marxism-digest V1 #6030

Jim Farmelant farmelantj at juno.com
Wed Jul 2 08:15:05 MDT 2003



On Wed, 02 Jul 2003 08:45:20 -0400 Les Schaffer <schaffer at optonline.net>
writes:
> Hey John:
>
> get a grip. i saw you had some interest in symbolic logic, which
> matched an interest of my own, and found one somewhat detailed
> article
> on Trotsy and logic.
>
> sure the article has flaws, it seems pretty intent on beating on
> Trotsky's presumed intellectual arrogance. but it does roughly
> discuss
> that a number of Soviet mathematicians tried to get some handle on
> Godel's results vis a vis dialetcical logic.
>
> and thats of interest to me.
>

The Australian logician and philosopher Graham Priest
has also attempted to address these issues through the
development of what he calls paraconsistent logics - that is
logics where the existence of a contradictions between two
propositions does not imply the truth of ALL propositions
(as is the case in classical logic).

Priest sees such logics as
providing a useful way for dealing with such paradoxes of logic
and set theory like the Liar's Paradox or Russell's Paradox,
as well as some of Zeno's Pardoxes (such as the Paradox
of the Arrow).

There are a couple of articles by him in the online
Stanford Encyclopedia of Philosophy:

http://plato.stanford.edu/entries/logic-paraconsistent/

http://plato.stanford.edu/entries/dialetheism/

He also, as I recall wrote a couple of articles on this
subject for SCIENCE & SOCIETY as well.  He maintains
that people like Hegel and Marx were what he calls
dialetheists, and seems to think that much of dialectical
logic can be understood in terms of dialetheism and
paraconsistent logics.

------------------------------------------

For Graham Priest, classical two-valued logic is what
would call "explosive" since there, whenever we have
a contradiction, then then that entails all possible
propositions.

The explosiveness of  classical two-value logic can
be shown by the following truth table where A and
B are two propositions which can have truth values
of either T or F. That point is illustrated in the
following truth table:

___________________________________
A	not-A	B  (A & not-A)  (A & not-A)-->B
____________________________________
T	F	T               F		T
T	F	F	  F		T
F	T	T	  F		T
F	T	F	  F		T
____________________________________

Classical logic is explosive because if we have both
A and not-A then B is always entailed regardless of its
own truth value.


Now it is possible to have a three-value logic where
propositions can have one of three truth-values:
T for true, F for false, and I for indeterminate.

 In this system negation
can be defined by the following truth table:

__________
A     not-A
___________
T	F
F	T
I	 I
___________

while for material implication, its
associated truth table might look
this:

_____________________________
A	B	(A-->B)
_____________________________
T	T	 T
T	F	  F
T	I	  I
F	T	 T
F	F	 T
F	I	 T
I	T	 T
I	F	  I
I	I	  I
__________________

For the conjunction of A and not-A and
what it entails:
 ___________________________________
 A	not-A	B	(A & not-A)	(A & not-A)-->B
 ____________________________________________
 T	F	T	   F		T
 T	F	F	   F		T
 T	F	I	   F		T
 F	T	T	   F		T
 F	T	F	   F		T
 F 	T	I	   F		T
 I	I	T	   I	                T
 I	I	F	   I		 I
 I	I	I	   I	                 I
 _________________________________________

Here the contradiction between A and not-A
is not explosive since in this logic, it does not
entail all propositions.



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