FOM: Contradiction-Free vs.Consistency

Robert Tragesser RTragesser at
Sun Nov 29 08:02:39 EST 1998

Topic: the identification of consistency with freedom-from-contradiction
seems now to be rather appalling,  hiding a comos,  or a cosmology,  of 
powerfully significant problems and phenomena.

        The curious and hardly intelligible current conflation of 
consistency and freedom-from-contradiction is an excellent example of: 
"Science does not always progress,  but sometimes regresses under the 
illusion of progress."  The positive interpretation of this,  Chairman 
Mao's three steps forward,  two steps backward, should only suggest 
that,  in the identification of consistency and contradiction-free,  we 
have taken two (regrouping,  simplifying steps backwards),  but now 
ought to recognize those as backward steps,  and move forward by setting
the problem of fully understanding "consistency" as a major of our 
        We have here a case of a resonant and cosmically rich notion -- 
"consistency"-- being egregiously suppressed,  displaced,  by,  
        For example,  it is absurd to say that contradiction-free-ness 
entails being,  but it is not so absurd to say that "consistency". 
[understood in the fully resonant and rather complex sense].
        "Consistency" carries with it a family of notions [and,  as 
always when one word is chosen for a complex of inter-related notions,  
no one of which is central,  it cannot be anything like simply defined].
 It carries with it the associations: coherent,  of a piece (a unified 
if not complete whole),  homogenous, standing-together,  standing-firm, 
senseful, staying true to something,  staying true. . .
        The firmest trait of beings,  entities,  even if like all the 
natural entities of Plato or Aristotle did,  they have ragged edges, or 
are dynmaical (like an organism as a life-cycle),  is consistency.  And 
indeed: It is pretty hard to find a trait that qualifies some "thing" as
existing other than consistency!
        The upper-half plane model for hyperbolic geometry,  Hilbert's 
polynomial interpretation of non-Archimedian geometry,  the geometrical 
interpretation(s) of _i_,  etc.,  establish consistency or we might 
better say relative consistency,  for they show that the theories or 
conceptions in questions can be _seen as_ (aspects of) some more 
strikingly consistent (= coherent,  together,  self-standing,  
sense-ful,  of a piece, . . . mathematics.
        The simplest illustration of the issue is that "dispute" between
Locke and Leibniz on the status of the law of of non-contradiction.   
For Leibniz,  this was an innate principle -- the serviging girl who 
blushes at a lie is vaguely grasping that she is violating this law. 
More to the point:
        Are there square circles?
        LOCKE:  No,  for one only has to try to understand 
"square-circle" and it immediately emerges that being and square and 
being a circle cannot be embodied by anything (in the intended sense),  
they are in that sense inconsistent with one another/incompatible. There
is not appeal whatsoever here to contradictoriness.
        LEIBNIZ: he required that "squar-circle" be resolved into a 
contradiction,  and the law of nontradiction implied.

Notes:  experimental and theoretical physics at the beginning of the 
century was inconsistent (wave/particle conflioct);  and this was not 
good quite independently of the inference of an outright contradiction. 
UNDERSTANDING EMERGES.  (Whereas a study of how it yielded a 
contradiction would have been of very little use indeed.)

Note:  Tait shows us how PRA is consistent.

Note:  the cumulative hierarchy conception powerfully suggests the 
consistency of ZF. . .but the CONCEPTUAL gaps among set theoretical 
practice,  the sense of the cumulative hierarchy,  and various 
conceptual gaps among these and the axioms suggests,  still,  that set 
theory is not fully consistent.

        The under the aspect of eternity important thing about "proofs 
of consistency" of T relative to seemingly finitary or concrete or 
concretely intuitive mathematics is not that they assure 
contradiction-freeness,  but that those limited bits of mathematics are 
strikingly consistent (coherent,  together,  not senseless,  of a piece,
etc.),  and one can believe by virtue of the relation the "consistenct 
proof of T" establishes with them that T is at least not-inconstent,  
even if T has not more directly won through to that kind of assurance 
that carries with it the stamp(s) of consistency in the 
mathematician's/physicist's/(and I should think? 
metaphysician's-epistemologist's important sense.   

Robert Tragesser
West(Running)Brook,  Connecticut.


Relative consitency.  

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