[hist-analytic] Vacuity

Roger Bishop Jones rbj at rbjones.com
Thu Jun 11 15:02:50 EDT 2009


On Wednesday 03 June 2009 19:50:16 Jlsperanza at aol.com wrote:

>(a) Grice -- 'vacuous names': two quotes by Grandy.
>(b) Strawson --  on 'all' -- citing Grice. -- and Carroll citing Jones.

There are a lot of variations of terminology and in other
matters in logic, depending on background (where you learnt
about logic), and mine is not particularly philosophical.
From time to time I find myself wondering what philosophers
are talking about.

e.g. what does "Existential Scope of singular terms" mean?
I know what the scope of an existentially quantified variable
is, but that's presumably not the same thing.

This term EXISTENTIAL GENERALISATION sounds like an oxymoron
to me! 
A much more transparent term is EXISTENTIAL INTRODUCTION,
which says exactly what it does.

I see reading on in Strawson from the passages
Speranza cited, that Strawson was critical of Russell's
theory of descriptions.  I am broadly in agreement with
his position insofar as it relates to natural languages,
and I have also in other contexts been inclined to the
view that statements often presuppose things on which
their having a truth value depends without properly
being held to assert them.
Carnap's internal/external distinction gives us very
many examples of this, for an answer to an internal
question may be said to presuppose but not assert
a positive answer to the relevant externsl questions.
Thus, to be more concrete, in mathematics, the assertion
that there exist infinitely many prime numbers
is a true mathematical statement which might be said
to presuppose but not to assert that numbers exist.

However, technically it is inconvenient to consider
statements whose presuppositions are not satisfied
as lacking a truth value.  This may be the right
way to read natural languages but for the purposes
of mathematics it is better to arrange that sentences
are always true or false. 

I checked out Strawson, chapter 6 in connection with
my Aristotelian modelling.
There is a minor problem in that it is "Traditional"
rather than "Aristotelian" and its not clear what really
came from Aristotle.

Strawson has these rules of direct inference, and I have
included a smattering of these in my formalisation.
These are supposed to be used to prove the syllogisms,
I haven't paid much attention to how they are derived
by Aristotle, but maybe another time.
There is a document by Spade out there with a thumbnail
history in which there is a concise story about this.

[Strawson]
>Since  there are four
>figures, there are altogether 256 possible moods of the  syllogism.
>Of these 256 only twenty-four are recognized as valid;

I had only 19, which I believe are the ones cited by Aristotle.
The extra 5 in Strawson all exhibit the existential fallacy
(easy to spot since all the premises are universal but the
conclusion is existential).

I have updated my document, and added in the extra five,
which are provable in my second model of the syllogism
using the same general proof tactic.
I managed to find the names for them, which I think
Strawson omits, which is a shame for they are informative.

The new version, at:

rbjones.com/rbjpub/pp/doc/t028.pdf

also now contains the two models which I promised
but had not included in the last version, viz:

    modal syllogism (with existential fallacy)
and
    the whole caboodle: modal sysllogism
    	with izz and hazz predication and existential fallacy..    

though still the emphasis is on the formal modelling and
the philosophical payoff is yet to come.

>THE  ORTHODOX CRITICISMS OF THE SYSTEM

I have to say that this seems rather laboured to me.
In constructing my models of the syllogism, I considered
very briefly adding existential import to the universal.
I immediately ruled this out because it violated the
square of opposition, and concluded that `empty' subjects
would have to be outlawed.
I never considered the other working alternative which
Strawson cites, it lacks plausibilty.

However, Strawson's interest is in the match with
ordinary language, mine is in the match with Aristotle.
I'm not sure that there is any point in approaching
the syllogism in this way these days, for it is clearly
hopeless for giving a good account of ordinary language,
and its interest lies in its place in the history
of logic and of analytic philosophy.
Furthermore, it seems pretty unlikely that Aristotle
ever though of it in terms of ordinary language.
It was for him a tool for doing demonstrative
science (and hence his attitude to logic appears
in this respect closer to Carnap's than the work
of contemporary mathematical or philosophical
logicians, who are primarily engaged in metatheory).

Strawson does not consider Aristotle's metaphysics,
and I believe that this provides support for (what I
suppose to be) the usual way of resolving the problem
of the existentials.

If you look only at the syllogism it seems natural
to read the terms in the syllogism as sets, and
in that case its no so easy to justify the exclusion
of the empty set.

However, when you look at the Metaphysics you get
a more complex idea of what terms are, which is
equally compatible with the syllogism and provides
a kind of explanation of the existential position.

In this view the terms are sets of individuals,
which latter it is convenient to consider as
singleton sets since this make the explanations
shorter.  In that case, "izz" is just set inclusion,
which is consistent with the obvious reading
of the syllogism.

But for Aristotle "izz" isn't merely set inclusion,
for the sets here are part of something like
a taxonomy (either of substance of some category
of attributes).  They do not correspond to
aribitrary descriptions but to definitions,
about which Aristotle is a bit picky.
When you consider this as a taxonomy it's natural
enought to expect all the defined concepts to
be non-empty.

When we look at "hazz" the situation is different.
Here we do not need to insist on non-empty extensions
(and my last model does not) because, in effect,
syllogisms never quantify over the extension of
an attribute.  If an individual attribute is
the subject of a predication then the predication
must be an "izz", and the syllogism rather than
saying anything about the objects in the extension
of the attribute is simply stating that the
attribute is IN some universal, e.g. red is a colour.
Aristotle's logic+metaphysics as I understand it at
present (mainly at second hand) doesn't allow you
to say "All red things are coloured".

So, by making clear that the sets whose extensions
must be non-empty are rather special and reasonably
required to be non-empty, the Metaphysics helps
in understanding the syllogism.  However, I think
futher analysis will reveal that the syllogism
interpreted as it is in the Metaphysics is even
more limited than one might otherwise have suspected.
For example, you can't really talk about universals.
To do this you would need a term whose extension
included only universals, but all terms are sets of
individuals.  You can have a predicate SUBSTANCE
which is truly predicable of every substance,
individual or otherwise. But it is just the set
of all individual substances.

VACUOUS

There is a terminological difficulty here which
still has me guessing the best way to talk about
the existential fallacy.

Depending on how you think about this the terminology
has opposite meaning.
If you think in terms of extension, then a predicate
with an empty extension may naturally be called
vacuous.
But a predicate with empty extension is always false,
and hence contradictory.

On the other hand, a concept which is empty of content
may also be called vacuous, and these are the concepts 
which are always true, and hence whose extension is
the Universe rather than the empty set.

Being devoid of content is exactly opposite to
having an empty extension, and there is a dilemma
about which of these we should consider vacuous.
My own inclination is to consider content as the
important stuff and count the everywhere true
predicate as vacuous.  Perhaps, by way of both
having and eating our cake, we should use the term
vacuous when speaking of predicates and empty only
when speaking of sets, so that a predicate can be
said to be vacuous iff its extension is the universe
and a predicate is said to be contradictory iff its
extension is empty, placing vacuous and empty at
opposite ends of the spectrum.

I have been greatly entertained and engrossed by
my efforts to formalise Aristotle's Logic and
Metaphysics. It seems a productive way of working
for me, and so I am inclined to do more, and to
make it a principle method of research for the
historical aspects of what I am trying to do for
Metaphysical Positivism.  Translating it into
more widely intelligible philosophy is a challenge
for the future.

Roger Jones




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