FOM: Re: Field on LEM and determinate truth value
sazonov at logic.botik.ru
Mon Mar 9 05:12:49 EST 1998
Neil Tennant wrote responding to Hartry Field:
> Let P be such a sentence: formally undecidable; objectively indeterminate
> (presumably, in truth value, not in meaning?); either true or false; but
> not determinately true or determinately false. [Please note that if there
> is any redundancy in this specification, it is from Hartry's description
> just quoted.]
> Hartry, can you suggest what kind of *justification* one might have for
> using, say, the classical law of excluded middle on P? I had thought that
> the only justification ever offered for the use of excluded middle was
> the principle of bivalence---that is, the belief/insistence that every
> declarative sentence enjoyed a *determinate truth value* (true or false),
> independently of our knowledge of that truth value, and independently
> of our means of coming to know what that truth value is.
> But, if I understand you correctly, you are suggesting that one would
> be justified in applying excluded middle to sentences *that do not have
> determinate truth values*.
> First question: Have I understood your position correctly?
> Second question: If so, how do you justify such uses of excluded middle?
Hartry Field will probably reply for himself. It seems that my
understanding of this subject is in some agreement with his. But
I use somewhat different argumentation.
Each of us may "believe" in anything. This is a delicate and
very personal matter. Each one likes his own beliefs and
illusions. Some kinds of beliefs may be very coherent and
widespread, and for any belief may exist a quite different,
contradictory and also internally sufficiently coherent and
attractive one. Moreover, the same person may *use* in different
situations different beliefs by his wish or even manipulate them.
However, objectively such mathematical concepts as those of
"the" infinite raw of "all" natural numbers or of "the"
continuum or of "the" universe of "all" sets (for ZFC or any
other version of set theory) are nothing but some extrapolation
of our conception of *finite* (small or big, but *feasible*, i.e.
really existing finite). Therefore infinity which exists only
in our imagination is inevitably rather vague idea. Wherefrom,
objectively, any "determinate truth value" of *arbitrary*
statement "independent of our knowledge of that truth value" may
arise at all? It is *our (sometimes inconscious) decision* which
way that extrapolation should be done (as soon as it is successful
in an appropriate sense.)
For example, it is very reasonable and straightforward to postulate
the *convenient* (I do not say "true"!) classical law of excluded
middle both for *infinite* and for *infeasible finite* structures
extrapolating it from feasible finite sets. This will give us very
attractive illusion of "the principle of bivalence" mentioned by
It is also reasonable to postulate induction axiom (IA) because
it is very nice, useful and, most important, gives us another
illusion that natural numbers are well ordered *even* with respect
to arbitrary properties A(n) which are *definable* by using unbounded
quantifiers over the *vague* set of "all" natural numbers. Of course
such properties may be also vague. Newertheless this gives us illusion
of the unique "standard" set of natural numbers which looks as nonvague.
Moreover, we postulate IA *despite* it contradicts to *empirical* Pi^0_1
log log n < 10 with n any (feasible) number in *unary* notation
and with base 2 logarithm; please check it on any real
(By the way, what is the maximal value of log log n or of log n ? Are
natural numbers "really" well ordered?)
Alternatively, it is interesting (and actually quite possible!) to
reject IA in its full strength (and even the classical logic in its
traditional formulation with the transitive implication, etc.) just
*because* of the above low of (double) logarithm. (I wrote on this in
my previous postings to FOM.)
However infinity is inevitably vague, our mathematical approaches
to it are very determinate in a sense because we are using *formal*
axioms and rules of inference. This is the main feature of
mathematics! It happens that the "ordinary" arithmetical statements
are usually decidable (i.e. provable or disprovable). Thus, we have
some, *almost objectively true* illusion of determinateness.
Classical logic and induction axiom are the main source for this
illusion. Of course, what is derivable in these systems is also not
very determinate because the set of "all" finite (feasible?)
derivations is vague. Moreover, we usually have corresponding
algorithmic undecidability results.
Therefore I believe that we should not give more value to our
illusions and beliefs (even to the strongest ones) than they deserve.
Program Systems Institute, | Tel. +7-08535-98945 (Inst.),
Russian Acad. of Sci. | Fax. +7-08535-20566
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