Re: logical paradoxes
From: Jeffrey Ketland (ketland_at_ketland.fsnet.co.uk)
Date: 08/18/04
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Date: Wed, 18 Aug 2004 15:20:49 +0100
JXStern wrote in message <05j5i0pcjmn9njc56v4ghg1ajbgtrdc3c3@4ax.com>...
>On Wed, 18 Aug 2004 02:30:03 +0100, "Jeffrey Ketland"
><ketland@ketland.fsnet.co.uk> wrote:
>>>... but, do you see natural language commiting itself to fixed
>>>interpretation?
>>
>>Even setting aside vagueness, ambiguity and indexicality, no.
>
>Good!
>
>>Not in the
>>sense of a fixed interpreted language, modelled as a pair (L, M), where M
is
>>the structure over which L-symbols are interpreted.
>
>M being the meta-language?
M is a *structure* over which L-symbols are interpreted.
For example, L might be a first-order language with a single binary
predicate symbol "<". The structure M might be (H, R), where H is the set of
humans and R is the relation of being taller than. Then, using Tarski's
inductive definition, we have truth conditions for arbitrary L sentences,
like:
"forall x, exist y (x < y)" is true in (L, M) iff,
for any human a, there is a human b taller than a
Or L might be the first-order language of arithmetic with a binary function
symbol "PLUS", and M might be the structure (N, 0, S, +, x). Then
"exists y, for all x (x PLUS y = x)" is true in (L, M) iff,
there is a number a such that for any number b, b+a = b.
>> The problem is Tarski's
>>indefinability result that genuine semantic self-representation appears to
>>be mathematically impossible. Such languages cannot express their own
>>semantical concepts (which natural languages at least appear to do), and
the
>>problem doesn't go away even if you consider many-valued logic. This
appears
>>to force a metalanguage/object-language distinction upon us, with the
>>metalanguage being "essentially richer" than the object language, as
Tarski
>>put it.
>
>Does nobody separate language and meta-language from axiomatics? My
>reading of Tarski has always been that he gave a good formalist
>doctrine of axiomatic truth, but was never the least bit convincing
>about any other variety of truth - correspondence, for example.
Tarski's work has nothing to do with "axiomatic truth", which perhaps just
means "theorem", a syntactical notion fully analysed in the 1920's by
Hilbert and co-workers. Tarski's work has nothing to do with axioms either.
The set of truths in an interpreted language has nothing to do with axioms.
For many languages, the set of truths cannot even be axiomatized.
Furthermore, people may state axioms which are false.
I don't see any connection between truth and axioms, and neither did Tarski.
The constant formalist confusion between axioms and truth, which was very
dominant in the 1920's and 30's amongst the positivists, was something that
Tarski had to overcome in developing his theory.
Tarski certainly took himself to be describing or perhaps refining the
correspondence theory, and many others have agreed with this estimation,
including Popper, Wolenski and others. Informally, he is attempting to
capture the idea that a statement S is true if what S says is the case, is
the case. I.e., if S says that p, and p. Tarski's examples were specifically
drawn from natural language, and he states the T-scheme with natural
language examples, such as "snow is white". Whether the statement "snow is
white" is true depends upon whether snow is white. As Koffa reported, Carnap
recalled that "the scales fell from my eyes" when Tarski explained the
T-scheme to him, sometime around 1933-4.
The T-scheme characterizes a central, disquotational, property of our use of
the semantical word "true".
>The trick is to make each meta-language *less* powerful than that
>which it interprets, this has been Minsky's lesson for reduction in
>AI, and is carried out in practice by the BNF specification of
>computing languages.
I don't understand this. By the usual definition, the metalanguage (L*, M*)
for an interpreted language (L, M) contains predicates for the notions of
truth, reference, denotation, etc. in (L, M). If the metalanguage is weaker
than (L, M), then you're just doing syntax for L: defining "formula in L",
"term in L", etc.
>The thing is, semantics cannot be an essential part of any language,
>and it's not even a technical issue. You speak English, so do I. One
>of your axioms is that "cats are nasty". One of my axioms is that
>"cats are nice." It is not a property of English that will resolve
>any sentence about cats that we interpret, it is a separable ontology.
Semantics is concerned with what a language is about: what entities,
objects, properties and relations in the world that the expressions of the
language refer to.
If one of my assertions is "cats are nasty", and cats are nasty, then my
assertion is true.
This follows from the T-sentence.
"cats are nasty" is true if and only if cats are nasty.
[snip]
>>Unfortunately, I have no idea how to respond to this argument. Maybe
humans
>>aren't smart enough to solve the problem of semantic self-representation.
>
>Fooey.
Well it's possible, given the amount of time the problem has been around.
Much as problems about the nature of the human mind may be beyond the
cognitive powers of the human mind to solve, as Chomsky has speculated.
>BTW, none of this helps with the liar paradox, IMHO.
Since no one knows, it's hard to say. The liar paradox is caused by the
inconsistent combination of syntactic richness, logical richness and
semantic richness. More exactly, it involves the inconsistency between the
disquotational T-scheme (which governs the word "true") and the
diagonalization property, which allows us to construct a term t such that t
denotes "t is not true".
This means that, for sufficiently rich languages, and for any property P
expressible in the language, we can construct a sentence S such that S is
equivalent to "S lacks property P". This is Goedel's Diagonal Lemma.
The T-sentence is then
(T) "S lacks property P" is true if and only if S lacks property P
and thus, since S is equivalent to "S lacks property P",
S is true if and only if S lacks property P
Now, if property P is the property of being true, then we have,
S is true if and only if S is not true.
This contradiction follows from the T-scheme and the diagonalization
property.
The liar paradox can be tamed within mathematical logic, where it leads to
useful and powerful indefinability theorems. For example, the set of
arithmetic truths is not definable in arithmetic. These cases lead to
interpreted languages which are semantically open: they cannot fully express
their own semantics. The attempt to provide semantically closed languages
has not yet succeeded, and appears to be mathematically impossible (although
Kripke's fixed point theory represents significant progress). And yet
English appears to be semantically closed. This is the problem, and why Barb
Knox originally wondered whether expressively rich semantically closed
languages are possible. Dialetheists claim that semantically closed
languages must be dialetheic languages---inconsistent languages. Perhaps
this is the right way, but I am sceptical.
--- Jeff
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