Jeff Rubard
2010-02-01 22:35:10 UTC
From Non Mussolini *muy* bastardo:
http://jarda.peregrin.cz/mybibl/PDFTxt/484.pdf
-----
The Laws of the Laws of Thought
“History continually effects totalisations of totalisations” — Sartre,
Critique of Dialectical Reason
One of the blogs I’m currently very interested in is Metalogic is
Ethics, run by a graduate student in Philadelphia. John and I agree
about the importance of formal concerns to “Continental” issues, and
we are both thankful for the liberalizing influence of Badiouianism on
that interface without quite having the grateful consciousness of
disciples. Something we’ve discussed is the significance of second-
order logic for considering dialectics: although I doubt anyone ever
completely agrees with what I say, hopefully this work-up of my
position on that topic will mark out an area broad enough to be
occupied by a group larger than a party of one.
To put it mildly, formal logicians are not Hegel fans; going back to
Russell’s turn away from British Idealism, formal logic has been
informally defined as everything Hegel’s “logic” was not. The closest
any formal thinkers have gotten to appropriating Hegelian themes is
“dialetheism”, the Australasian philosophical movement which holds
that paraconsistent logics (which have rules for reasoning with
contradictions that are more sophisticated than the traditional
“principle of explosion”) demonstrate that it’s coherent to believe
there are real contradictions, “contradictions in the object” as a
traditional dialectician might say. People like Graham Priest have
mentioned Hegel in connection with this project, as well they might;
but I think the real story of Hegel and logic is a little bit more
complicated than simply accepting dialetheism. The story begins, as
well it might, with Plato.
I’m no Plato scholar, but I imagine it’d be an uncontroversial
observation that Platonic dialogues operate in this fashion: Socrates
gets one of his interlocutors to produce a description of an Idea, and
then they collectively reason about the consequences of that Idea for
reasoning with Ideas generally, and the consequences of reasoning with
Ideas generally for the employment of that Idea. This is clearly a
“second-order” process of reasoning, but those less familiar with
formal logic may not know there’s no need to leave “second-order” as
an inexact descriptor: there is “second-order” logic. First-order
logic allows the reasoner to quantify over objects in the universe of
discourse, which produces universal and existential statements about
the application of predicates to those objects: second-order logic
allows one to quantify over those predicates, producing universal and
existential statements about predication in general.
Sounds great, huh? In fact, using second-order logic one can describe
all mathematical concepts without resorting to set-theoretic axioms,
as Frege did with his second-order logic, his “laws of the laws of
nature”. Or at least you could, if that didn’t produce paradoxes like
Russell’s “set of all barbers that shave themselves”. Some people have
recently tried to salvage Frege’s logicism from the paradoxes (by
restricting his Basic Law V), but that’s not quite what I want to talk
about here — although his mathematical “platonism” may shed some light
on the original article, he was certainly no dialectician. No, what I
aim to talk about is the relationship between Platonic and Hegelian
dialectics in light of second-order considerations.
Between Plato and Hegel, we have Kant’s “Transcendental Dialectic”,
his logic of metaphysical illusion. Unlike the understanding, which
operates by subsuming intuitions under concepts (much as constants are
included in the extension of predicates), Kant’s Reason works with
Ideas (concepts involving totality, the unconditioned, and the
perfect) and gets entangled in antinomies and contradictions on
account of their character. I guess you could anachronistically
characterize Kant as a Quinean of sorts, interested in restricting
theoretical cognition to “first-order” concepts of the understanding,
and I think that would not be an unreasonable way to gloss the
influence of modern science on modern philosophy which culminated in
his work.
Hegel accepts the results of Newtonian physics, and the constraints of
experimental method on philosophy of nature: but unlike Kant he held
no truck with skepticism, and wanted a modern version of Plato’s
productive dialectic. Consequently, Hegel returned to the second-
order, and his dialectic is much more nearly a process of moving back
and forth between orders of abstraction than cookie-cutter application
of a “thesis-antithesis-synthesis” schema. That contradictions seem to
occur in this process is perhaps an indication of the fact that this
second-order reasoning falls prey to the affliction of any formal
system powerful enough to generate the principles of mathematics,
incompleteness: the interrelation between dialectically demonstrated
truths cannot be systematized axiomatically.
Maybe that’s tendentious — but there’s an interesting feature of
second-order logic which I think is quite illuminating for considering
Hegel and Hegelianism. “Standard semantics” for second-order logic is
incomplete, but the logician Leon Henkin (who developed the proof of
first-order completeness which is commonly taught today) devised a
“Henkin semantics” for second-order logic which is complete. Henkin
semantics only allows second-order statements over defined first-order
totalities: this is similar to the restriction in the Zermelo-Frankel
“axiom of comprehension” which replaced Frege’s unrestricted law of
comprehension. If we consider Hegel as second-order, perhaps Henkin
semantics is a good “model” of Left Hegelianism (where dialectic
reasoning is preserved but only in reference to “real abstractions”,
concepts that are incarnated by material realities); Right Hegelianism
preserves the full expressive power of second-order dialectic (Henkin
second-order logic is no more expressive than first-order logic), but
at the cost of rational cogency and lack of “mystification”.
I’m definitely floating with the universe by the end of this line of
thought, but there is a more recent “dialectical” concept that reminds
me very much of another thought I independently had about two
approaches to logic and geo-linguistic correspondences to them. One of
Sartre’s major concepts in his Critique of Dialectical Reason is the
“practico-inert”, elements of social organization which are resistant
to the subjective projects of praxis and form the ground of social
struggle. Now, in 1970s logic a distinction was made between “Western
model theory” and “Eastern model theory”; the former being exemplified
by the work of Alfred Tarski and his students at Berkeley, the latter
being exemplified by the work of Abraham Robinson and his students at
Yale.
In pure logic there’s not much heavy weather to be made over this
distinction; both Tarski and Robinson were from Central Europe (by way
of Palestine and Britain in Robinson’s case), and Robinson had quite
happily taught at UCLA. However, I think the distinction is not
without its geographical aptness. Western model theory was more
heavily “logical”, and focused on the significance of models for
definitions of logical consequence and other “abstract” features of
logic. Eastern model theory was more heavily “mathematical”, and
focused on the significance of models for proving things about
mathematical theories and other “concrete” formal phenomena. This
parallels a distinction in discursive styles between the western and
eastern US. In the West, people have traditionally been quite fond of
solecism and sophistry as devices for getting points across
indirectly, whereas in the East there’s more of a focus on
ineliminable realities bound up with noble sentiment: a Western
raconteur might be “temporarily disabled” by a stunning woman, whereas
an Easterner might be discomfited by the plight of personally
aggressive people with “disabilities”.
It seems to me that this “Eastern model theory” seems to capture the
presence in language of the practico-inert which Sartre touches on at
one point, the crude and tasteless plays on words you just can’t get
away from, low “interpretations” of signifiers which distract us from
drawing the appropriate conclusions. But lest this seem to be mere
provincial boastfulness, I will mention that the reason people don’t
use this distinction in logic anymore is that all new work in model
theory today is “Eastern” model theory, leading to Wilfrid Hodges’
redefinition of the subject as “universal algebra minus fields” rather
than Chang and Keisler’s “universal algebra plus logic”. And, like
Sartre says, as undesirable as many aspects of the practico-inert are
from the standpoint of revolutionary subjectivity it’s a fundamental
existentiale of sociality which you can’t get away from.
-----
http://en.wikipedia.org/wiki/Georg_Wilhelm_Friedrich_Hegel
http://en.wikipedia.org/wiki/Karl_Marx
http://jarda.peregrin.cz/mybibl/PDFTxt/484.pdf
-----
The Laws of the Laws of Thought
“History continually effects totalisations of totalisations” — Sartre,
Critique of Dialectical Reason
One of the blogs I’m currently very interested in is Metalogic is
Ethics, run by a graduate student in Philadelphia. John and I agree
about the importance of formal concerns to “Continental” issues, and
we are both thankful for the liberalizing influence of Badiouianism on
that interface without quite having the grateful consciousness of
disciples. Something we’ve discussed is the significance of second-
order logic for considering dialectics: although I doubt anyone ever
completely agrees with what I say, hopefully this work-up of my
position on that topic will mark out an area broad enough to be
occupied by a group larger than a party of one.
To put it mildly, formal logicians are not Hegel fans; going back to
Russell’s turn away from British Idealism, formal logic has been
informally defined as everything Hegel’s “logic” was not. The closest
any formal thinkers have gotten to appropriating Hegelian themes is
“dialetheism”, the Australasian philosophical movement which holds
that paraconsistent logics (which have rules for reasoning with
contradictions that are more sophisticated than the traditional
“principle of explosion”) demonstrate that it’s coherent to believe
there are real contradictions, “contradictions in the object” as a
traditional dialectician might say. People like Graham Priest have
mentioned Hegel in connection with this project, as well they might;
but I think the real story of Hegel and logic is a little bit more
complicated than simply accepting dialetheism. The story begins, as
well it might, with Plato.
I’m no Plato scholar, but I imagine it’d be an uncontroversial
observation that Platonic dialogues operate in this fashion: Socrates
gets one of his interlocutors to produce a description of an Idea, and
then they collectively reason about the consequences of that Idea for
reasoning with Ideas generally, and the consequences of reasoning with
Ideas generally for the employment of that Idea. This is clearly a
“second-order” process of reasoning, but those less familiar with
formal logic may not know there’s no need to leave “second-order” as
an inexact descriptor: there is “second-order” logic. First-order
logic allows the reasoner to quantify over objects in the universe of
discourse, which produces universal and existential statements about
the application of predicates to those objects: second-order logic
allows one to quantify over those predicates, producing universal and
existential statements about predication in general.
Sounds great, huh? In fact, using second-order logic one can describe
all mathematical concepts without resorting to set-theoretic axioms,
as Frege did with his second-order logic, his “laws of the laws of
nature”. Or at least you could, if that didn’t produce paradoxes like
Russell’s “set of all barbers that shave themselves”. Some people have
recently tried to salvage Frege’s logicism from the paradoxes (by
restricting his Basic Law V), but that’s not quite what I want to talk
about here — although his mathematical “platonism” may shed some light
on the original article, he was certainly no dialectician. No, what I
aim to talk about is the relationship between Platonic and Hegelian
dialectics in light of second-order considerations.
Between Plato and Hegel, we have Kant’s “Transcendental Dialectic”,
his logic of metaphysical illusion. Unlike the understanding, which
operates by subsuming intuitions under concepts (much as constants are
included in the extension of predicates), Kant’s Reason works with
Ideas (concepts involving totality, the unconditioned, and the
perfect) and gets entangled in antinomies and contradictions on
account of their character. I guess you could anachronistically
characterize Kant as a Quinean of sorts, interested in restricting
theoretical cognition to “first-order” concepts of the understanding,
and I think that would not be an unreasonable way to gloss the
influence of modern science on modern philosophy which culminated in
his work.
Hegel accepts the results of Newtonian physics, and the constraints of
experimental method on philosophy of nature: but unlike Kant he held
no truck with skepticism, and wanted a modern version of Plato’s
productive dialectic. Consequently, Hegel returned to the second-
order, and his dialectic is much more nearly a process of moving back
and forth between orders of abstraction than cookie-cutter application
of a “thesis-antithesis-synthesis” schema. That contradictions seem to
occur in this process is perhaps an indication of the fact that this
second-order reasoning falls prey to the affliction of any formal
system powerful enough to generate the principles of mathematics,
incompleteness: the interrelation between dialectically demonstrated
truths cannot be systematized axiomatically.
Maybe that’s tendentious — but there’s an interesting feature of
second-order logic which I think is quite illuminating for considering
Hegel and Hegelianism. “Standard semantics” for second-order logic is
incomplete, but the logician Leon Henkin (who developed the proof of
first-order completeness which is commonly taught today) devised a
“Henkin semantics” for second-order logic which is complete. Henkin
semantics only allows second-order statements over defined first-order
totalities: this is similar to the restriction in the Zermelo-Frankel
“axiom of comprehension” which replaced Frege’s unrestricted law of
comprehension. If we consider Hegel as second-order, perhaps Henkin
semantics is a good “model” of Left Hegelianism (where dialectic
reasoning is preserved but only in reference to “real abstractions”,
concepts that are incarnated by material realities); Right Hegelianism
preserves the full expressive power of second-order dialectic (Henkin
second-order logic is no more expressive than first-order logic), but
at the cost of rational cogency and lack of “mystification”.
I’m definitely floating with the universe by the end of this line of
thought, but there is a more recent “dialectical” concept that reminds
me very much of another thought I independently had about two
approaches to logic and geo-linguistic correspondences to them. One of
Sartre’s major concepts in his Critique of Dialectical Reason is the
“practico-inert”, elements of social organization which are resistant
to the subjective projects of praxis and form the ground of social
struggle. Now, in 1970s logic a distinction was made between “Western
model theory” and “Eastern model theory”; the former being exemplified
by the work of Alfred Tarski and his students at Berkeley, the latter
being exemplified by the work of Abraham Robinson and his students at
Yale.
In pure logic there’s not much heavy weather to be made over this
distinction; both Tarski and Robinson were from Central Europe (by way
of Palestine and Britain in Robinson’s case), and Robinson had quite
happily taught at UCLA. However, I think the distinction is not
without its geographical aptness. Western model theory was more
heavily “logical”, and focused on the significance of models for
definitions of logical consequence and other “abstract” features of
logic. Eastern model theory was more heavily “mathematical”, and
focused on the significance of models for proving things about
mathematical theories and other “concrete” formal phenomena. This
parallels a distinction in discursive styles between the western and
eastern US. In the West, people have traditionally been quite fond of
solecism and sophistry as devices for getting points across
indirectly, whereas in the East there’s more of a focus on
ineliminable realities bound up with noble sentiment: a Western
raconteur might be “temporarily disabled” by a stunning woman, whereas
an Easterner might be discomfited by the plight of personally
aggressive people with “disabilities”.
It seems to me that this “Eastern model theory” seems to capture the
presence in language of the practico-inert which Sartre touches on at
one point, the crude and tasteless plays on words you just can’t get
away from, low “interpretations” of signifiers which distract us from
drawing the appropriate conclusions. But lest this seem to be mere
provincial boastfulness, I will mention that the reason people don’t
use this distinction in logic anymore is that all new work in model
theory today is “Eastern” model theory, leading to Wilfrid Hodges’
redefinition of the subject as “universal algebra minus fields” rather
than Chang and Keisler’s “universal algebra plus logic”. And, like
Sartre says, as undesirable as many aspects of the practico-inert are
from the standpoint of revolutionary subjectivity it’s a fundamental
existentiale of sociality which you can’t get away from.
-----
http://en.wikipedia.org/wiki/Georg_Wilhelm_Friedrich_Hegel
http://en.wikipedia.org/wiki/Karl_Marx