Jeff Rubard
2010-02-06 20:01:47 UTC
A Marxist Theory of Truth
If you were in a charitable mood, I suppose you might describe my
intellectual project as moving analytic philosophy to its Marxist
moment — with the understanding that this project might not go off
without a hitch. At any rate, one of my core intellectual principles
is something like a Marxist theory of truth, and I’d like to explain
what that entails. Although I think it’s a minor scandal that “new
theories of truth” like Kripke’s and the revision theory are not
widely studied for their philosophical implications, as opposed to
going around and around disquotation, I don’t myself have anything
clever to say about them at the moment and so my remarks here will be
“philosophical” rather than formal.
Donald Davidson once tried (rather unsuccessfully, in the eyes of
most) to show that “coherence implies correspondence”, that the idea
of truth as adequatio falls out rather trivially from a logically well-
integrated web of belief. However, I think the implication in the
opposite direction is not quite as trivial as people make it out to
be: getting any use out of the idea of true statements as mirroring
facts actually requires going quite far into “ideological” features of
discourse. Why? When we assess an utterance for its truth or falsity,
we are interested in “what is said” — not just the surface form of the
words, but the concrete and relevant meaning of them. In formal
theories what is said is stipulatively clear, but in natural language
a whole host of “pragmatic” phenomena combine to make figuring out the
contribution of a statement to communication difficult.
Principles for dealing with these phenomena model social life — what
it is rational to think someone said in a context depends on one’s
model of their position within society and one’s model of their
understanding of that position. So much so, in fact, that I think
there is some point to simply identifying the real meaning of a
statement with its role as a “move” within society, as a contribution
to social action. This is not a “sceptical solution to sceptical
doubts” along the lines of Kripke’s interpretation of Wittgenstein:
although custom obviously plays a role in the role of words as an
element of practices, we are very far from being required to view
tradition as the fundament of “social meaning” and to exclude the role
of novelty (think of Wittgenstein’s picture of language as a city,
with narrow old streets and regularly plotted suburbs). We are also
far from having to view language as “smooth and homogenous” in the
words of Richard Rorty: the structure of “social practice” involving
meaning is not any simpler than the structure of society in general,
with its various divisions and conflicts.
Obviously, for communication to occur an utterance must coordinate ego
and alter: someone making an “offer you can’t understand” is
subverting the function of communication for “reasons” that are at
least eccentric. In traditional Marxism and sociological theories
following on from it, this coordination is the function of ideology,
and what I am saying is that there is not another “alethic” dimension
beyond communicative coordination — truth is good ideology, which
coordinates people in constructive ways. This is a redefinition of
traditional problems of truth and truth-telling, not a restriction of
it to a particular “materialist” province; it’s not any easier to see
what will really constitute a good ideological program at a particular
point in time than what is “really real”. But it is at least coherent
to look at things this way, and to think that there are no
“politically incorrect” truths, or politically correct falsehoods.
---- [!!!!]
Set text but:
Coherence in Cartesian Closed Categories and the Generality of Proofs
Author(s): M. E. Szabo
Source: Studia Logica: An International Journal for Symbolic Logic,
Vol. 48, No. 3 (1989), pp.
285-297
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015444
Accessed: 06/02/2010 14:57
Your use of the JSTOR archive indicates your acceptance of JSTOR's
Terms and Conditions of Use, available at
http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms
and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire
issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-
commercial use.
Please contact the publisher regarding any further use of this work.
Publisher contact information may be obtained at
http://www.jstor.org/action/showPublisher?publisherCode=springer.
Each copy of any part of a JSTOR transmission must contain the same
copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit service that helps scholars, researchers,
and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology
and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact
***@jstor.org.
Springer is collaborating with JSTOR to digitize, preserve and extend
access to Studia Logica: An
International Journal for Symbolic Logic.
M. E. Szabo
Coherence in Cartesian Closed Categories and the Generality of Proofs
1 Abstract. We introduce the notion of an alphabetic trace of a cut-
free intuitionistic propositional proof and show that it serves to
characterize the equality of arrows in cartesian closed categories. We
also show that alphabetic traces improve on the notion of the
generality of proofs proposed in the literature. The main theorem of
the paper yields a new and considerably simpler solution of the
coherence problem for cartesian closed categories than those in [11,
14].2
1. Introduction The fundamental contribution of Gentzen to the
analysis of formal proofs is the observation that most proofs contain
redundancies and information not reflected in the conclusions of the
proofs (cf. [10], Chapter 3). It is quite remarkable that in the case
of several basic axiomatic systems such as classical and
intuitionistic logic, these redundancies can be systematically
eliminated and that all valid formulas have direct proofs. On the
other hand, even direct proofs are usually not unique and a given
formula may have infinitely many distinct direct proofs. In addition,
different formulas may have proofs that are substitution instances of
the same deductive structure. Given these insights, it is natural to
speculate that certain formal proofs have generalizations that yield
proofs of more general results. The idea of defining a functorial
generality of proofs and of using this notion to characterize
algebraic congruences goes back to Lambek [3] and was used by Szabo
[11] to construct an algorithm for deciding the commutativity of
diagrams in cartesian closed categories. Unfor? tunately, the
intuition underlying the idea of the generality of proofs was
considerably weakened by the dominance of algebraic objectives. In
this paper we present a new non-functorial definition of generality
for intuitionistic propositional logic, i.e., cartesian closed
categories, that is based on deductive considerations, but is still
strong enough to distinguish algebraically non-equivalent proofs. This
work should be considered as a continuation of Lambek's program of
establishing the essential unity of algebra and logic. Twenty years
ago, Lambek wrote a paper on the mathematics of sentence structure
(cf. [3]) in which he 2 - Studia L?gica 3/89 1 This research was
supported in part by N.S.E.R.C. Grant A-8224 and F.C.A.R. Grant
EQ-3491. The author is attached to the Centre interuniversitaire en ?
tudes cat?goriques, McGill University, Montreal. 2 1980 Mathematics
Subject Classification (1985): 03B40, 03F05, 18F15, 68S05, 68S20.
286 M. E. Szabo initiated the use of cut-free derivations Tto
represent algebraic objects. In two recent articles ([8] and [7]), he
returned to this theme and argued that the links between proofs and
algebraic operators were such that algebra and logic were in fact two
different aspects of the same mathematical phenomenon. The
relationship between algebra and logic becomes precise if algebra is
conceived of as dealing with many-sorted operations and if logic takes
account of the equality of proofs. One of the tools for distinguishing
members of different congruence classes of proofs is that of the
generality of proofs (cf. [5, 6, 11]). The basic difference between
the -notion ^developed in this paper and its precursor in [11] is the
fact that the former was derived from a top-down functorial analysis
of proofs, whereas the present notion is in the spirit of Gentzen-
style proof theory (cf. [10] and [14]) and is based on a bottom-up
approach. The result is a more intuitive and more widely applicable
notion of generality.
2. Intuitionistic proofs The proofs considered in this paper are
proofs of the theorems of intuitionistic propositional logic, in
sequent form. We denote the deductive system involved by A(V) and base
its formulas on a fixed countable set V of propositional variables,
the constant truth (J), and the operations of conjunc? tion (a) and
implication (=>). The axioms and rules of inference of A(V) are slight
variants of those presented in Chapter 9 of [14]. As usual, sequents
are written in the form T ^A, with capital Greek letters denoting
finite sequences of formulas and capital Roman letters denoting
individual formulas.
2.1. Axioms and rules of inference
2.1.1. Axioms
The axioms of A(V) are the following:
1. If X is a propositional variable, then X-+X is an axiom.
2. The sequent ->T is an axiom.
3. There are no other axioms.
If you were in a charitable mood, I suppose you might describe my
intellectual project as moving analytic philosophy to its Marxist
moment — with the understanding that this project might not go off
without a hitch. At any rate, one of my core intellectual principles
is something like a Marxist theory of truth, and I’d like to explain
what that entails. Although I think it’s a minor scandal that “new
theories of truth” like Kripke’s and the revision theory are not
widely studied for their philosophical implications, as opposed to
going around and around disquotation, I don’t myself have anything
clever to say about them at the moment and so my remarks here will be
“philosophical” rather than formal.
Donald Davidson once tried (rather unsuccessfully, in the eyes of
most) to show that “coherence implies correspondence”, that the idea
of truth as adequatio falls out rather trivially from a logically well-
integrated web of belief. However, I think the implication in the
opposite direction is not quite as trivial as people make it out to
be: getting any use out of the idea of true statements as mirroring
facts actually requires going quite far into “ideological” features of
discourse. Why? When we assess an utterance for its truth or falsity,
we are interested in “what is said” — not just the surface form of the
words, but the concrete and relevant meaning of them. In formal
theories what is said is stipulatively clear, but in natural language
a whole host of “pragmatic” phenomena combine to make figuring out the
contribution of a statement to communication difficult.
Principles for dealing with these phenomena model social life — what
it is rational to think someone said in a context depends on one’s
model of their position within society and one’s model of their
understanding of that position. So much so, in fact, that I think
there is some point to simply identifying the real meaning of a
statement with its role as a “move” within society, as a contribution
to social action. This is not a “sceptical solution to sceptical
doubts” along the lines of Kripke’s interpretation of Wittgenstein:
although custom obviously plays a role in the role of words as an
element of practices, we are very far from being required to view
tradition as the fundament of “social meaning” and to exclude the role
of novelty (think of Wittgenstein’s picture of language as a city,
with narrow old streets and regularly plotted suburbs). We are also
far from having to view language as “smooth and homogenous” in the
words of Richard Rorty: the structure of “social practice” involving
meaning is not any simpler than the structure of society in general,
with its various divisions and conflicts.
Obviously, for communication to occur an utterance must coordinate ego
and alter: someone making an “offer you can’t understand” is
subverting the function of communication for “reasons” that are at
least eccentric. In traditional Marxism and sociological theories
following on from it, this coordination is the function of ideology,
and what I am saying is that there is not another “alethic” dimension
beyond communicative coordination — truth is good ideology, which
coordinates people in constructive ways. This is a redefinition of
traditional problems of truth and truth-telling, not a restriction of
it to a particular “materialist” province; it’s not any easier to see
what will really constitute a good ideological program at a particular
point in time than what is “really real”. But it is at least coherent
to look at things this way, and to think that there are no
“politically incorrect” truths, or politically correct falsehoods.
---- [!!!!]
Set text but:
Coherence in Cartesian Closed Categories and the Generality of Proofs
Author(s): M. E. Szabo
Source: Studia Logica: An International Journal for Symbolic Logic,
Vol. 48, No. 3 (1989), pp.
285-297
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015444
Accessed: 06/02/2010 14:57
Your use of the JSTOR archive indicates your acceptance of JSTOR's
Terms and Conditions of Use, available at
http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms
and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire
issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-
commercial use.
Please contact the publisher regarding any further use of this work.
Publisher contact information may be obtained at
http://www.jstor.org/action/showPublisher?publisherCode=springer.
Each copy of any part of a JSTOR transmission must contain the same
copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit service that helps scholars, researchers,
and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology
and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact
***@jstor.org.
Springer is collaborating with JSTOR to digitize, preserve and extend
access to Studia Logica: An
International Journal for Symbolic Logic.
M. E. Szabo
Coherence in Cartesian Closed Categories and the Generality of Proofs
1 Abstract. We introduce the notion of an alphabetic trace of a cut-
free intuitionistic propositional proof and show that it serves to
characterize the equality of arrows in cartesian closed categories. We
also show that alphabetic traces improve on the notion of the
generality of proofs proposed in the literature. The main theorem of
the paper yields a new and considerably simpler solution of the
coherence problem for cartesian closed categories than those in [11,
14].2
1. Introduction The fundamental contribution of Gentzen to the
analysis of formal proofs is the observation that most proofs contain
redundancies and information not reflected in the conclusions of the
proofs (cf. [10], Chapter 3). It is quite remarkable that in the case
of several basic axiomatic systems such as classical and
intuitionistic logic, these redundancies can be systematically
eliminated and that all valid formulas have direct proofs. On the
other hand, even direct proofs are usually not unique and a given
formula may have infinitely many distinct direct proofs. In addition,
different formulas may have proofs that are substitution instances of
the same deductive structure. Given these insights, it is natural to
speculate that certain formal proofs have generalizations that yield
proofs of more general results. The idea of defining a functorial
generality of proofs and of using this notion to characterize
algebraic congruences goes back to Lambek [3] and was used by Szabo
[11] to construct an algorithm for deciding the commutativity of
diagrams in cartesian closed categories. Unfor? tunately, the
intuition underlying the idea of the generality of proofs was
considerably weakened by the dominance of algebraic objectives. In
this paper we present a new non-functorial definition of generality
for intuitionistic propositional logic, i.e., cartesian closed
categories, that is based on deductive considerations, but is still
strong enough to distinguish algebraically non-equivalent proofs. This
work should be considered as a continuation of Lambek's program of
establishing the essential unity of algebra and logic. Twenty years
ago, Lambek wrote a paper on the mathematics of sentence structure
(cf. [3]) in which he 2 - Studia L?gica 3/89 1 This research was
supported in part by N.S.E.R.C. Grant A-8224 and F.C.A.R. Grant
EQ-3491. The author is attached to the Centre interuniversitaire en ?
tudes cat?goriques, McGill University, Montreal. 2 1980 Mathematics
Subject Classification (1985): 03B40, 03F05, 18F15, 68S05, 68S20.
286 M. E. Szabo initiated the use of cut-free derivations Tto
represent algebraic objects. In two recent articles ([8] and [7]), he
returned to this theme and argued that the links between proofs and
algebraic operators were such that algebra and logic were in fact two
different aspects of the same mathematical phenomenon. The
relationship between algebra and logic becomes precise if algebra is
conceived of as dealing with many-sorted operations and if logic takes
account of the equality of proofs. One of the tools for distinguishing
members of different congruence classes of proofs is that of the
generality of proofs (cf. [5, 6, 11]). The basic difference between
the -notion ^developed in this paper and its precursor in [11] is the
fact that the former was derived from a top-down functorial analysis
of proofs, whereas the present notion is in the spirit of Gentzen-
style proof theory (cf. [10] and [14]) and is based on a bottom-up
approach. The result is a more intuitive and more widely applicable
notion of generality.
2. Intuitionistic proofs The proofs considered in this paper are
proofs of the theorems of intuitionistic propositional logic, in
sequent form. We denote the deductive system involved by A(V) and base
its formulas on a fixed countable set V of propositional variables,
the constant truth (J), and the operations of conjunc? tion (a) and
implication (=>). The axioms and rules of inference of A(V) are slight
variants of those presented in Chapter 9 of [14]. As usual, sequents
are written in the form T ^A, with capital Greek letters denoting
finite sequences of formulas and capital Roman letters denoting
individual formulas.
2.1. Axioms and rules of inference
2.1.1. Axioms
The axioms of A(V) are the following:
1. If X is a propositional variable, then X-+X is an axiom.
2. The sequent ->T is an axiom.
3. There are no other axioms.