> The failure of the law of non-contradiction is very important and
> further discussion of NAFL would be meaningful only after this failure
> (as described below) is understood. So please take some time to go
> through this elementary post and then we can get down to the business
> of actually seeing how NAFL theories can be constructed in detail.

> I want to thank George for starting the "FOL versus NAFL" thread in
> the past. I could not revive that thread, so I am starting a new one
> here.

> This thread is intended to be a continuation from the thread "Non
> standard models of PA" started in sci.logic on Oct. 22, 2007. In that
> thread, I had explained the NAFL truth definition and why it requires
> a failure of the law of non-contradiciton.

> Here is a recap of the NAFL truth definiton (to skip the recap, go
> straight to "\end{recap}" below and to the section titled "Failure of
> the law of non-contradiction in NAFL"). I have answered a couple of
> George's quereis (from the previous thread) after this recap.

> \begin{recap}

> I will discuss the language and rules of inference of NAFL theories
> later on. There are no absolute truths for formal propositions in the
> language of NAFL theories. The truths for these propostions must
> always be with respect to NAFL theories. In other words, there is no
> inherent pre-assigned meanings for formal propostions in the language
> of NAFL theories. The Main Postulate of NAFL provides the truth
> definition and I will briefly discuss it here.

> Define an "intepretation" T* of T as follows.

> Here T* is a NAFL theory that, like T, is held in the human mind and
> proves all the axioms of T. Further, T* could change with time, i.e.,
> the human mind could choose different theories for T* (i.e., different
> interpretations of T) at different times, and different human minds
> could choose different theories for T* at any given time.. E.g. I
> could take T*=T+P today, while another individual could take T*=T+~P.
> Tomorrow I could change my mind and take T*=T+~P irrespective of what
> another individual chooses. T* is chosen by the free will of the human
> mind.

> The NAFL model of T resides temporarily in the human mind and is
> generated by that particular T* which the human mind specifies, in the
> following sense. Only propositions provable/refutable in T* are
> assigned the truth values "True"/"False" in the NAFL model of T. All
> other propositions are "neither true nor false" in the NAFL model of T
> (generated by T*) in a sense to be explained below.

> A propostion P in the language of a consistent NAFL theory T is true/
> false with respect to T if P is provable/refutable in T, and hence
> provable/refutable in T*.

> If P is undecidable in T, then P is true/false with respect to T if
> and only if P is provable/refutable in the interpretation T* of T
> which the human mind specifies. In other words, truth for formal
> propostions P which are undecidable in a NAFL theory T must be
> *axiomatic* in nature, i.e. such truths are (temporary) axiomatic
> declarations in the human mind.

> It follows that if T* is specified such that P is undecidable in T*,
> then P must be "neither true nor false" according to the above truth
> definition (i.e., neither P nor ~P is provable in T*). We will argue
> in what follows that in this case T* generates a non-classical model
> in which P&~P is the case. In this non-classical model, 'P' is
> interpreted as "~P is not provable in T*" and '~P' is interpreted as
> "P is not provable in T*". So both P and ~P are both non-classically
> "true" in this sense and what P&~P expresses is that neither P nor ~P
> is provable in T* (i.e., neither P nor ~P is *classically* true with
> respect to T according to the above definition).

> The above is the truth definition for formal propostions in the
> language of NAFL theories. There are absolute (Platonic) truths in
> NAFL, but these are truths *about* NAFL theories and cannot be
> formalized in the language of NAFL theories. For example, "T is
> consistent" and "P is undecidable in T" are propostiions about NAFL
> theories that are taken to have absolute (classical) truth values,
> i.e., these propositons are either true or false without any reference
> to NAFL theories and independently of the human mind.

> \end{recap}

> Before gettting to the failure of the law of non-contradiction, let me
> address an objection raised by George, who does not like the fact that
> T, T* and the NAFL model are all (temporarily) resident in the human
> mind. He asks why this should be the case. Note that there is no
> unique value of T*, given some theory T. It is the human mind that
> "fixes" T* (temporarily) and hence fixes the truth values for
> undecidable propostions of T. I am asserting that there is no
> objective criterion for fixing T* other than the free will of the
> human mind and that is what NAFL is all about. In other words, if all
> human beings are wiped out, there is simply no "truth" for formal
> propositons of NAFL theories. The theories themselves exist in an
> abolute sense (independently of the human mind), but only the human
> mind can "interpret" theories according to NAFL. The NAFL models for
> NAFL theories will not exist without human beings.

> Another objection of George is that T* is a time-dependent variable
> and should not be given the name of a constant. OK, but it is
> something like the velocity of light "c" or other physical "constants"
> which are taken to be "fixed" at a given instant, but could have
> different values over a period of time. Of course T* could also be a
> function of time in the usual sense, but let us deal with that later
> on.

> FAILURE OF THE LAW OF NON-CONTRADICTION IN NAFL

> Meta-theorem

> If a proposition P is undecidable in a consistent NAFL theory T, then
> there must exist a model for T in which "P&~P" is the case (in other
> words, T does not prove Pv~P, which is equivalent to ~(P&~P) in NAFL).

> \begin{Proof}:

> This proof assumes that the *only* arguments for ~(P&~P) are either
> classical or intuitionistic in nature and both of these are refuted in
> NAFL as follows.

> Classically, Pv~P and ~(P&~P) are equivalent (as is the case in NAFL)
> and the arguments for these can be expressed as:

> If P (~P) is the case then ~P (P) cannot be the case.

> But according to the Main Postulate of NAFL, the hypothesis that "If P
> (~P) is the case...." is actually an axiomatic declaration of truth
> with respect to T, and basically the human mind has taken T*=T+P (T
> +~P). The conclusion that "...then ~P (P) cannot be the case" only
> follows in these theories and not in T. In other words, the classical
> argument for Pv~P (or ~(P&~P)) fails in NAFL.

> Let us now consider the intuitionistic argument. Take a system of
> natural deduction, for convenience. The intuitionistic argument for
> ~(P&~P) starts with the hypotheses P and ~P, finds an "absurdity",
> discharges the hypotheses and then concludes ~(P&~P).

> But note that intuitionistically, any proposition can be deduced from
> the contradictiory hypotheses P and ~P. The intuitionistic argument
> for ~(P&~P) crucially depends on the claimed "aburdity", which enables
> the "discharge" of the contradictory assumptions P and ~P.

> I am asserting that this discharge is essentially vacuous from the
> NAFL point of view and should not be permitted. Because one cannot see
> any absurdity from P&~P without first *presuming* ~(P&~P). The
> intuitionistic argument for ~(P&~P) really assumes what it wants to
> prove.

> The claim that P&~P is absurd because of the *inherent* meaning of the
> negation symbol is rejected in NAFL. The whole point of NAFL is that
> there is no such inherent meaning for formal propositions in the
> language of NAFL theories. Any such meaning can be only with respect
> to NAFL theories (or in other words, with respect to axiom-sets).
> Hence the intuitionistic argument for an inherent absurdity in P&~P
> (without any reference to axiomatic theories) is rejected in NAFL, for
> formal propositions P in the language of NAFL theories. In fact in
> NAFL, P&~P is indeed "impossible" wtih respect to NAFL theories that
> prove P or prove ~P, for these theories also prove ~(P&~P). If P is
> undecidable in T, there is no absurdity according to the meanings of
> "P" and "~P" assigned in the non-classical model as described above,
> and such a NAFL theory cannot prove ~(P&~P).

> \end{Proof}

> In conclusion, NAFL negation (for formal propositions) is different
> from classical and intuitionistic negations, both of which require pre-
> assigned meanings to P and ~P independent of axiomatic theories. This
> is rejected in NAFL.

> As I mentioned at the beginning of this post, further discussion of
> how NAFL theories are formulated can only proceed after the NAFL truth
> definition and the consequent failure of the law of non-contradiction
> are understood.

> Regards, RS

> > The claim that P&~P is absurd because of the *inherent* meaning of the
> > negation symbol is rejected in NAFL. The whole point of NAFL is that
> > there is no such inherent meaning for formal propositions in the
> > language of NAFL theories. Any such meaning can be only with respect
> > to NAFL theories (or in other words, with respect to axiom-sets).

> There is a simply 4-valued logic, that gives sense
> to P & ~P. Its Belnaps 4-valued logic, which is
> based on the following lattic:

>               t
>              / \
>             e   u
>              \ /
>               f

> Here we might define:

>     A |~A       A B | A & B
>     -----       -----------
>     f | t       f f |   f
>     u | u       f u |   f
>     e | e       f e |   f
>     t | f       f t |   f
>                 u f |   f
>                 u u |   u
>                 u e |   f
>                 u t |   u
>                 e f |   f
>                 e u |   f
>                 e e |   e
>                 e t |   e
>                 t f |   f
>                 t u |   u
>                 t e |   e
>                 t t |   t

> One can easily see that P & ~P is not contradictory,
> as it does not always evaluate to false:

>     P | P & ~P
>     ----------
>     f |   f
>     u |   u
>     e |   e
>     t |   f

> A nice property of belnaps four is that we can turn
> the lattic clockwise by 90 degree, and we will get
> another lattic. This lattice has its own ~' and &'.

> The u can be interpreted as underspecification, and
> the e can be interpreted as overspecification. The ~
> and & can be interpreted as logical reasoning, whereby
> the ~' and &' can be interpreted as knowledge fusion
> operators.

> There are even some nice applications in logic
> programming, as the u and e can deal with pathological
> kinds of recursion.

> Questions:
>     - What's the difference between NAFL and using
>       u (or even e).
>     - Do you have also the dual &' to & in NAFL? Do
>       you have something else?
>     - Why should I care about NAFL, given the fact
>       that Belanp published 1977.

> > Define an "intepretation" T* of T as follows.

> And then you don't define it. You ramble and ramble on about it, but
> without actually getting around to DEFINING it.

> Please just define:

> X is a an interpretation of a theory T iff [fill in here the exact
> property X must have to be an interpretation of a theory T].

> Meanwhile, would you please tell me what I'm allowed in proving
> theorems of NAFL theories.

> You say syntax is as classical first order logic. So what non-logical
> axioms are allowed or disallowed by NAFL?

> But what if we just define ~A  as   A-> falsum or _|_

> Not A means that A leads to (the general) contradiction

> then A & ~A suddenly makes sense
> A & ~A
> A & A -> _|_     df ~

> _|_  mp

> Or please explain what you do mean by ~A.

> > But what if we just define ~A  as   A-> falsum or _|_

> > Not A means that A leads to (the general) contradiction

> > then A & ~A suddenly makes sense
> > A & ~A
> > A & A -> _|_     df ~

> > _|_  mp

> > Or please explain what you do mean by ~A.

> What you have given is basically intuitionistic negation. NAFL
> negation is more complex.

> In NAFL, what ~A means cannot be fixed independently of NAFL theories.

> To put it in a nutshell, with respect to a NAFL theory T that proves
> either A or ~A, NAFL negation is the same as classical negation, i.e.,
> A and ~A have classical meanings.

> But with respect to a theory T in which A is undecidable, A and ~A can
> take on either classical or non-classical meanings depending on how
> the human mind chooses the interpretation T* of T. If "A" or "~A" is
> provable in T*, then again they have classical meanings in the model
> of T generated by T*.

>  If A is undecidable in T*, then both A and ~A have non-classical
> meanings in the non-classical model of T generated by T* (as explained
> in my previous post). "A" expresses that "~A is not provable in T*"
> and "~A" expresses that "A is not provable in T*". Thus A&~A is the
> case in the non-classical model of T generated by T* and you can see
> that both "A" and "~A" (and hence "A&~A") are true according to the
> non-classical interpretations given above.

> I will explain all of this further in ensuing posts. Please let me
> know if you follow the NAFL truth definition (the Main Postulate of
> NAFL) given in my previous post  and the arguments therein for the
> failure of the law of non-contradiction in NAFL.

> Regards, RS

> > > Define an "intepretation" T* of T as follows.

> > And then you don't define it. You ramble and ramble on about it, but
> > without actually getting around to DEFINING it.

> > Please just define:

> > X is a an interpretation of a theory T iff [fill in here the exact
> > property X must have to be an interpretation of a theory T].

> An interpretation T* of a NAFL theory T  is also a NAFL theory with
> the property that T* must at least prove all the axioms of T.

> I will get to these. The Main Postulate of NAFL, which (as explained
> in my previous post) provides its truth definition, is sacred and
> cannot be violated. It is this postulate that tells us that the law of
> non-contradition must, in general, fail in NAFL theories.

> > Here T* is a NAFL theory that, like T, is held in the human mind and
> > proves all the axioms of T. Further, T* could change with time, i.e.,
> > the human mind could choose different theories for T* (i.e., different
> > interpretations of T) at different times, and different human minds
> > could choose different theories for T* at any given time.. E.g. I
> > could take T*=T+P today, while another individual could take T*=T+~P.
> > Tomorrow I could change my mind and take T*=T+~P irrespective of what
> > another individual chooses. T* is chosen by the free will of the human
> > mind.

> But the point is, this is oversimplified from birth.   If all that is
> really
> true then there is simply no such thing as T*.  INSTEAD, there is a
> TERNARY FUNCTION *, with THREE arguments, a mind m, a theory T,
> and a time t,  such that AmTt[ *(mTt) = <what you have been calling
> T*>].

> Your account raises the immediate *philosophical* problem
> here involving "perdurantism".  I am quoting this from the
> Wikipedia article but you need to begin by picking a side.

> Perdurantists break into two distinct sub-groups. The former are 'worm
> theorists'. They believe that a persisting object is composed of the
> various temporal parts that it has. So all persisting objects are four-
> dimensional 'worms' that stretch across space-time, and that you are
> mistaken in believing that chairs, mountains and people are actually
> three-dimensional. This is to be contrasted to a more recent twist
> called 'stage theory'. Stage theorists take you to be identical with a
> particular temporal part at any given time. So, in a manner of
> speaking, I only exist for an instantaneous period of time. However
> there are other temporal parts at other times which I am related to in
> a certain way (Sider talks of 'modal counterpart relations', whilst
> Hawley talks of 'non-Humean relations') such that when I say that I
> was a child, or that I will be an OAP, these things are true because I
> am related to a temporal part that is a child (that exists in the
> past) or a temporal part that is an OAP (that exists in the future).
> Stage theorists are sometimes called 'exdurantists'.

> My point is, abstract objects are generally considered completely
> outside this debate.
> Abstract objects take no note whatsoever of time; a symbol is always
> the same symbol, irrespective of time.
> Time simply is not meaningful as applied to an abstract string, such
> as "abc".  "abc" does not need a human
> mind, or a time-continuum, in order to exist.  T, IN CONTRAST to T*,
> was historically/previously considered an abstract object.  T*, by
> contrast, since *you* are defining it, is something whose abstraction-
> status is yet underdetermined.
> You have to engage on THIS SUBpoint BEFORE you go pontificating about
> your superstructure.

> > The NAFL model of T resides temporarily in the human mind and is
> > generated by that particular T* which the human mind specifies, in the
> > following sense. Only propositions provable/refutable in T* are
> > assigned the truth values "True"/"False" in the NAFL model of T.

> How many different NAFL models of T might there be?  Doesn't every
> different
> T*, in every different mind at every different time, offer a
> potentially DIFFERENT
> NAFL model of T?  Why is there even any need to distinguish between a
> NAFL
> model of T and the-NAFL-model-of-T-generated-by-T*?  Could we have two
> different
> T*'s generating the same NAFL model of T?
> Given that all rich T's are going to have non-isomorphic models, does
> it
> *ever* even make *any* sense to speak of *the* NAFL model of T?
> Won't NAFL allow T to have many different models?

> > The above is the truth definition for formal propostions in the
> > language of NAFL theories. There are absolute (Platonic) truths in
> > NAFL, but these are truths *about* NAFL theories and cannot be
> > formalized in the language of NAFL theories. For example, "T is
> > consistent" and "P is undecidable in T" are propostiions about NAFL
> > theories that are taken to have absolute (classical) truth values,
> > i.e., these propositons are either true or false without any reference
> > to NAFL theories and independently of the human mind.

> So these classically true statements cannot be formalized/asserted by
> NAFL theories?
> That is a good reason to ashcan the whole paradigm right there.
> Please!

> > Before gettting to the failure of the law of non-contradiction, let me
> > address an objection raised by George, who does not like the fact that
> > T, T* and the NAFL model are all (temporarily) resident in the human
> > mind. He asks why this should be the case. Note that there is no
> > unique value of T*, given some theory T. It is the human mind that
> > "fixes" T* (temporarily) and hence fixes the truth values for
> > undecidable propostions of T. I am asserting that there is no
> > objective criterion for fixing T* other than the free will of the
> > human mind and that is what NAFL is all about. In other words, if all
> > human beings are wiped out, there is simply no "truth" for formal
> > propositons of NAFL theories.

> Well, that is a serious weakness.  You have got to be clear about the
> realm you are in.  "If al human beings were wiped out" is kind of
> stupid
> anyway.

> > The theories themselves exist in an
> > abolute sense (independently of the human mind),

> Oh, bullshit.  That is PRECISELY what you JUST denied!

> > but only the human
> > mind can "interpret" theories according to NAFL. The NAFL models for
> > NAFL theories will not exist without human beings.

> OK, fine, I get it.
> Theories are abstract.
> Models are concrete.
> But that is not as big a difference as you think.
> In the classical paradigm, theories are abstract in the language of
> your choice,
> but models are abstract in ZFC.  My point is, there remains, in both
> cases,
> a distinction of levels.  That is all that is really needed.

> > On Nov 12, 8:24 pm, translogi <wilem...@googlemail.com> wrote:

> > > But what if we just define ~A  as   A-> falsum or _|_

> > > Not A means that A leads to (the general) contradiction

> > > then A & ~A suddenly makes sense
> > > A & ~A
> > > A & A -> _|_     df ~

> > > _|_  mp

> > > Or please explain what you do mean by ~A.

> > What you have given is basically intuitionistic negation. NAFL
> > negation is more complex.

> > In NAFL, what ~A means cannot be fixed independently of NAFL theories.

> > To put it in a nutshell, with respect to a NAFL theory T that proves
> > either A or ~A, NAFL negation is the same as classical negation, i.e.,
> > A and ~A have classical meanings.

> > But with respect to a theory T in which A is undecidable, A and ~A can
> > take on either classical or non-classical meanings depending on how
> > the human mind chooses the interpretation T* of T. If "A" or "~A" is
> > provable in T*, then again they have classical meanings in the model
> > of T generated by T*.

> >  If A is undecidable in T*, then both A and ~A have non-classical
> > meanings in the non-classical model of T generated by T* (as explained
> > in my previous post). "A" expresses that "~A is not provable in T*"
> > and "~A" expresses that "A is not provable in T*". Thus A&~A is the
> > case in the non-classical model of T generated by T* and you can see
> > that both "A" and "~A" (and hence "A&~A") are true according to the
> > non-classical interpretations given above.

> > I will explain all of this further in ensuing posts. Please let me
> > know if you follow the NAFL truth definition (the Main Postulate of
> > NAFL) given in my previous post  and the arguments therein for the
> > failure of the law of non-contradiction in NAFL.

> > Regards, RS

> Sorry i don't understand it

> A  is true if (and only if) it is provable that ~A is untrue?
> That looks like intuitionistic logic.

> But also
> ~A is true if (and only if) A is not provable ?

> I am getting confused here.
> Sorry

> > On Nov 13, 4:53 am, MoeBlee <jazzm...@hotmail.com> wrote:> On Nov 12, 5:47 am, "R. Srinivasan" <sradh...@in.ibm.com> wrote:

> > > > Define an "intepretation" T* of T as follows.

> > > And then you don't define it. You ramble and ramble on about it, but
> > > without actually getting around to DEFINING it.

> > > Please just define:

> > > X is a an interpretation of a theory T iff [fill in here the exact
> > > property X must have to be an interpretation of a theory T].

> > An interpretation T* of a NAFL theory T  is also a NAFL theory with
> > the property that T* must at least prove all the axioms of T.

> Okay, thanks.

> You've said that the syntax of NAFL is just as in classical first
> order logic. So, what is an NAFL theory as distinct from just a
> classical first order theory (as 'T is a classical first theory' can
> be defined as 'T is a set of first order sentences closed under
> classical provability' (actually, the property is usually closure
> under entailment, but courtesy of the completeness theorem, and for
> sake of convenience here, we can use provability rather than
> entailment for this particular purpose)).

> So, would you please say what is the definition of 'T is an NAFL
> theory' as opposed to just the definition of 'T is a classical first
> order theory' (as I just gave a particular definition (for this
> particular ad hoc purpose) of 'classical first order theory')?

> > The point is that the NAFL model of T generated by T*
> > will only contain truths that correspond to provability in T*.

> What is the definition of 'the NAFL model of T generated by T*'?

> > > You say syntax is as classical first order logic. So what non-logical
> > > axioms are allowed or disallowed by NAFL?

> > I will get to these. The Main Postulate of NAFL, which (as explained
> > in my previous post) provides its truth definition, is sacred and
> > cannot be violated. It is this postulate that tells us that the law of
> > non-contradition must, in general, fail in NAFL theories.

> Is the main postulate of NAFL expressible as a first order sentence
> that is added to the logical axioms of the classical first order
> predicate calculus?  Or is it rather a  general informal thesis of
> yours? And are there any logical axioms or rules of inference of the
> classcial first order predicate calculus that are not allowed by NAFL?

> > I will explain how to formulate NAFL theories to satisfy the Main
> > Postulate in subsequent posts. Basically one starts with classical
> > theories and then suitably restricts/modifies them to make them
> > conform to the Main Postulate.

> And you will give this as a formal syntactical operation?

> > Do you have any comments on the failure of the law of non-
> > contradiction in NAFL?

> Not until I understand what exactly NAFL is.

> > It is very important for you to understand and accept my arguments
> > given in my previous post before I proceed further.

> I can't accept or reject them until I get precise defintions of your
> terminology. To put it in your words adapated: It is necessary that
> you give me precise definitions before I can proceed further to
> evaluate your arguments couched in that terminology.

> > I think this post
> > is more or less self-contained and you should be able to understand
> > and get an intuitive feel for my arguments therein.

> I'll see what intutive feel I get once you give me precise definitions
> as I've asked for now in this post.

> For the time being, NAFL itself can be thought of as the meta-theory
> for NAFL theories, i.e., it tells us how NAFL theories have to be
> constructed and what constraints are to be satisfied. It is an
> informal description in the sense that I do not consider any
> "totality" of NAFL theories or  the totality of the models of a theory
> and the like. These totalities are not needed to understand how NAFL
> theories are constructed and in fact such totalities cannot even be
> legally formulated within NAFL theories.

> > On Nov 14, 1:58 pm, "R. Srinivasan" <sradh...@in.ibm.com> wrote:

> > > On Nov 12, 8:24 pm, translogi <wilem...@googlemail.com> wrote:

> > > > But what if we just define ~A  as   A-> falsum or _|_

> > > > Not A means that A leads to (the general) contradiction

> > > > then A & ~A suddenly makes sense
> > > > A & ~A
> > > > A & A -> _|_     df ~

> > > > _|_  mp

> > > > Or please explain what you do mean by ~A.

> > > What you have given is basically intuitionistic negation. NAFL
> > > negation is more complex.

> > > In NAFL, what ~A means cannot be fixed independently of NAFL theories.

> > > To put it in a nutshell, with respect to a NAFL theory T that proves
> > > either A or ~A, NAFL negation is the same as classical negation, i.e.,
> > > A and ~A have classical meanings.

> > > But with respect to a theory T in which A is undecidable, A and ~A can
> > > take on either classical or non-classical meanings depending on how
> > > the human mind chooses the interpretation T* of T. If "A" or "~A" is
> > > provable in T*, then again they have classical meanings in the model
> > > of T generated by T*.

> > >  If A is undecidable in T*, then both A and ~A have non-classical
> > > meanings in the non-classical model of T generated by T* (as explained
> > > in my previous post). "A" expresses that "~A is not provable in T*"
> > > and "~A" expresses that "A is not provable in T*". Thus A&~A is the
> > > case in the non-classical model of T generated by T* and you can see
> > > that both "A" and "~A" (and hence "A&~A") are true according to the
> > > non-classical interpretations given above.

> > > I will explain all of this further in ensuing posts. Please let me
> > > know if you follow the NAFL truth definition (the Main Postulate of
> > > NAFL) given in my previous post  and the arguments therein for the
> > > failure of the law of non-contradiction in NAFL.

> > > Regards, RS

> > Sorry i don't understand it

> > A  is true if (and only if) it is provable that ~A is untrue?
> > That looks like intuitionistic logic.

> > But also
> > ~A is true if (and only if) A is not provable ?

> > I am getting confused here.
> > Sorry

> No problem. First of all "A is true" or "A is provable", etc. are not
> meaningful sentences in NAFL. Truth and provability are always with
> respect to NAFL theories.

> Basically, in NAFL, the following hold:

> A is true (false) with respect to a NAFL theory T if and only if the
> interpretation T* of T proves (refutes) A.

> In particular, if T proves (refutes) A, then T* is constrained to
> prove (refute) A for T* is defined to be a NAFL theory that must prove
> all the axioms (and hence all the theorems) of A.

> With the above choices of T* that make A true or false with respect to
> T, NAFL negation is the same as classical negation.

> The next question that arises would be what if T* does not prove or
> refute A, i.e., what if A is undecidable in T*?

> Such a T* would generate a non-classical model of T in which A&~A is
> the case. Here A is interpreted as "~A is not provable in T*" and ~A
> is interpreted as "A is not provable in T*". Note that both A and ~A
> are indeed true according to this intepretation. But now ~A is not
> *really* the negation of A in this non-classical model. For "A is not
> provable in T*" is not the negation of "~A is not provable in T*".

> In a nutshell, NAFL negation is both theory dependent and could be
> model dependent (within a given theory). I.e., the meaning of ~A for a
> given A could possibly change according to the theory T in which it is
> formulated, and according to whether T decides A or not. Within a
> given theory T in which A is undecidable, ~A will have different
> meanings in the classical models (in which A is either true or false)
> and in the non-classical models (in which ~A is not the true negation
> of A).

> If this is not clear at this point, I hope it will become clear later
> on when I (hopefully) get to explaining NAFL in detail.

> Regards, RS- Hide quoted text -

> - Show quoted text -

> > OK, I will get around to answering your questions to the best of my
> > ability shortly as I get started on how to formulate NAFL theories.

> > For the time being, NAFL itself can be thought of as the meta-theory
> > for NAFL theories, i.e., it tells us how NAFL theories have to be
> > constructed and what constraints are to be satisfied. It is an
> > informal description in the sense that I do not consider any
> > "totality" of NAFL theories or  the totality of the models of a theory
> > and the like. These totalities are not needed to understand how NAFL
> > theories are constructed and in fact such totalities cannot even be
> > legally formulated within NAFL theories.

> As you say, you'll get around to defining these things. I'll just have
> to wait for that, because until you do, I can't make sense of 'NAFL as
> the meta-theory for NAFL theories' until you define 'T is an NAFL
> theory' in some form such as:

> T is an NAFL theory iff [fill in here an exact condition that T must
> fulfill to be an NAFL theory]

> > The truths in a NAFL model of T generated by an interpretation T* are
> > precisely those propostiions that are provable in T*.

> But until you define 'T is an NAFL theory', there's not much for me to
> work with here.

> > A proposition P
> > that is undecidable in T* is "neither true nor false" in the
> > corresponding non-classical NAFL model, where P&~P holds. The NAFL
> > model can thus be thought of as a structure that fixes the truth
> > values of the various propostions of T. A formal description of a NAFL
> > model cannot be made within NAFL, for "P&~P" is not even a legitimate
> > propostion in (the theory syntax of) NAFL theories. You will need a
> > paraconsistent logic (which allows P&~P to be provable within its
> > theories) to give a formal description of NAFL models.

> I know, since you've told me, that an NAFL interpretation of an NAFL
> theory T is an NAFL theory T* such that T* proves the axioms of T.
> (So, unless you tell me otherwise, I'll take that as equivalent to:

> T* is an NAFL interpretation of the NAFL theory T iff T* is an NAFL
> theory such that T is a subset of T*.

> So what we need now is a definition of 'T is an NAFL theory'.

> So please let me know when you have a definition of 'T is an NAFL
> theory'.

> > On Nov 14, 8:29 pm, translogi <wilem...@googlemail.com> wrote:

> > > On Nov 14, 1:58 pm, "R. Srinivasan" <sradh...@in.ibm.com> wrote:

> > > > On Nov 12, 8:24 pm, translogi <wilem...@googlemail.com> wrote:

> > > > > But what if we just define ~A  as   A-> falsum or _|_

> > > > > Not A means that A leads to (the general) contradiction

> > > > > then A & ~A suddenly makes sense
> > > > > A & ~A
> > > > > A & A -> _|_     df ~

> > > > > _|_  mp

> > > > > Or please explain what you do mean by ~A.

> > > > What you have given is basically intuitionistic negation. NAFL
> > > > negation is more complex.

> > > > In NAFL, what ~A means cannot be fixed independently of NAFL theories.

> > > > To put it in a nutshell, with respect to a NAFL theory T that proves
> > > > either A or ~A, NAFL negation is the same as classical negation, i.e.,
> > > > A and ~A have classical meanings.

> > > > But with respect to a theory T in which A is undecidable, A and ~A can
> > > > take on either classical or non-classical meanings depending on how
> > > > the human mind chooses the interpretation T* of T. If "A" or "~A" is
> > > > provable in T*, then again they have classical meanings in the model
> > > > of T generated by T*.

> > > >  If A is undecidable in T*, then both A and ~A have non-classical
> > > > meanings in the non-classical model of T generated by T* (as explained
> > > > in my previous post). "A" expresses that "~A is not provable in T*"
> > > > and "~A" expresses that "A is not provable in T*". Thus A&~A is the
> > > > case in the non-classical model of T generated by T* and you can see
> > > > that both "A" and "~A" (and hence "A&~A") are true according to the
> > > > non-classical interpretations given above.

> > > > I will explain all of this further in ensuing posts. Please let me
> > > > know if you follow the NAFL truth definition (the Main Postulate of
> > > > NAFL) given in my previous post  and the arguments therein for the
> > > > failure of the law of non-contradiction in NAFL.

> > > > Regards, RS

> > > Sorry i don't understand it

> > > A  is true if (and only if) it is provable that ~A is untrue?
> > > That looks like intuitionistic logic.

> > > But also
> > > ~A is true if (and only if) A is not provable ?

> > > I am getting confused here.
> > > Sorry

> > No problem. First of all "A is true" or "A is provable", etc. are not
> > meaningful sentences in NAFL. Truth and provability are always with
> > respect to NAFL theories.

> > Basically, in NAFL, the following hold:

> > A is true (false) with respect to a NAFL theory T if and only if the
> > interpretation T* of T proves (refutes) A.

> > In particular, if T proves (refutes) A, then T* is constrained to
> > prove (refute) A for T* is defined to be a NAFL theory that must prove
> > all the axioms (and hence all the theorems) of A.

> > With the above choices of T* that make A true or false with respect to
> > T, NAFL negation is the same as classical negation.

> > The next question that arises would be what if T* does not prove or
> > refute A, i.e., what if A is undecidable in T*?

> > Such a T* would generate a non-classical model of T in which A&~A is
> > the case. Here A is interpreted as "~A is not provable in T*" and ~A
> > is interpreted as "A is not provable in T*". Note that both A and ~A
> > are indeed true according to this intepretation. But now ~A is not
> > *really* the negation of A in this non-classical model. For "A is not
> > provable in T*" is not the negation of "~A is not provable in T*".

> > In a nutshell, NAFL negation is both theory dependent and could be
> > model dependent (within a given theory). I.e., the meaning of ~A for a
> > given A could possibly change according to the theory T in which it is
> > formulated, and according to whether T decides A or not. Within a
> > given theory T in which A is undecidable, ~A will have different
> > meanings in the classical models (in which A is either true or false)
> > and in the non-classical models (in which ~A is not the true negation
> > of A).

> > If this is not clear at this point, I hope it will become clear later
> > on when I (hopefully) get to explaining NAFL in detail.

> > Regards, RS- Hide quoted text -

> > - Show quoted text -

> Thanks for your reply

> It is quite a strange to interpret
> A as "~A is not provable in T*"
> and
> ~A as "A is not provable in T*"

> It is not only the question of are they circular?
> (possibly they are not)

> But also the interpretations like
> A as "A is provable in T*"
> and
> ~A as "~A is provable in T*"
> seem much more "natural"
> I know there is no good ultimate reason to go for the natural feel.
> But the oppostite needs a real advantage before using it, and that
> advantage i do not see.

> 1. Given the NAFL truth definition (the Main Postulate) as axiomatic
> declarations in the human mind, which identifies NAFL truth for formal
> propostions with provability in NAFL theories.

> > On Nov 15, 2:59 pm, "R. Srinivasan" <sradh...@in.ibm.com> wrote:

> > > On Nov 14, 8:29 pm, translogi <wilem...@googlemail.com> wrote:

> > > > On Nov 14, 1:58 pm, "R. Srinivasan" <sradh...@in.ibm.com> wrote:

> > > > > On Nov 12, 8:24 pm, translogi <wilem...@googlemail.com> wrote:

> > > > > > But what if we just define ~A  as   A-> falsum or _|_

> > > > > > Not A means that A leads to (the general) contradiction

> > > > > > then A & ~A suddenly makes sense
> > > > > > A & ~A
> > > > > > A & A -> _|_     df ~

> > > > > > _|_  mp

> > > > > > Or please explain what you do mean by ~A.

> > > > > What you have given is basically intuitionistic negation. NAFL
> > > > > negation is more complex.

> > > > > In NAFL, what ~A means cannot be fixed independently of NAFL theories.

> > > > > To put it in a nutshell, with respect to a NAFL theory T that proves
> > > > > either A or ~A, NAFL negation is the same as classical negation, i.e.,
> > > > > A and ~A have classical meanings.

> > > > > But with respect to a theory T in which A is undecidable, A and ~A can
> > > > > take on either classical or non-classical meanings depending on how
> > > > > the human mind chooses the interpretation T* of T. If "A" or "~A" is
> > > > > provable in T*, then again they have classical meanings in the model
> > > > > of T generated by T*.

> > > > >  If A is undecidable in T*, then both A and ~A have non-classical
> > > > > meanings in the non-classical model of T generated by T* (as explained
> > > > > in my previous post). "A" expresses that "~A is not provable in T*"
> > > > > and "~A" expresses that "A is not provable in T*". Thus A&~A is the
> > > > > case in the non-classical model of T generated by T* and you can see
> > > > > that both "A" and "~A" (and hence "A&~A") are true according to the
> > > > > non-classical interpretations given above.

> > > > > I will explain all of this further in ensuing posts. Please let me
> > > > > know if you follow the NAFL truth definition (the Main Postulate of
> > > > > NAFL) given in my previous post  and the arguments therein for the
> > > > > failure of the law of non-contradiction in NAFL.

> > > > > Regards, RS

> > > > Sorry i don't understand it

> > > > A  is true if (and only if) it is provable that ~A is untrue?
> > > > That looks like intuitionistic logic.

> > > > But also
> > > > ~A is true if (and only if) A is not provable ?

> > > > I am getting confused here.
> > > > Sorry

> > > No problem. First of all "A is true" or "A is provable", etc. are not
> > > meaningful sentences in NAFL. Truth and provability are always with
> > > respect to NAFL theories.

> > > Basically, in NAFL, the following hold:

> > > A is true (false) with respect to a NAFL theory T if and only if the
> > > interpretation T* of T proves (refutes) A.

> > > In particular, if T proves (refutes) A, then T* is constrained to
> > > prove (refute) A for T* is defined to be a NAFL theory that must prove
> > > all the axioms (and hence all the theorems) of A.

> > > With the above choices of T* that make A true or false with respect to
> > > T, NAFL negation is the same as classical negation.

> > > The next question that arises would be what if T* does not prove or
> > > refute A, i.e., what if A is undecidable in T*?

> > > Such a T* would generate a non-classical model of T in which A&~A is
> > > the case. Here A is interpreted as "~A is not provable in T*" and ~A
> > > is interpreted as "A is not provable in T*". Note that both A and ~A
> > > are indeed true according to this intepretation. But now ~A is not
> > > *really* the negation of A in this non-classical model. For "A is not
> > > provable in T*" is not the negation of "~A is not provable in T*".

> > > In a nutshell, NAFL negation is both theory dependent and could be
> > > model dependent (within a given theory). I.e., the meaning of ~A for a
> > > given A could possibly change according to the theory T in which it is
> > > formulated, and according to whether T decides A or not. Within a
> > > given theory T in which A is undecidable, ~A will have different
> > > meanings in the classical models (in which A is either true or false)
> > > and in the non-classical models (in which ~A is not the true negation
> > > of A).

> > > If this is not clear at this point, I hope it will become clear later
> > > on when I (hopefully) get to explaining NAFL in detail.

> > > Regards, RS- Hide quoted text -

> > > - Show quoted text -

> > Thanks for your reply

> > It is quite a strange to interpret
> > A as "~A is not provable in T*"
> > and
> > ~A as "A is not provable in T*"

> > It is not only the question of are they circular?
> > (possibly they are not)

> There is no circularity. In the non-classical models this
> interpretation must hold. Basically the non-classical model upholds
> the only facts that we know about P and ~P; that neither of these are
> provable in T, and neither of these have been asserted by the human
> mind, via provability in T*. Note that for a given T, the
> interpreation T* is chosen by the free will of the human mind, and
> truth (in the classical sense) is identified with provability in T*
> (which generates the NAFL models). So what P&~P signifies is that P
> and ~P are both "neither true nor false" in this classical sense.

> There are two points here that I have made clear in my parallel reply
> to Moeblee. This non-classical interpretation is more or less forced
> on us via the Main Postulate of NAFL, i.e. it is the Main Postulate
> that is sacred in NAFL. What we are sacrificing is the classical
> meaning of the negation symbol "~" as strict negation. For clearly in
> a non-classical NAFL model of T in which P&~P is the case, the
> intepretation I have given makes both "P" and "~P" non-classically
> true in the sense that both P and ~P are indeed unprovable in T*. So
> clearly one cannot be the strict negation of the other.

> To be specific, take T to be the NAFL version of quantum mechanics and
> let "P" be the proposition that "The Schrodinger cat is alive". Here P
> is undecidable in T. When P is provable or refutable in T* (which is
> the interpretation of T fixed by the human mind, say when the box is
> opened and the state of the cat in the real world is observed by the
> human mind), we may take the (informal) meanings of P or ~P to be the
> classical meanings, i.e., the cat is ":really" alive or the cat is
> "really" dead; clearly only one of these cases can be "observed" in
> the real world, and correspondingly in NAFL, only one of these can be
> asserted as an axiom of T* (remember that consistent NAFL theories do
> not allow provability of P&~P).

> Note the importance of the temporal nature of T* and of NAFL truth for
> T-undecidable propostions in the above example of the Schrodinger cat.
> We do not commit ourselves to the "existence" of a totality of NAFL
> theories T* as a time-dependent function. Instead we only commit
> ourselves to one T* at any given time, which exists by *definition* of
> T*, and does make enough sense to understand NAFL without commiting
> ourselves to any totality of T*'s.

> As for *formal* existence of objects within NAFL

> > On Nov 18, 3:33 am, translogi <wilem...@googlemail.com> wrote:

> > > On Nov 15, 2:59 pm, "R. Srinivasan" <sradh...@in.ibm.com> wrote:

> > > > On Nov 14, 8:29 pm, translogi <wilem...@googlemail.com> wrote:

> > > > > On Nov 14, 1:58 pm, "R. Srinivasan" <sradh...@in.ibm.com> wrote:

> > > > > > On Nov 12, 8:24 pm, translogi <wilem...@googlemail.com> wrote:

> > > > > > > But what if we just define ~A  as   A-> falsum or _|_

> > > > > > > Not A means that A leads to (the general) contradiction

> > > > > > > then A & ~A suddenly makes sense
> > > > > > > A & ~A
> > > > > > > A & A -> _|_     df ~

> > > > > > > _|_  mp

> > > > > > > Or please explain what you do mean by ~A.

> > > > > > What you have given is basically intuitionistic negation. NAFL
> > > > > > negation is more complex.

> > > > > > In NAFL, what ~A means cannot be fixed independently of NAFL theories.

> > > > > > To put it in a nutshell, with respect to a NAFL theory T that proves
> > > > > > either A or ~A, NAFL negation is the same as classical negation, i.e.,
> > > > > > A and ~A have classical meanings.

> > > > > > But with respect to a theory T in which A is undecidable, A and ~A can
> > > > > > take on either classical or non-classical meanings depending on how
> > > > > > the human mind chooses the interpretation T* of T. If "A" or "~A" is
> > > > > > provable in T*, then again they have classical meanings in the model
> > > > > > of T generated by T*.

> > > > > >  If A is undecidable in T*, then both A and ~A have non-classical
> > > > > > meanings in the non-classical model of T generated by T* (as explained
> > > > > > in my previous post). "A" expresses that "~A is not provable in T*"
> > > > > > and "~A" expresses that "A is not provable in T*". Thus A&~A is the
> > > > > > case in the non-classical model of T generated by T* and you can see
> > > > > > that both "A" and "~A" (and hence "A&~A") are true according to the
> > > > > > non-classical interpretations given above.

> > > > > > I will explain all of this further in ensuing posts. Please let me
> > > > > > know if you follow the NAFL truth definition (the Main Postulate of
> > > > > > NAFL) given in my previous post  and the arguments therein for the
> > > > > > failure of the law of non-contradiction in NAFL.

> > > > > > Regards, RS

> > > > > Sorry i don't understand it

> > > > > A  is true if (and only if) it is provable that ~A is untrue?
> > > > > That looks like intuitionistic logic.

> > > > > But also
> > > > > ~A is true if (and only if) A is not provable ?

> > > > > I am getting confused here.
> > > > > Sorry

> > > > No problem. First of all "A is true" or "A is provable", etc. are not
> > > > meaningful sentences in NAFL. Truth and provability are always with
> > > > respect to NAFL theories.

> > > > Basically, in NAFL, the following hold:

> > > > A is true (false) with respect to a NAFL theory T if and only if the
> > > > interpretation T* of T proves (refutes) A.

> > > > In particular, if T proves (refutes) A, then T* is constrained to
> > > > prove (refute) A for T* is defined to be a NAFL theory that must prove
> > > > all the axioms (and hence all the theorems) of A.

> > > > With the above choices of T* that make A true or false with respect to
> > > > T, NAFL negation is the same as classical negation.

> > > > The next question that arises would be what if T* does not prove or
> > > > refute A, i.e., what if A is undecidable in T*?

> > > > Such a T* would generate a non-classical model of T in which A&~A is
> > > > the case. Here A is interpreted as "~A is not provable in T*" and ~A
> > > > is interpreted as "A is not provable in T*". Note that both A and ~A
> > > > are indeed true according to this intepretation. But now ~A is not
> > > > *really* the negation of A in this non-classical model. For "A is not
> > > > provable in T*" is not the negation of "~A is not provable in T*".

> > > > In a nutshell, NAFL negation is both theory dependent and could be
> > > > model dependent (within a given theory). I.e., the meaning of ~A for a
> > > > given A could possibly change according to the theory T in which it is
> > > > formulated, and according to whether T decides A or not. Within a
> > > > given theory T in which A is undecidable, ~A will have different
> > > > meanings in the classical models (in which A is either true or false)
> > > > and in the non-classical models (in which ~A is not the true negation
> > > > of A).

> > > > If this is not clear at this point, I hope it will become clear later
> > > > on when I (hopefully) get to explaining NAFL in detail.

> > > > Regards, RS- Hide quoted text -

> > > > - Show quoted text -

> > > Thanks for your reply

> > > It is quite a strange to interpret
> > > A as "~A is not provable in T*"
> > > and
> > > ~A as "A is not provable in T*"

> > > It is not only the question of are they circular?
> > > (possibly they are not)

> > There is no circularity. In the non-classical models this
> > interpretation must hold. Basically the non-classical model upholds
> > the only facts that we know about P and ~P; that neither of these are
> > provable in T, and neither of these have been asserted by the human
> > mind, via provability in T*. Note that for a given T, the
> > interpreation T* is chosen by the free will of the human mind, and
> > truth (in the classical sense) is identified with provability in T*
> > (which generates the NAFL models). So what P&~P signifies is that P
> > and ~P are both "neither true nor false" in this classical sense.

> Thanks for yiour explaination here.
> Was reading about FDE First  degree entailment (sorry no authoritive
> weblink here  it is mentioned inhttp://en.wikipedia.org/wiki/Paraconsistent_logic
> but not very succint)
> In what differs NAFL from FDE

> > Note also that in the non-classical models, ~~P has the same meaning
> > as P (i.e., both P and ~~P are interpreted as "~P is not provable in
> > T*, or "~P is not true with respect to T in the classical sense").
> > Similarly ~P, ~~~P, etc. all express the same thing, namely, that "P
> > is not provable in T*" or that "P not true in the classical sense with
> > respect to T".

> NO NO sorry this is not true

> in Intuitionistic logic  for example

> P -> ~~P
> but not the other way around.
> So they do not have the same meaning.
> (or maybe you mean something else with Non-classical models?)

> > > But also the interpretations like
> > > A as "A is provable in T*"
> > > and
> > > ~A as "~A is provable in T*"
> > > seem much more "natural"
> > > I know there is no good ultimate reason to go for the natural feel.
> > > But the oppostite needs a real advantage before using it, and that
> > > advantage i do not see.

> > There are two points here that I have made clear in my parallel reply
> > to Moeblee. This non-classical interpretation is more or less forced
> > on us via the Main Postulate of NAFL, i.e. it is the Main Postulate
> > that is sacred in NAFL. What we are sacrificing is the classical
> > meaning of the negation symbol "~" as strict negation. For clearly in
> > a non-classical NAFL model of T in which P&~P is the case, the
> > intepretation I have given makes both "P" and "~P" non-classically
> > true in the sense that both P and ~P are indeed unprovable in T*. So
> > clearly one cannot be the strict negation of the other.

> Don't understand this here

> What do you mean by Main Postulate of NAFL?

> > NAFL is a paraconsistent logic in the sense that it does not allow
> > deduction of an arbitrary proposition from P&~P.

> Would you please give the logical axioms and rules of inference for
> NAFL already? Sheesh, you keep talking and talking and talking about
> NAFL, but I don't know how anyone except you can rally know what's up
> with it if you won't specify such basic things.

> How would you respond if someone kept telling you about some notion,
> call it 'Finitary Non Fregean Logic', aka 'FNFL' , but wouldn't tell
> you its axioms, inference rules, or even define what a FNFL theory IS?