Re: syllogism
From: David Longley (David_at_longley.demon.co.uk)
Date: 10/01/04
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Date: Fri, 1 Oct 2004 14:45:04 +0100
In article <%z17d.20748$MD5.1110057@news20.bellglobal.com>, Wolf
Kirchmeir <wwolfkir@sympatico.ca> writes
>patty wrote:
>
>[...]
>> Well i see where we are stumbling over the rather ambiguous (A--
>>>B). I was interpreting it as a first order term with *one* property
>>and no quantification (Fa => Fb); but now i see you meant it as
>>(for *all* properties F, Fa <=> Fb) which *is* a coding of the
>>definition of the identity of a and b:
>
>Sorry, I thought A-->B was a commonly understood representation
>of "If A, then B", or "A implies B." I also think that since I used this
>symbolism in the context of implication, it should've been clear to
>you that's what I intended. I use => only as "equal to or greater
>than", never as a logic operator.
>
>Thus A-->B is false if "A true", and "B false", and true otherwise. If
>A, B are truthfunctions, then sometimes (A-->B ANDF B-->A), and
>sometimes not - depends on A, B. IOW, [(A==B) iff (A-->B AND B--
>>A] When it comes to truthfunctions, that does not mean A, B are
>indistinguishable. It means that either can be transformed into the
>other, but that operation is meaningless unless A, B were
>distinguishable to start with. Right?
>
>Whether it makes sense to say that A, B have the same properties
>in this case I'll leave to other thinkers. I'm getting leery of the term
>"property."
>
The law of extensionality is what's missing in this discussion, and I
suspect Patty of a little obfuscation or at least a little foggy writing
here as I suspect she does know the intensional nature of properties.
'The notion of a property is one of various notions,
called INTENSIONAL, that depend thus on the nebulous
notion of meaning. Other examples are necessity,
possibility, and idioms of propositional attitude such
as belief, hope, regret.'
Quine (1985)
The Time of My Life
Quine does a nice comparison of properties vs classes in Quiddities:
'If it makes sense to speak of properties, it should
make clear sense to speak of sameness and differences of
properties; but it does not. If a thing has this
property and not that, then certainly this property and
that are different properties. But what if everything
that has this property has that one as well, and vice
versa? Should we say that they are the same property? If
so, well and good; no problem. But people do not take
that line. I am told that every creature with a heart
has kidneys, and vice versa; but who will say that the
property of having a heart is the same as that of having
kidneys?
In short, coextensiveness of properties is not seen as
sufficient for their identity. What then is? If an
answer is given, it is apt to be that they are identical
if they do not just happen to be coextensive, but are
necessarily coextensive. But NECESSITY, q.v., is too
hazy a notion to rest with.
We have been able to go on blithely all these years
without making sense of identity between properties,
simply because the utility of the notion of property
does not hinge on identifying or distinguishing them.
That being the case, why not clean up our act by just
declaring coextensive properties identical? Only because
it would be a disturbing breach of usage, as seen in the
case of the heart and kidneys. To ease that shock, we
change the word; we speak no longer of properties, but
of CLASSES......
We must acquiesce in ordinary language for ordinary
purposes, and the word 'property' is of a piece with it.
But also the notion of property or its reasonable
facsimile that takes over, since these contexts never
hinge on distinguishing coextensive properties. One
instance among many of the use of classes in mathematics
is seen under DEFINITION, in the definition of number.
For science it is classes SI, properties NO.'
W. V. O. Quine (1987)
Classes versus Properties
QUIDDITIES:
See "Fragments..." for more details, but the following should give the
basic idea:
'The new logic is distinguished from the old not only by the
form in which it is presented but chiefly also by the
increase of its range....The only form of statements
(sentences) in the old logic was the predicative form:
"Socrates is a man," "All (or some) Greeks are men." A
predicate-concept or property is attributed to a subject-
concept. Leibniz had already put forward the demand that
logic should consider sentences of relational form. In a
relational sentence such as, for example, "a is greater than
b," a relation is attributed to two or more objects, (or, as
it might be put, to several subject-concepts). Liebniz's idea
of a theory of relations has been worked out in the new
logic. The old logic conceived relational sentences as
sentences of predicative form. However, many inferences
involving relational sentences thereby become impossible. To
be sure, one can interpret the sentence "a is greater than b"
in such a way that the predicate "greater than b" is
attributed to the subject a. But the predicate then becomes a
unity; one cannot extract b by any rule of inference.
Consequently, the sentence "b is smaller than a" cannot be
inferred from this sentence. In the new logic, this inference
takes place in the following way: The relation "smaller than"
is defined as the "converse" of the relation "greater than."
The inference in question then rests on the universal
proposition: If a relation holds between x and y, its
converse holds between y and x. A further example of a
statement that cannot be proved in the old logic: "Wherever
there is a victor someone is vanquished." In the new logic,
this follows from the logical proposition: If a relation has
a referent, it also has a relatum. Relational statements are
especially indispensable for the mathematical sciences. Let
us consider as an example the geometrical concept of the
three-place relation "between" (on an open straight line).
The geometrical axioms "If a lies between b and c, b does not
lie between c and a" can be expressed only in the new logic.
According to the predicative view, in the first case we would
have the predicates "lying between b and c" and "lying
between c and a". If these are left unanalyzed, there is no
way of showing how the first is transformed into the second.
If one takes the objects b and c out of the predicate, the
statement "a lies between b and c" no longer serves to
characterise only one object, but three. It is therefore a
three-place relational statement....
Restriction to predicate-sentences has had disastrous effects
on subjects outside logic. Perhaps Russell is right when he
made this logical failing responsible for certain
metaphysical errors.....Above all, we may well assume that
this logical error is responsible for the concept of absolute
space. Because the fundamental form of a proposition had to
be predicative, it could only consist in the specification of
the position of a body. Since Leibniz had recognized the
possibility of relational sentences, he was able to arrive at
a correct conception of space: the elementary fact is not
position of a body but its positional relations relative to
other bodies. He upheld the view on epistemological grounds:
there is no way of determining the absolute position of a
body, but only its positional relations. His campaign in
favor of the relativistic view of space, as against the
absolutistic views of the followers of Newton, had as little
success as his program for logic.
Only after two hundred years were his ideas on both subjects
taken up and carried through: in logic with the theory of
relations (De Morgan 1858; Pierce 1870), in physics with the
theory of relativity (anticipatory ideas in Mach 1883;
Einstein 1905).'
R. Carnap
The Old and the New Logic (1930)
In A.J. Ayer (ed) Logical Positivism (1959)
'.. consists in characterizing the predicates by their
extension instead of according to their content. To each
predicate corresponds a certain "class" of objects,
consisting of all objects for which the predicate holds. The
case of a class containing no object is of course not
excluded here. Classes are now to be taken as the entities
dealt with by the calculus, which in this interpretation will
be called the calculus of classes.
D. Hilbert & W. Ackermann (1950)
The Principles of Mathematical Logic p.46
'We think of a science as comprising those truths which are
expressible in terms of 'and', 'not', quantifiers, variables,
and certain predicates appropriate to the science in
question....To specify a science, within the described mold,
we still have to say what the predicates are to be, and what
the domain of objects is to be over which the variables of
quantification range.'
W.V.O. Quine (1954)
The Scope and Language of Science
The Ways of Paradox and other essays p.242
'Thus we have arrived at something fundamental: our
conventions regarding the use of the words "not" and "or" is
such that in asserting the two propositions "object A is
either red or blue" and "object A is not red," I have
implicitly already asserted "object A is blue." This is the
essence of so-called *logical deduction*. It is not then, in
any way based on real connections between states of affairs,
which we apprehend in thought. On the contrary, it has
nothing at all to do with the nature of things, but drives
from our manner of speaking about things. A person who
refused to recognize logical deduction would not thereby
manifest a different belief from mine about the behaviour of
things, but he would refuse to speak about things according
to the same rules as I do. I could not convince him, but I
could refuse to speak with him any longer, just as I should
refuse to play chess with a partner who insisted on moving
the bishop orthogonally.
What logical deduction accomplishes, then, is this: it makes
us aware of all that we have implicitly asserted - on the
basis of conventions regarding the use of language - in
asserting a system of propositions, just as, in the above
example, "object A is blue" is implicitly asserted by the
assertion of the two propositions "object A is red or blue"
and "object A is not red."
In saying this we have already suggested the answer to the
question, which naturally must have forced itself on the mind
of every reader who has followed our argument: if it is
really the case that the propositions of logic are
tautologies, that they say nothing about objects, what
purpose does logic serve?
..logical propositions, though being purely tautologous, and
logical deductions, though being nothing but tautological
transformations, have significance for us because we are not
omniscient. Our language is so constituted that in asserting
such and such propositions we implicitly assert such and such
other propositions - but we do not see immediately all that
we have implicitly asserted in this manner. It is only
logical deduction which makes us conscious of it.
If I have succeeded in clarifying somewhat the role of logic,
I may now be brief about the role of mathematics. The
propositions of mathematics are of exactly the same kind as
the propositions of logic: they are tautologous, they say
nothing at all about the objects we want to speak about, but
concern only the manner in which we want to speak of
them....We become aware of meaning the same by "2+3" and by
"5", by going back to the meanings of "2," "3," "5," "+," and
making tautological transformations until we just see that
"2+3" means the same as "5". It is such successive
tautological transformation that is meant by "calculating";
the operations of addition and multiplication which are
learned in school are directives for such tautological
transformation; every mathematical proof is a succession of
such tautological transformations. Their utility, again, is
due to the fact that, for example, we do not by any means see
immediately that we mean by "24 x 31" the same as by "744";
but if we calculate the product "24 x 31", then we transform
it step by step, in such a way that in each individual
transformation we recognize that on the basis of the
conventions regarding the use of the signs involved (in this
case numerals and the signs "+" and "x") what we mean after
the transformation is still the same as what we meant before
it, until finally we became consciously aware of meaning the
same by "744" and by "24 x 31."
..at first glance it is difficult to believe that the whole
of mathematics, with its theorems that it cost such labour to
establish, with its results that so often surprise us, should
admit of being resolved into tautologies. But there is just
one little point which this argument overlooks: it overlooks
the fact that we are not omniscient. An omniscient being,
indeed, would at once know everything that is implicitly
contained in the assertion of a few propositions. IT would
know immediately that on the basis of the conventions
concerning the use of the numerals and the multiplication
sign, "24 x 31" is synonymous with "744". An omniscient being
has no need for logic and mathematics. We ourselves, however,
first have to make ourselves conscious of this by successive
tautological transformations, and hence it may prove quite
surprising to us that in asserting a few propositions we have
implicitly also asserted a proposition which seemingly is
entirely different from them, or that we do mean the same by
two complexes of symbols which are externally altogether
different.'
H Hahn (1933)
Logic, Mathematics and Knowledge of Nature
In Ayer (Ed) Logical Positivism (1959)
'At first the problem of mind was ontological and linguistic.
With the passing of mind as substance, there remained a
twofold problem of mentalistic language: syntactic and
semantic. The distinctive syntactic trait of mentalistic
discourse was the content clause 'that p'. This obstructed
extensionality: that is, the substitutivity of identity and
more generally the interchangeability of all coextensive
terms and clauses salva veritate. It obstructed classical
predicate logic as a universal theoretical framework. Now
this quarter of the mind problem is in a fair way to
dissolution. Quotational treatment of propositional attitudes
de dicto delivers them to the extensional domain of predicate
logic, thanks to the reduction of quotation to spelling.
Propositional attitudes de re, on the other hand, we
downgraded.
So we see the attitudes de dicto reconciled syntactically
with extensional logic. A single language, regimented in
predicate logic, can take them in stride along with natural
science. The residual oddity of these mentalistic predicates
de dicto is purely semantic: they do not interlock
productively with the self-sufficient concepts and causal
laws of natural science.
Still the mentalistic predicates, for all their vagueness,
have long interacted with one another, engendering age-old
strategies for predicting and explaining human action. They
complement natural science in their incommensurable way, and
are indispensable both to the social sciences and our
everyday dealings. Read Dennett and Davidson.'
W. V. O. Quine (1992)
Intension
The Pursuit of Truth p.72-73
Note - "incommensurable way" - this is the part of "the double standard"
of anomalous monism (and research) that few really grasp the
significance of - hence my frequent references to "Two Dogmas of
Empiricism".
I thought the following worth repeating too:
'The first-order predicate calculus is an extensional logic
in which Leibniz's Law is taken as an axiomatic principle.
Such a logic cannot admit 'intensional' or 'referentially
opaque' predicates whose defining characteristic is that they
flout that principle.'
U. T. Place (1987)
Skinner Re-Skinned P. 244
In B.F. Skinner Consensus and Controversy
Eds. S. Modgil & C. Modgil
But I bet none of this will make any difference to what is posted by
most folk here. It has all been posted in the past, as has much else
besides but they insist on having it rehashed. Here's just one example:
<http://groups.google.com/groups?selm=spr961206123219-4437@kauri.vuw.ac.n
z>
What does this tell one other than that people have very short memories
and aren't really here for much more than post to post verbal jousting?
<g>
Kind regards,
-- David Longley http://www.longley.demon.co.uk/Frag.htm
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