FOM: Re: Field on LEM and determinate truth value

[Vladimir Sazonov]

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  
Neil Tennant. 

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 
lows like

        log log n < 10 with n any (feasible) number in *unary* notation 
        and with base 2 logarithm; please check it on any real
computer!.

(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. 

Vladimir Sazonov
--
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