches an infinitely large memory. What is confused about
>>this idea?
>But, unless you place the computer *outside* the world, the world
>itself keeps getting larger and larger as the computer does. So the
>computer never gets to omniscience.
>It is not at all clear that it is sensible to talk about something
>existing outside the world.
It's not clear, but neither is it inconceivable. One could try to
imagine what the world would look like from the outside. This idea
is no more bizarre than eg. the idea of a self-membered set, which
is also in conflict with our common sense intuitions. There are
lots of ideas that are counter-intuitive. Think of the first time
you heard about time dilation, for instance. God might know all
truths, including all truths about himself; if this were possible,
it would eliminate the difficulty you cite.
Alternatively, one could settle for near-omniscience. I was only
trying to show that one could try to imagine what a collection of
all truths would be like in the real world. There is no need to
cling to the computer metaphor--it was only an illustration.
Marko Amnell
>And sticking to actual languages, how do you propose to explain what
>"expresses a true proposition" means applied to English sentences, in a
>way that makes this locution any clearer than those sentences themselves?
Propositions could be any number of things: Abstract objects like Fregean
senses, the meanings of sentences, or they could be identified with some
class of sentences. I don't have an opinion on which of these possibilities
(or others) is the best. Each has its difficulties. What is an abstract
object? How can it be the sense of a sentence? In what does the meaning
of a sentence consists? How do we pick out the class of sentences in the
case of ambigious locutions and assertions?
>After all, to speak of "the totality of true English sentences" is not to
>make any obvious sense, unless one is able to provide an explication (based
>on whatever considerations) of every possible English sentence.
This is a job for the linguist. I don't think I have to give an exhaustive
characterization of every possible English sentence before I can refer
to such a totality in a philosophical theory. Nor, for that matter, is
it necessary for me to provide a full account of the nature of propositions
before I can make use of them in my theorizing.
Torkel Franzen
Helsinki.FI (Marko Amnell) writes:
>What is an abstract object?
As far as my questions are concerned, it doesn't matter what an
abstract object is. I have no problem with abstract objects in
this context.
>This is a job for the linguist. I don't think I have to give an exhaustive
>characterization of every possible English sentence before I can refer
>to such a totality in a philosophical theory.
You're ignoring the point at issue. Take a set S of English
sentences, say the set of sentences found in the writings of Bertrand
Russell. What subset of S is specified by "the set of true sentences
in S"? If you hold that there is in fact such a well-defined subset,
please say so, in which case further arguments can be brought to bear.
If you do not, please take notice of the fact that nothing is gained
by switching to "the set of true propositions expressed by sentences
in S".
>Nor, for that matter, is
>it necessary for me to provide a full account of the nature of propositions
>before I can make use of them in my theorizing.
How true. However, nobody has asked you to provide a full account of
the nature of propositions. Rather, I have asked how you propose to make
sense of the locution "the totality of true propositions". So far it
appears that you are content to "theorize" about this supposed totality
in hot-air style.
Marko Amnell
(Torkel Franzen) writes:
>You're ignoring the point at issue. Take a set S of English
>sentences, say the set of sentences found in the writings of Bertrand
>Russell. What subset of S is specified by "the set of true sentences
>in S"? If you hold that there is in fact such a well-defined subset,
>please say so, in which case further arguments can be brought to bear.
>If you do not, please take notice of the fact that nothing is gained
>by switching to "the set of true propositions expressed by sentences
>in S".
Here is a sketch of an argument for the existence of a set of all true
propositions expressible in English:
Let's consider your example. We start out with a set S of sentences in
English, say n of them (all the sentences in Russell's writings, I guess
at least several million). We define a sentence to be any string of
words ending in a full stop that is syntactically correct (given the
clarity of Russell's style we needn't worry here).
Now, of these n sentences only some will be descriptive ie. they assert
something about the world (including mathematical sentences, and I guess
logical truths too). Which sentences qualify would be determined by some
group of rational persons who were familiar with the subject matter of the
sentences, say, the readers of this newsgroup. So, given enough time, we
could pick out from S all those sentences of which it would make sense to
ask whether they were true or false. Call this subset SD (for descriptive
sentences). Let us say there are m of them, with m < n.
Given this set SD, we could, again given enough time, go through each
SDi (i=1,2,...,m) and check whether it is true or not. The method of
verification would vary from sentence to sentence and there is no
simple rule for how in general it would be done. But, a group of
rational persons could in principle check at least most of the SDi.
Some might be left over for various reasons, but this ignorance does
not detract from the fact that in principle each SDi could be so checked.
We collect all the true sentences in SD into the set ST, the set of all
true sentences in Russell's writings. The cardinality of this subset
would again be smaller than SD, say r < m.
Now, since we can take the relation between sentences and propositions
for granted, we say that each STi expresses some proposition. The
number of true propositions expressed would be smaller than the
number of sentences, since Russell presumably asserts the same thing
several times in various places. In any case, we get a further set,
call it SP, of all true propositions found in Russell's writings.
Let us say there are s of them, with s < r.
We arrived at SP in stages from S and each stage seems legitimate. So,
we are justified in saying that Russell's writings express some set SP
of true propositions. We could in principle do this for any set of English
sentences, including the set of all possible sentences A, which would
express the set of all true propositions T (expressible in English).
There is nothing different about S and A, nor hence about SP and T.
Well, this is something to start from, anyway. Where are my flaws?
Randall Holmes
Marko, I suggest that you fall back from "proposition" to "sentence"
in a suitable language. I think that "sentence of first-order logic
in a language which has constants for every object" is a good
approximation to "proposition". It can be established that this is a
consistent totality (it exists in NFU), while the set of "truths"
(true sentences in a language of this kind) is an inconsistent
totality under any reasonable assumptions.
This would answer the objection of vagueness in the notion of
proposition. It also shows that Grim's argument for the inconsistency
of the totality of propositions is wrong; his argument appeals to no
feature of propositions which is not found in my nonce notion of
"proposition".
In ZFC, the collection of sentences of a language of this kind is a
definable proper class; the collection of true sentences of a language
of this kind will fail to be even definable. Once again, the notion
of the set of all propositions seems to make sense, while Truth
remains inconceivable. (Once again, without posing any problem for
the truth of particular sentences in this language or for definability
of truth for sentences of smaller languages in our large language).
The problem with Truth (the class of true sentences) is that its
definition for an initial segment of the iterative hierarchy cannot be
carried out predicatively (without use of quantifiers ranging over
classes) in the next level of the hierarchy. Definable "Absolute
Infinite" notions are those which can be defined predicatively in the
"next" level (this is how proper classes work). Truth can be thought
of as present in the metaphorical realm of the Absolute Infinite, but
it is more elusive than the universe or the Russell class, which are
definable proper classes. The collection of _sentences_ of a language
with a constant for every object in the universe is predicatively
definable, while the collection of _true_ sentences is not. Making
the theory of classes impredicative does not help; one can make truth
for predicative sentences (those without quantification over classes)
definable (I think) but one now cannot define truth for the new
impredicative sentences. It does appear that Truth can be constructed
iteratively ("subsists in the Absolute Infinite", Mikhail) but the
construction is in principle indescribable. This is frustrating, but
true (with a small "t", note).
In NFU, there are analogous considerations which I will not bore you
with; they rest on the fact that in TT it takes at least n+1 types to
define truth for sentences which use n types.
Torkel Franzen
FI (Marko Amnell) writes:
>Here is a sketch of an argument for the existence of a set of all true
>propositions expressible in English:
So let us consider your requirements. First,
>We define a sentence to be any string of
>words ending in a full stop that is syntactically correct
The problems of syntax may reasonably be set aside here as inessential.
Your next requirement, however, is that a sentence expressing a true
proposition must "assert something about the world":
>Which sentences qualify would be determined by some
>group of rational persons who were familiar with the subject matter of the
>sentences, say, the readers of this newsgroup. So, given enough time, we
>could pick out from S all those sentences of which it would make sense to
>ask whether they were true or false.
Already at this point your definition collapses. It is a laughable notion
that the readers of this newsgroup - which readers? at which point in
time? - could collectively agree on a definite set of descriptive
sentences among those occurring in Russell's writings. In brief: not
only are the terms used in your definition - "rational persons
familiar with the subject matter" - exceedingly vague and
indeterminate, but the idea that there is some set of sentences which
any such group of persons would single out as "descriptive" has no
basis in fact.
Next you stipulate that the sentences in S are those descriptive
sentences that we - presumably these rational and informed people -
could in principle "verify":
>SDi (i=1,2,...,m) and check whether it is true or not. The method of
>verification would vary from sentence to sentence and there is no
>simple rule for how in general it would be done. But, a group of
>rational persons could in principle check at least most of the SDi.
This is quite absurd. Are you saying that these hypothetical persons could
verify the truth of e.g. all true mathematical sentences or sentences of
physical theory occurring in Russell's writings? Again, are you saying that
there is a method of "verifying" the truth of any "true" sentences concerning
human existence occurring in Russell's writings?
To introduce some reality into this discussion, here are a few sentences
occurring in Russell's writings:
This question is one of great importance, since it introduces us to
the whole problem of how knowledge can transcend personal experience.
If we confine ourselves to spoken words in one language, a word is
a class of closely similar noises produced by breath combined with
movements of the throat and tongue and lips.
What cannot be verified or falsified is meaningless.
The simplest imaginable facts are those which consist in the possession
of a quality by some particular thing.
To understand the Renaissance, it is necessary first to review briefly
the political condition of Italy.
Schopenhauer's gospel of resignation is not very consistent and not
very sincere.
Before philosophy began, the Greeks had a theory or feeling about the
universe, which may be called religious or ethical.
The Son of Man shall send forth His angels, and they shall gather out
of His kingdom all things that offend, and them which do iniquity,
and shall cast them into a furnace of fire; there shall be wailing
and gnashing of teeth.
To be a nice person it is necessary to be protected from crude contact
with reality, and those who do the protecting cannot be expected to
share the niceness that they preserve.
I submit that many people will reject as simple-minded in the extreme
any proposed task of dividing these sentences into those that do and
those that do not "assert something about the world", as well as the task
of deciding which of these sentences are "true". Each of the sentences may
be questioned in many ways, elucidated or analyzed in many ways, argued
for or against in many ways, accepted or rejected on different grounds
and interpretations. If we accept one of these sentences as true or
reject it as false, what we mean by this can be understood only in the
context of our interpretations and explanations.
I submit further that even if a bunch of people accept your two tasks
as meaningful, there is no reason whatever to believe that they will arrive
at any consensus, or that their decisions will agree with those of a
different group, or that they will arrive at the same decisions the next time
you ask them.
Marko Amnell
>In brief: not only are the terms used in your definition - "rational persons
>familiar with the subject matter" - exceedingly vague and
>indeterminate, but the idea that there is some set of sentences which
>any such group of persons would single out as "descriptive" has no
>basis in fact.
Well, there's no way to get around the vagueness of natural language,
and this would include deciding when one in asserting something
substantive.
>Next you stipulate that the sentences in S are those descriptive
>sentences that we - presumably these rational and informed people -
>could in principle "verify":
This is one way of defining truth.
>Are you saying that these hypothetical persons could
>verify the truth of e.g. all true mathematical sentences or sentences of
>physical theory occurring in Russell's writings? Again, are you saying that
>there is a method of "verifying" the truth of any "true" sentences concerning
>human existence occurring in Russell's writings?
Well, not "verifying", necessarily, but there would be a lot of
sentences about whose truth or falsehood there was general agreement.
>I submit that many people will reject as simple-minded in the extreme
>any proposed task of dividing these sentences into those that do and
>those that do not "assert something about the world", as well as the task
>of deciding which of these sentences are "true". Each of the sentences may
>be questioned in many ways, elucidated or analyzed in many ways, argued
>for or against in many ways, accepted or rejected on different grounds
>and interpretations. If we accept one of these sentences as true or
>reject it as false, what we mean by this can be understood only in the
>context of our interpretations and explanations.
Sure. Again, the vagueness of natural language.
>I submit further that even if a bunch of people accept your two tasks
>as meaningful, there is no reason whatever to believe that they will arrive
>at any consensus, or that their decisions will agree with those of a
>different group, or that they will arrive at the same decisions the next time
>you ask them.
But there is a consensus about the truth of lots of sentences in everyday
life. I'll admit that the sketch isn't too great, but then I just made
it up to give you an idea of what I had in mind. I think much of your
criticism might be avoided if one concentrated on formal systems, instead.
That would probably be the only way of making the picture work, as in
the Tractatus. If we stay with natural language, there's no way to
eliminate the vagueness. And like you said, if we stick to formal
language, the connection to reality is severed.
Torkel Franzen
FI (Marko Amnell) writes:
>Well, not "verifying", necessarily, but there would be a lot of
>sentences about whose truth or falsehood there was general agreement.
What is this reflection supposed to contribute to the explanation of
"the set of true sentence occurring in Russell's writings"?
>I think much of your
>criticism might be avoided if one concentrated on formal systems, instead.
What does this have to do with the set of true sentences occurring in
Russell's writings?
Marko Amnell
(Torkel Franzen) writes:
>In article <1992Nov10....@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
>
> >Well, not "verifying", necessarily, but there would be a lot of
> >sentences about whose truth or falsehood there was general agreement.
>
> What is this reflection supposed to contribute to the explanation of
>"the set of true sentence occurring in Russell's writings"?
But, you're the one who picked this example from natural language, to
which my lame argument was a response. If one is formulating some
kind of ultra-Platonistic metaphysical theory, one can do as Ludwig did
and just stick stubbornly to formal logic. See Randall's recent comments
in this thread for where this leads.
> >I think much of your
> >criticism might be avoided if one concentrated on formal systems, instead.
>
> What does this have to do with the set of true sentences occurring in
>Russell's writings?
See above.
Gary Merrill
In article <1992Nov10....@guinness.idbsu.edu>, hol...@garnet.idbsu.edu (Randall Holmes) writes:
|>
|> Marko, I suggest that you fall back from "proposition" to "sentence"
|> in a suitable language. I think that "sentence of first-order logic
|> in a language which has constants for every object" is a good
|> approximation to "proposition". It can be established that this is a
|> consistent totality (it exists in NFU), while the set of "truths"
|> (true sentences in a language of this kind) is an inconsistent
|> totality under any reasonable assumptions.
The problem with Randall's suggestion is that this concept of
proposition is quite inadequate as even an "approximation" if
a proposition is taken to be something like the "meaning" of
a sentence, or the "intension", or some other Platonistic thingamabob
that a sentence "expresses". However, given this (common)
meaning of 'proposition', Marko will have some problems. However,
never mind whether Randall is correct in his "approximation" claim
since that's really immaterial. I believe he certainly is correct
in the suggestion that appealing to propositions will just cause
Marko *more* trouble.
For example, it is fairly common to construe a proposition as a
*function* from a set K onto the truth values. K may be the set of
possible worlds, the set of models of the language, or some similar
set. If S is a sentence, then the-proposition-expressed-by S is
that function which for every element k of its domain assigns the
value T (true) just in case S is true at k. The problem is that
there are *more* propositions than sentences since the cardinality
of the set of propositions (so construed) is that of the power set
of the set of the set of possible worlds, models, or whatever.
You have countably many sentences, but uncountably many propositions
to express.
This exposes a problem with the argument Marko posted earlier when
he urged that the set of propositions expressed in Russell is smaller
than the set of sentences in Russell. While this is true when the
domain is restricted to "in Russell", it is not true in general (under
common representations of propositions).
Randall Holmes
"Appended to its own quotation is not a true sentence of
English" appended to its own quotation is not a true sentence of
English.
Is this a true sentence of English?
Randall Holmes
>
>In article <1992Nov10....@guinness.idbsu.edu>, hol...@garnet.idbsu.edu (Randall Holmes) writes:
>|>
>|> Marko, I suggest that you fall back from "proposition" to "sentence"
>|> in a suitable language. I think that "sentence of first-order logic
>|> in a language which has constants for every object" is a good
>|> approximation to "proposition". It can be established that this is a
>|> consistent totality (it exists in NFU), while the set of "truths"
>|> (true sentences in a language of this kind) is an inconsistent
>|> totality under any reasonable assumptions.
>
>The problem with Randall's suggestion is that this concept of
>proposition is quite inadequate as even an "approximation" if
>a proposition is taken to be something like the "meaning" of
>a sentence, or the "intension", or some other Platonistic thingamabob
>that a sentence "expresses".
Of course, _I_ admit the existence of no such thing!
Torkel Franzen
Helsinki.FI (Marko Amnell) writes:
>But, you're the one who picked this example from natural language, to
>which my lame argument was a response.
In other words, you don't in fact want to claim that "the set of
true English sentences" is well-defined? In that case I revert to
my earlier question: how do you propose to explain what you are
referring to in your speculations about the "totality of true
propositions"?
Marko Amnell
in its present form. This is not so surprising; as Kripke said, most
philosophical theories end up being false. But, this doens't mean
we should stop developing them--the method of failure invariably
teaches us something.
Gary Merrill
|>
|> "Appended to its own quotation is not a true sentence of
|> English" appended to its own quotation is not a true sentence of
|> English.
|>
|> Is this a true sentence of English?
Yes and no. (Ha! Ha!) This is a version of the Grelling-Nelson
paradox. It can be made to fail in some semantically closed
systems. In fact, in the system T* of my dissertation it fails
to be paradoxical. However, this is at the expense of invalidating
certain interchange principles in quotation contexts that
otherwise seem to be reasonable. I omit the details here. It's
all pretty ugly and complex.
Frank Adams
>In <1992Nov09.2...@Cookie.secapl.com> fr...@Cookie.secapl.com
>(Frank Adams) writes:
Marko Amnell (I think):
>>>Can you not even _imagine_ what it would be like for
>>>there to be a collection of all true propositions about the world?
>>>Suppose eg. that there is an omniscient being, would it not know all
>>>true propositions? Apparently so, and this means that you are saying
>>>that the very notion of omniscience is confused. I disagree. I think
>>>I can conceive what an omniscient being would be like. Just imagine
>>>a succesion of larger and larger computers, with the limit as the
>>>series approaches an infinitely large memory. What is confused about
>>>this idea?
>
>>But, unless you place the computer *outside* the world, the world
>>itself keeps getting larger and larger as the computer does. So the
>>computer never gets to omniscience.
>
>>It is not at all clear that it is sensible to talk about something
>>existing outside the world.
>
>It's not clear, but neither is it inconceivable.
If the world is everything that exists (a reasonable if not very informative
definition), then existing outside the world *is* inconceivable.
> God might know all
>truths, including all truths about himself; if this were possible,
>it would eliminate the difficulty you cite.
No absolute proof is available, but there are all sorts of reasons to think
that this is impossible. Informally, in my objection to your computer
model. Formally, in such things as Tarski's theorem on the undefinability
of truth in formal systems.
>Alternatively, one could settle for near-omniscience.
Near-omniscience is as different from omniscience as near-pregnancy is from
pregnancy.
> I was only
>trying to show that one could try to imagine what a collection of
>all truths would be like in the real world. There is no need to
>cling to the computer metaphor--it was only an illustration.
But if all the metaphors fail in the same way, one has to wonder if the
problem isn't in the underlying concept.
Leo Smith
>In article <1992Nov5.2...@klaava.Helsinki.FI> amn...@klaava.Helsinki.
>FI (Marko Amnell) writes:
> >I think we already covered all this two months ago. I don't see
> >why realism about scientific theories (which is what I have in
> >mind here) can't include a belief that science approaches some
> >sort of ultimate truth about reality.
> Realism about scientific theories "can include" all sorts of things,
>for example the belief that the universe is a banana being eaten by
>the Composite Principle of Aloofness. But why associate these ideas
>with realism?
Is this academic put-down game restricted to members of scandinavian
teaching establishments, or can anyone join in?
BTW Torkel, you might some time care to look at one or two logical
schemas that don't rely on binary logic. I am sure you are
intelligent enough to concoct one that is reasonably consistent with
reality and itself.
You know. Not just dividing the world into 'Torkel' and '~Torkel' but
including such categories as 'Almost Torkel' 'Somewhat Torkel' and
'Not exactly Torkel, but nevertheless _not [Not Interesting] for all
that....'
Who knows what dizzying revelations might result from such wild
theorising?
Leo Smith
amn...@klaava.Helsinki.FI (Marko Amnell) writes..
In article <TORKEL.92...@lludd.sics.se> tor...@sics.se
(Torkel Franzen) writes:
[....]
>Allow me to quote Peirce's definition of reality in "The Fixation of
>Belief".
> Such is the method of science. Its fundamental hypothesis, restated
> in more familiar language, is this: There are Real things, whose
> characters are entirely independent of our opinions about them;
> those Reals affect our senses according to regular laws, and, though
> our sensations are as different as are our relations to the objects,
> yet, by taking advantage of the laws of perception, we can ascertain
> by reasoning how things really and truly are; and any man, if he have
> sufficient experience and he reason enough about it, will be led to
> the one True conclusion. The new conception involved here is that
> of Reality.
Neatly put. I agree that that is exactly the way Western thought has
gone: It pre-supposes that reality, operates as if that reality were
de-facto, and uses the functionality derived from that assumption
(the fact that science WORKS) to support the notion that the de-facto
assumption is valid.
In fact it is merely internal consistency: It is not its truth that
gives science its power, it is its utility: Ideas of the planet being
consumed by a giant banana (or whatever Torkels fevered imagination
conjures up) are not useful in adding to that utility.
I think it is not so much a question of what is Real or True, but
what works.
But then I am only a humble engineer, and my degree was in Cambridge,
and not in Scandinavia.
[..]
>If we begin with Peirce's definition of reality, then you see why I
>am so interested in the problem of the existence of Truth. If there
>is no totality of all true propositions (if, say, Tarski's result
>about formal logics with the usual configuration of quantifiers --
>Hintikka changes the way quantifiers work, and it is this that
>enables him to avoid Tarski) then there is no Truth as the ideal end
>of inquiry, and the whole Peircean metaphysics crumbles. We are left
>with some kind of radical constructivism (that reality is a purely
>social construct) or with some kind of relativism (a variety of
>relatvism that avoids most of the old knock-down arguments can be
>found in Margolis' book _The Truth about Relativism_)
Surely Marko, Truth is in one sense just a mental concept: the game
goes like this.
IF you assume that there is an objective reality independent of our
ability to intellectualise it
THEN
it ought to be possible to approach a perfect intellectualised
conception of it which for the purposes of this philosophy we can
call the 'Truth'
That is to say, that Reality and Truth are aspects of the same model.
The model that Western thought is based on, the model of the
independent perceiver of the exterior universe.
As an engineer, I am used to approximations. I assume that for all
practical purposes, the 'physical world' is not affected by my
assessment of it. Hesienberg represents one aspect of the limitation
of that assumption. Psychology another. So I suppose that Truth and
Reality become truth and reality. Only relative.
>It is because all these metaphysical theories are living options to
>me, and because I more or less follow Peirce's definition of reality
I respect that. I am probably in a minority of one though :-(
>and his linkage of it to Truth, that I am interested in the problem
>of whether a truth predicate can be defined internally in a formal
>system that does the things we want it to do (like arithmatic).
>Note that this problem is not the same problem as the metaphysical
>problem of whether there is a totality of all truths or not (what
>is usually called simply Truth). But, there is a linkage here,
>whose strength depends on how far we follow, eg. Hintikka or
>Penrose that human thought can be modelled by a formal system.
Surely formal systems are just tools: we are looking to generalise and
simplify our experience such that we can condense it into general
rules. This has utility. It is easier to teach a child that 'all
moving cars are dangerous' than it is to recount the deatails of all
the automobile accidents one has witnessed... but that doesn't make
it nearly as TRUE as the recounting of all that data...and the data
itself is only a reference to the presupposed actual events....
.. perhaps I could say that for me, it is not such a binary thing as
correct or not correct: There is existence. Maybe. I don't and can't
know. What I DO know is what experience I have. This is the nearest
to the Truth that I can get, and having been accused of being a
mystic on this group, I suppose that means that using the experience
as the starting point of the model. Then the memory of the experience
is another layer, and further away from one sort of reality, the
reality of direct experience, is The Model. That is the series of
generalised internal constructs and guidelines that we use to
manipulate our experience. The way we store it and refer to it
internally. THAT is what a realist calls Reality IMHO. He calls what
I call Reality mystical babblings :-)
What Peirce (Pierce?) seems to be saying, is that the assumption of
an independent reality is the basis for an inquiry into its nature:
The endpoint of that inquiry is the Truth. But Science itself, in its
attempt to simplify that reality into objects and rules, is in itself
a part of that reality. As we reduce complexity in the exterior
world, we introduce it in the interior! It seems likely to me that it
would require an infinite description to simplify the world
completely.
So I consider that such a search for Truth is meaningless in absolute
terms: You can for utilitarian reasons exploit the modelling ability
in your mind to produce object/rule arrangements of the world. This
gives you funtional efficiency - but it doesn't add to the 'Truth'.
All there is is all there is, and restating it in other terms IS just
philosopho-babble - amusing as an intellectual exercise - but not
useful in answering the basic questions of 'what am I?' and 'what the hell
is going on?'...:-)
Because in the final analysis, the Zen like answer, and I am afraid
it IS the only answer to those questions is:
'whatever you think' [you are/is going on]
The questions are a product of a model, and the answers are in terms
of that model. Mystics might say that the way out of that Gordian
knot is to practice Not Thinking. i.e. transcendental mentalism of
some sort. But then although you may experience Reality a little more
directly, you are [literally and in a very pragmatic sense]
completely at a loss for words...since these too are a product of
thinking, of that rational model. So this experience, whilst
illuminating, can't help answer those mental questions either.
Metaphysicist: 'Which is more powerful - God or an H-Bomb'
Engineer: 'If you can put God there at Ground Zero we'll find out......'
:-)
Torkel Franzen
(Leo Smith) writes:
>Is this academic put-down game restricted to members of scandinavian
>teaching establishments, or can anyone join in?
Which Scandinavian teaching establishments are those?
Greg Restall
*It is possible to reason quite precisely about the iterative hierarchy
*(the Absolute Infinite), and, similarly, it is possible to reason
*about the Truth. But one can only refer to it indirectly.
Isn't this self-refuting?
The use of proper names such as "the Absolute Infinite",
"the Truth" and forms such as "it" seem to indicate
that a rather direct means of reference is in use.
I wonder what this phrase looks like when translated
into something that doesn't refer to the Truth directly.
Does anyone wish to make a stab?
The odd nature of this reminds me of the last sentence in
a manuscript by Graham Priest that I've just been reading.
Whereof we cannot speak, thereof
we have just contradicted ourselves.
with due apologies to L.W.
Best wishes,
Greg Restall
--
------------------+------------------------------------------------
Greg Restall | Philosophy Department, University of Queensland
g...@cltr.uq.oz.au | Queensland, 4072 Australia.
------------------+------------------------------------------------
Redemption rips through the surface of time in the cry of a tiny babe.
- Bruce Cockburn, from the album "Nothing but a Burning Light"
William Tucker
Leo Smith says in response to Torkel's and Marko's dialogue:
[stuff deleted]
Surely formal systems are just tools: we are looking to generalise and
simplify our experience such that we can condense it into general
rules. This has utility. It is easier to teach a child that 'all
moving cars are dangerous' than it is to recount the deatails of all
the automobile accidents one has witnessed... but that doesn't make
it nearly as TRUE as the recounting of all that data...and the data
itself is only a reference to the presupposed actual events....
I (William Tucker) agree:
An abstraction is necessarily that which removes from consideration what
need not be considered NOW. However when you proceed with multiple abstractions
you get the same results as you would in mathematics or science when you have
a "neglected" amount(s) or "margins of error," the error multiplies. Perhaps I'm mistaken but, I don't remember seeing this discussed before, nor am I familiar
with any regular method for dealing with it. Is there?
Leo continues:
.. perhaps I could say that for me, it is not such a binary thing as
correct or not correct: There is existence. Maybe. I don't and can't
know. What I DO know is what experience I have. This is the nearest
to the Truth that I can get, and having been accused of being a
mystic on this group, I suppose that means that using the experience
as the starting point of the model.
I (William Tucker) say:
Whether they realize it or not all philosophers deal with their own personal
experiences exclusively. Information enters via sensorial means period.
What we percieve of as truth is that which matches our experience. We can
link groups of past experiences together into patterns, this cutting and
pasting of events to get a "story" is called intelligence...the ability to see similarity in seemingly unrelated experiences.
Leo continues:
Then the memory of the experience
is another layer, and further away from one sort of reality, the
reality of direct experience, is The Model. That is the series of
generalised internal constructs and guidelines that we use to
manipulate our experience. The way we store it and refer to it
internally. THAT is what a realist calls Reality IMHO. He calls what
I call Reality mystical babblings :-)
[stuff deleted]
I (William Tucker) think:
If that is what the realist calls reality then he again defines his world
in terms of his tools and as such drops a signicant portion of the "out there"
onto the floor. Your mystical babblings are IMUHO lucidity in prose.
ps.
Kudos to mister M. Zeleny and M. Rooney for your brief but hilarious
burst of pretentious/outrageous and french innuendo, my apologies to those
who took it seriously.
Wm T.
.standard disclaimer
Michael Zeleny
tuc...@mitre.org (William Tucker) writes:
>ps.
>
>Kudos to mister M. Zeleny and M. Rooney for your brief but hilarious
>burst of pretentious/outrageous and french innuendo, my apologies to those
>who took it seriously.
Comrade Tucker, in the name of the legality of equality, I solemnly demand
that you distribute your honorifics in a fair and equitable manner. Be it
`mister' or `arsehole', if you award it to me, you must likewise award it
to my esteemed interlocutor. I further differ from you in condemning with
the utmost gravity all those tiresomely ubiquitous, sniveling metapedantic
footnote chasers who would take seriously any subject involving a public
display of personal peccadilloes.
I mean it!
>Wm T.
>
>.standard disclaimer
cordially,
mikhail zel...@husc.harvard.edu
" -- I shall speak bluntly, because life is short."
Torkel Franzen
(Greg Restall) writes:
>Isn't this self-refuting?
>The use of proper names such as "the Absolute Infinite",
>"the Truth" and forms such as "it" seem to indicate
>that a rather direct means of reference is in use.
This is an old problem which essentially has no solution. In many
religious traditions what is recognized as self-refuting is rather to
try to refer to what cannot be referred to. (As in the Tao Te Ching
and the Diamond Sutra.) However, by the same token, it doesn't really
matter what words we use: "the absolute infinite", "the truth", "the
buddha mind": such terms are all quite arbitrary, and the essential thing is
only to keep this in mind.
This traditional point of view carries over naturally to the
cumulative hierarchy. The cumulative hierarchy is of course ineffable,
and any attempt to refer to it is doomed to failure! Of course those
who lack the requisite religious faith may well take a dim view of
this notion, but it has a mathematical point as well, since it leads us
to the reflection principle "any mathematically sensible assertion
true of the universe is true of some set". Of course this principle
itself vainly attempts to refer to the universe! But from it we get
mathematically formulated reflection principles.
Randall Holmes
No, it is not self-refuting. Look in any textbook on set theory to
see how the iterative hierarchy, which cannot be discussed as a whole,
is actually discussed. The direct references to the Absolute Infinite
or Truth could be understood as being made to actual objects standing
in sutiable relations to incomplete models of the whole hierarchy
(initial segments of the iterative hierarchy which look more or less
like the whole thing from the inside). But it is a dangerous way of
speaking.
Randall Holmes
I concur.