FOM: Re: Having truth-values

[Roger Bishop Jones]

In response to: William Tait Wednesday, October 18, 2000 8:30 PM


<snip>
> On these grounds, I think that the assertion that CH, or its
> truth-value is `indeterminate' can be nothing more than a
> sociological prediction that such axioms will never be found. I don't
> know how risky that prediction is.

There is a quite precise technical sense in it is possible for a sentence
in a classical logical language to have no definite truth value.

The semantics of an applied first order language can be given by
identifying the interpretations which correspond to the
intended meaning of the non-logical symbols.
A sentence will then be true (simpliciter) in this language
if it is true (under the semantics of first order logic)
in the intended interpretation.

In the case that there is just one intended interpretation,
this will result in all sentences (by which is meant "closed formula")
will have a truth value, even if it is not decided by any reasonable
set of axioms and its truth value is not known.

However, in many cases we may prefer to have a "loose" semantics
(or may find it difficult to come up with anything else).
To do this we identify more than one intended interpretation.
For example I might give a semantics to the language of first order
set theory by specifying that the intended interpretations are the
well-founded models of ZFC (hypothesising that there are some).

Sentences are then to be understood as asserting some condition
under an unspecified member of this collection of interpretations.
In this example there will be statements which have a definite truth value
but which are not provable or disprovable in ZFC
(I believe all arithmetic claims will have a definite truth value).
There will also be statements that are have no definite truth value
(e.g. large cardinal axioms, and CH) since they will be true in some of the
"intended interpretations" but not in others.
We still have "CH or not CH"
but the truth value of CH really is indeterminate under this semantics.

The use of precise but loose semantics is practised
in formal specification of computer systems, its not something I came
up with to be argumetative.
It is a practice which is encouraged by the Z specification language
(but generally, "overspecification" is thought to be a bad thing, there is
something analogous to occams razor operating - don't say any more
than is strictly necessary)
When we were using the Cambridge HOL tool for the
formal verification of hardware (back in the 1980's) against specifications
written in Z a new feature was introduced into the methods of definition
supported by the HOL proof tool.
This allowed the introduction of a set of new constants satisfying some
arbitrary but provably consistent property to be treated as if it were a
definition
rather than a new axiom.

When your logical system is a "foundation system" you are able
to extend the language by introducing new constants using definitions.
In practice (e.g. in engineering applications) the use of arbitrary
conservative
extensions is more useful, these can be understood as "loose" definitions.
The effect of allowing such extensions is to extend the indeterminacy
which affects formulae with free variables
(e.g. the indeterminacy of the formula "x=y" where x and y are variables)
to sentences involving loosely defined constants
(constants introduced by conservative extension rather than by definition).

The cultural prejudice in favour of loose specification in some application
circles is matched in more mathematical cultures by the attitudes of
category theorists and structuralists.
For example, category theorists in foundational debates may criticise the
use in set theory of a specific arbitrary method of representing ordered
pairs.
When developing set theory using a tool like Cambridge HOL,
ordered pairs can be introduced in a way which leaves loose the specific
construction but obtains only the required properties.
(unfortunately "<v,w>=<x,y> iff v=x and w=y" is not enough)
If this is done the truth value of <x,y> = {{x},{x,y}} becomes
indeterminate.

Roger Jones
RBJones at RBJones.com