ONT Re: Model Theory
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John Collier wrote (JC):
Jon Awbrey wrote (JA):
JA, quoting Chang & Keisler:
| We shall say that @S@ is 'inconsistent'
| iff we have @S@ |- q for all sentences q.
| Otherwise, we say that @S@ is 'consistent'.
JC: There are a number of reasons _not_ to adopt this terminology for general logic:
This is the main definition of inconsistency that is used by
just about every standard source that I have ever read, since
they all treat the presence of a contradictory pair in the set
as as a corollarial equivalent. For instance, C&K 1.2.8.3.
| Proposition 1.2.8 [Deductive Closure Properties of Consistent Sets]
|
| 1. If @S@ is consistent and @G@ is the set
| of all sentences deducible from @S@,
| then @G@ is consistent.
|
| 2. If @S@ is maximal consistent and @S@ |- q,
| then q is an element of the set @S@.
|
| 3. @S@ is inconsistent if and only if
| @S@ |- (S & (~S)) for any S in $S$.
|
| 4. Deduction Theorem.
| If @S@ |_| {r} |- q, then @S@ |- (r => q).
|
| C.C. Chang & H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973,
| pages 9-10.
JC: 1) A set of sentences (in the natural, not artificial, sense) can be
inconsistent, even if the inconsistency is not derivable within
any formal system.
I do not get this. Still, I am only concerned with classical logic at this point,
and I would not want to entertain any proof system that is so weak that it cannot
derive all sentences from a contradiction.
JC: To assume otherwise is to beg the question about formalizability of the logic.
This is also a good reason not to refer to logical WFFs as sentences. An example
might be the formalization of Bohm's theory of the atom in paraconsistent logic,
which is arguably a correct representation of Bohm's intentions, but it still
is not satisfiable, and is inconsistent in any intuitive sense.
Does not sound all that relevant here.
I do not see any hard and fast connection
between consistency and satisfiability
unless the formal system is complete,
and I do not see that you can say much
of anything definite about consistency
except in relation to a formalization.
JC: We would want it that any consistent set of sentences is satisfiable, and
allow that if each member is satisfiable, then all the members of the set
are satisfiable, but this usage does not sanction this in general.
Again, I only get a connection between consistent and satisfiable if the system is complete.
Also, the transfer of satisfiability from finite sets to the whole depends on compactness.
JC: This problem turns up in second order problems, and is one of the reasons
that constructivists often do not consider second order logic to be logic.
I don't think this can be decided by definition. As long as this consequence
is recognized, though, and we replace "consistent" above with "satisfiable",
and recall that we are really talking about the intuitive notion of consistency,
then the objection dissolves, since the semantic properties of 2o logic remain
the same, as do all the implications and validity relations. However, one has
to give a clear idea of what one means by satisfiability in 2o logic. Note that
there is no justification in C&K of the notion of satisfiability except the
Extended Completeness Theorem, but nothing analogous is possible for 2o theories,
so it is not scalable to higher order logics. This begs the question against them
from the beginning, despite the fact that we have used them reasonably well at least
since the time of Frege (I would go back to Leibniz, myself). There is a clear idea,
through the concept of an "erasure". Prop Calc, by analogy, should use a similar concept
in order to achieve scalability. In other words, C&K adopt an unduly restrictive paradigm
of what it is to be true.
JC: 2) Traditionally, the law forbidding what C&K call inconsistent sets of sentences
is called the Law of Non-Contradiction, and the formation of the simplest case,
q & ~q is called a contradiction, not an inconsistency.
Nothing is ever forbidden.
Somethings are just false.
The 'Law of Non-Contradiction' is a statement about single sentences,
and all it says is that a sentence of the form q & ~q is a falsehood.
Inconsistency is a property of a theory (a set of sentences), earned
by virtue of claiming among its premisses just a little bit too much.
JC: 3) Starting with Prop Calc (I dislike the term sentential logic; it isn't),
which is complete, points 1) and 2) are unproblematic, but it becomes deceptive
as soon as you introduce second order concepts like identity, and it has led to
a lot of really silly things being said about identity. In particular, these
are confusions between the consequence relation and the deducibility relation.
'A is a consequence of B' is not in general coextensive with 'A is deducible from B',
given the long standing intuitive meanings of both. This confusion reaches laughable
proportions when, for example, someone like S. Kellert argues that determinism is false
because the consequences of deterministic chaos are not fully deducible.
All my long standing intuitive meanings for these things
had their knees lock up and fell over a long time ago.
That is why I must lean on the crutch of formalization.
JC: 4) Tarski's satisfaction notion of truth ("Snow is white" if and only if snow is white.)
is the most widely accepted notion of truth (though some would want to add more,
nobody wants to require less, even deflationists like my friend Paul Horwich).
Now, C&K say:
| We shall say that A is a 'model' of @S@, in symbols, A |= @S@,
| iff every sentence q in @S@ is true in A.
|
| The set @S@ of sentences is said to be 'satisfiable'
| iff it has at least one model.
The circularity is a little too immediate for me,
but at least we can applaud them for not being
inconsistent :-)
Sorry, I just do not see the circle.
$S$ is a nonempty set of primitive sentences.
Pow($S$) is the set of all subsets A c $S$.
<$S$, ~, &> is the set of compound sentences.
The relation (A |= q) c Pow($S$) x <$S$, ~, &>
is defined for model A and sentence q.
The relation (A |= @S@) c Pow($S$) x Pow(<$S$, ~, &>)
is defined for model A and theory @S@.
The property Sat : Pow(<$S$, ~, &>) -> B
is defined for theory @S@.
Where do you see a circle?
Jon Awbrey
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