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\begin{document}
\title{Comment on ``On the logical consistency of special relativity
theory and non-Euclidean geometries: Platonism versus formalism''}
\author{Radhakrishnan Srinivasan\thanks{R \& D Group,
IBM India Software Labs, 5th floor, Golden Enclave,
Airport Road, Bangalore~560017, India.
E-Mail:~\texttt{sradhakr@in.ibm.com}}}
\date{}
\maketitle
\begin{abstract}
As observed in the PhilSci preprint ID Code~1255~[1], consistency in the
author's proposed non-Aristotelian finitary logic~(NAFL) demands that
Euclid's fifth postulate must necessarily be provable from the first four,
and that diagrammatic reasoning with Euclidean concepts must necessarily be
admitted into the rules of inference for plane neutral geometry in order
to argue for the said provability. Two important consequences, namely,
the indispensable role of diagrams as formal objects of Euclidean
geometry in NAFL and the resulting NAFL concept of `line' as an infinite
proper class of line segments are highlighted and elaborated upon.
A misleading comment in Remark~6 of [1], regarding negation for
undecidable propositions in the theory of special relativity~(SR), is
corrected. This comment is unrelated to the main argument in [1] and
the resulting analysis reinforces the conclusions of [1] that negation
and implication are problematic concepts for undecidable propositions in SR.
\end{abstract}
\section{The essential role of diagrammatic reasoning in NAFL and the
resulting concept of `line' as an infinite proper class in Euclid's geometry}
Consider plane neutral geometry~(NG) as formalised in classical
first-order predicate logic~(FOPL). Our goal in this section is
to highlight and further elaborate upon two important results deduced
in~\cite{srneg}, namely, the \emph{essential} role of diagrams as formal
geometric objects in the non-Aristotelian finitary logic~(NAFL) proposed
by the author~\cite{nafl, nafl2} and the consequent NAFL concept of
`line' as an infinite proper class of line segments. In contrast, classical
logic admits a line as a mathematical object and considers diagrams to
be informal (and sometimes unreliable) reasoning tools.
An excellent reference for the formalization of diagrammatic reasoning
is the work of Miller~\cite{miller}. The modern attitude
towards diagrams is neatly summarized by the following quote
from Chapter~1 of~\cite{miller}:
\begin{quote}
It has often been asserted that proofs like this, which make crucial
use of diagrams, are inherently informal. The comments made by Henry
Forder in \emph{The Foundations of Geometry} are typical: `Theoretically,
figures are unnecessary; actually they are needed as a prop to human
infirmity. Their sole function is to help the reader to follow the
reasoning; in the reasoning itself they must play no part.' \dots
The transition from mathematics with geometry at its core to
mathematics with arithmetic at its core had a profound influence
on the way in which people viewed geometric diagrams. When geometric
diagrams were seen as the foundation of mathematics, the geometric
diagrams used in these proofs had an important role to play. Once
geometry had come to be seen as an extension of arithmetic, however,
geometric diagrams could be viewed as merely being a way of trying to
visualize underlying sets of real numbers. It was in this context
that it became possible to view diagrams as being ``theoretically
unnecessary'' and mere ``props to human infirmity''.
\end{quote}
Miller~\cite{miller} proceeds to develop the thesis that diagrams can
indeed be considered as formal objects in plane Euclidean geometry and
that diagrammatic proofs are as rigorous as their sentential counterparts.
NAFL takes this a step further: diagrams are \emph{necessarily} formal
objects and diagrammatic reasoning must \emph{necessarily} be admitted
into the rules of inference of classical NG in order to argue for the
provability, required by consistency in NAFL, of Euclid's fifth postulate
from the first four; see Sec.~2 of~\cite{srneg}.
The theory NG(NAFL) is the NAFL version of NG with diagrammatic reasoning
(admitting only Euclidean concepts) introduced formally into the rules of
inference. For an idea of how this can be done, see~\cite{miller}.
In order to argue for the provability of the fifth postulate
in NG(NAFL), `parallel' was re-defined in~\cite{srneg} as follows:
\begin{definition}\label{df1}
Two distinct coplanar lines are parallel if and only if they are
equidistant at all points, where distance between the lines at a
point (on either line) is defined as the length of the
perpendicular to the other line dropped from that point.
Similarly, a line segment AB that does not lie on a line $l$ is
parallel to $l$ if and only if AB is equidistant from $l$ at
every point of AB.
\end{definition}
It was stated in \cite{srneg} that Definition~\ref{df1} is first due
to Posidonius in the first century B.C.; however, it appears that
Archimedes~(287--212~B.C.) in his treatise \emph{On parallel lines}
had already defined parallelism similarly. Definition~\ref{df1} can
be extended in an obvious manner to define parallelism between two
coplanar line segments and we assume that this has been done.
Proposition~\ref{pr1} of \cite{srneg} is stated below for
convenience:
\begin{proposition}\label{pr1}
Given a line $l$ and a point \emph{P} at an arbitrary non-zero distance
$D$ from $l$, there exists a unique line segment \emph{M}
through \emph{P} parallel to $l$, such that \emph{P} is
at the midpoint of \emph{M} and \emph{M} is of a given arbitrary non-zero
length $L$. Here $D$ and $L$ are (standard) finite lengths.
The line segment \emph{M} will remain parallel to
$l$ when extended by an arbitrary (standard) finite length
such that \emph{P} continues to remain at the midpoint of M.
Here `parallel' is defined in Definition~\ref{df1}.
\end{proposition}
A diagrammatic proof of Proposition~\ref{pr1} was given in~\cite{srneg},
with a scaling argument to represent line segments~M of arbitrary length
in a single diagram drawn to scale; different line segments will imply
different scale factors for the diagram. Here we emphasize that Tarski's
axioms for NG do not permit the consideration of `arbitrary' entities, such
as, line segments; if NG is suitably modified to permit this, it becomes
classically incomplete, since NG would then interpret classical Peano
Arithmetic~\cite{simpson}. NAFL, however, rejects G\"odel's incompleteness
theorems~\cite{nafl2} and so NG(NAFL) can admit arbitrary line segments
without becoming incomplete~(equivalently in NAFL, inconsistent). In fact
consistency \emph{demands} that NG(NAFL) interpret Peano Arithmetic,
for the notion of `arbitrary' line segment is essential in the above
diagrammatic proof.
The rationale for the diagrammatic proof is that NAFL permits only
constructive Euclidean concepts, and the diagram represents precisely
the constructions we must have in mind when we think of `line' or
`line segment'. The scaling argument is simply justified by noting
that the ideal `lengths' of line segments in NG(NAFL) have nothing to do
with the \emph{physical} length of the line segment drawn in the
diagram; we are completely free to attribute an arbitrary scale factor
to represent arbitrary \emph{ideal} lengths, which are \emph{mental}
constructions, by a single \emph{physical} diagram. Let `\emph{idlg}'
represent the unit ideal length of a line segment in NG(NAFL), and let the
(usual, real-world) `inch' be the corresponding unit length in the diagram.
One might take the scale factor for the line segment~M of
Proposition~\ref{pr1} in the diagram as ``one inch = one
hundred~\emph{idlgs}''; one could also take the scale factor as,
for example, ``one inch = one hundred thousand~\emph{idlgs}'', etc.
It should be emphasized that these two scale factors correspond to two
\emph{different} line segments~M in NG(NAFL), represented in a
single diagram. The point here is that there is no reality for the
question of how many \emph{idlgs} are `actually' contained in one
inch, since NAFL does not require our mental constructions of line
segments and their unit length scale \emph{idlg} to have any connection
whatsoever with the real world; formally, this question is undecidable
in NG(NAFL), and the Main Postulate for NAFL~\cite{nafl,nafl2} makes
truth for such a proposition purely \emph{axiomatic} in nature. It follows
that in the absence of any axiomatic declaration as to the value of the
length $L$ (in \emph{idlgs}) of the line segment $M$ represented in the
diagram, \emph{all} ideal length scales are present in one inch in a
superposed state, as required by NAFL. It is this fact that
permits the scaling argument in NAFL, and the consequent representation of
arbitrarily long (mentally constructed) line segments superposed in a single
(physical) diagram. In fact this scaling argument is absolutely essential for
consistency of NG(NAFL), as will be argued shortly. In contrast, FOPL
requires that the line segment~M in the diagram represents one and only one
instance of Proposition~\ref{pr1}, which prevents the diagram from being a
direct proof of Proposition~\ref{pr1} and amounts to a Platonic assertion
that the line segments of NG `really' exist. Clearly, FOPL requires that
there is a fact of the matter as to the conversion factor between the ideal
length scale~\emph{idlg} and the physical length scale of one inch in the
diagram, despite the undecidability of this proposition in NG; different
conversion factors only succeed in redefining \emph{idlg} in terms
of inches from the FOPL point of view, and we are looking at one and only
one line segment~M of a fixed length. Of course,
similar diagrammatic proofs can be given in NG(NAFL) for each of the other
four postulates of Euclid which must also be asserted as tautologously
true in NG(NAFL). The fifth postulate was perceived to be
`counter-intuitive' precisely because FOPL does not permit such a
diagrammatic proof, as will be explained below.
Playfair's postulate, logically equivalent to Euclid's fifth
postulate~$\psi$, asserts that for every line $l$ and for every
point P that does not lie on $l$, there exists a unique line $m$
through P that is parallel to $l$. We use Playfair's postulate
interchangeably with $\psi$. It was argued in \cite{srneg} that
Proposition~\ref{pr1} provides a \emph{direct, constructive} proof of
$\psi$ in NG(NAFL). An indirect proof of $\psi$ may also be obtained
in NG(NAFL) by dropping perpendiculars from the end-points of the line
segment~M to the line $l$ of Proposition~\ref{pr1} and exhibiting
diagrammatically the rectangle bounded by $M$, $l$ and the two
perpendiculars. Since only Euclidean concepts are permitted in
diagrammatic proofs, the existence of such a rectangle is equivalent to
and proves $\psi$ in NG(NAFL). It was noted in \cite{srneg} that such a
diagrammatic proof is not legitimate in classical NG, because
FOPL requires that even infinitely many instances of
Proposition~\ref{pr1} (with Euclidean concepts) do \emph{not} prove
$\psi$ in NG; none of these instances represent the line $m$, which has
infinite length and is a separate mathematical object in classical logic.
Thus we have the classical anomaly that infinitely many instances
of Proposition~\ref{pr1} should not prove $\psi$ in NG, but if
the diagrammatic proof (with Euclidean concepts) is permitted, $\psi$
can be inferred \emph{indirectly} from a single such instance.
It was concluded in~\cite{srneg} that consistency in FOPL
necessarily requires the uninterpreted entities of NG to have
a non-constructive existence, and consequently, $\psi$ must
necessarily be undecidable in NG with non-Euclidean concepts permitted.
The above classical scenario is diametrically opposite
to the notion of consistency in NG(NAFL), as noted in \cite{srneg}.
The first observation is that the scaling argument in the diagrammatic
proof of Proposition~\ref{pr1} in NG(NAFL) neatly removes the
classical anomaly noted above; a single instance of Proposition~\ref{pr1},
as represented in the diagram, can be \emph{interpreted} via the
scaling argument to represent infinitely many instances and hence
provides a direct as well as indirect proof (via the rectangle
construction noted above) of Proposition~\ref{pr1}. Hence it is
absolutely essential to incorporate the scaling argument into the
rules of inference of NG(NAFL). Once this is done, we might as well
rename the resulting theory as EG(NAFL), where `EG' stands for Euclidean
geometry; all the axioms of classical EG are theorems of EG(NAFL),
provable by diagrammatic constructions. Here we have in mind a
finitely axiomatizable version of EG proposed by Tarski~\cite{simpson},
all of whose theorems are provable by elementary ruler and compass
constructions. EG(NAFL) has only diagrammatic rules of inference and no
axioms. It follows that diagrams of EG(NAFL) are formal objects that come
prior to and are more fundamental than the axioms of Tarski's elementary EG.
Indeed, since all of the theorems of EG are provable by ruler and
compass constructions, it is obvious that the axioms of EG must also be
so provable. It is striking that diagrammatic reasoning, which plays a
crucial and indispensable role in preventing inconsistency in EG(NAFL),
would be considered as `inconsistent' and `unreliable' in classical
neutral geometry. Note that the diagrams are physical (real-world)
devices by which we communicate our mental geometric constructions
as `proofs' of the Euclidean postulates to our fellow-human beings.
As shown in Sec.~2 of \cite{srneg}, we can only have these Euclidean
constructions in mind; the classical non-Euclidean models neccessarily
have to assume the metamathematical (Platonic) truth of the Euclidean
postulates and so are not models at all by the NAFL truth definition.
If there were only one human being in the whole world, such an
individual need not construct any diagrams as formal devices to
prove the Euclidean postulates; the mental constructions would
suffice for this purpose.
The second observation is that the line segment~M of
Proposition~\ref{pr1} is of \emph{arbitrary} length~$L$, which is to be
interpreted in NAFL as being in a superposed state of (quantification
over) all possible \emph{standard} values for $L$. As
noted in \cite{srneg}, this amounts to an explicit construction
for the line $m$ because there are no nonstandard models for
arithmetic (and hence, for EG) in NAFL. Thus the line~$m$
of Proposition~\ref{pr1} is modelled as the union
of an infinite proper class of line segments~M, as represented
by the superposed state. Here we wish to make a few remarks on the
notion of infinite class, which is a \emph{proper class} and not a
mathematical object in NAFL~\cite{nafl}.
In NAFL, whenever infinitely many mathematical objects identified by
a given property (such as, that of being a natural number) exist within
a theory, that theory must also necessarily admit the corresponding
infinite class of such objects. The class comprehension scheme
is necessarily a theorem of NAFL theories which admit infinitely many
objects in the universe of discourse; quantification is
restricted to be only over objects that belong to classes.
NAFL interprets the axiom of extensionality for classes
(which is also necessarily a theorem of such NAFL theories) to mean
that an infinite class must be identified by \emph{all}
and \emph{only} its elements; the infinite proper class is by
itself not a mathematical object. The existence and uniqueness
of an infinite class can be inferred from, and is equivalent to,
the existence and uniqueness of every element of that class.
In the example of Proposition~\ref{pr1},
EG(NAFL) requires it to be universally quantified over all possible
lengths $L$; that there are infinitely many such line segments~M of
arbitrarily large lengths results from Euclid's second postulate,
which is a theorem of EG(NAFL). It immediately follows in NAFL that
an infinite proper class of such line segments, identified
constructively by a given property, exists; call this class $C$.
The assertion that $m$ is parallel to $l$ is to be
interpreted in NAFL as meaning that every line segment of the infinite
class $C$ is parallel to every line segment of the corresponding
infinite class constituting $l$. The `existence' and `uniqueness' of
the line $m$ is to be identified with, and may be inferred from,
the existence and uniqueness of every element of $C$.
At this stage the reader might wonder \emph{why} NAFL requires
quantification to be restricted to classes when infinitely
many objects are involved. This has been formally established
in~\cite{nafl2}; here we will give an intuitive explanation as follows.
Let us take the example of a line segment
of initial length $L_0$, and successively extended (equally on both
sides of the segment) to lengths $2L_0$, $3L_0$, $4L_0$, etc.
NAFL asserts that the process of extending the line segment may be said
to have been `completed', or equivalently, reached an `arbitrary'
length $nL_0$, if and only if the infinite proper class
$C=\{L_0, 2L_0, 3L_0, \dots, nL_0, (n+1)L_0, \dots,\}$ exists.
This infinite class represents \emph{all} extensions of the original
line segment \emph{at the same time}. If $C$ did not
exist, one can only imagine the line segment being extended, say, one
instance (in general, at most finitely many instances) at a time, but
by induction, such a process can never ever be `completed' and it would be
wrong to quantify over `all' such extensions. Equivalently, one can never
exhaust the class $N$ of natural numbers by counting them one at a time;
induction says that there will always be infinitely many natural numbers
left to be counted. Therefore quantification over infinitely many natural
numbers automatically implies the existence of $N$ in NAFL~\cite{nafl2}.
However, $N$ is a paradoxical entity because \emph{any} element of $N$ is
clearly accessible by counting one at a time. It is clear that `any'
in the preceding sentence does not translate to `all', for then the above
induction (that $N$ cannot be exhausted by counting one at a time) would be
violated. This is the intuitive explanation for why $N$ must be a proper
class in NAFL; for the formal argument that results from the axiomatic
nature of NAFL truth, see Sec.~3 of ~\cite{nafl}. To summarize,
for the purposes of quantification over infinitely many mathematical
objects, NAFL requires that $N$ must necessarily exist as a `completed'
infinite class, but the `incomplete' or `potential' nature of $N$ is
recognized by not admitting it as a mathematical object~(i.e., a set)
in NAFL theories. Thus NAFL provides the correct logical framework for
assertions denying the existence of a `completed' infinity by a
long list of famous logicians/mathematicians/philosophers, including
Aristotle, Gauss, Kronecker, Poincar\'e and Brouwer; it is somewhat
ironic that, in order to accomplish this feat, NAFL has to deny
two of the most fundamental laws of classical logic enunciated by
Aristotle, namely, the laws of non-contradiction and the
excluded middle~\cite{nafl}.
Classical logic~(FOPL), however, insists that we can talk about `all'
natural numbers or `all' elements of $C$ without ever invoking the
existence of $N$ or $C$; the induction mentioned above that such a
counting process is always incomplete then leads us to nonstandard models
of arithmetic and the corresponding Platonism which is rejected by NAFL
as inconsistent and unacceptable~\cite{nafl2}. That NAFL should consider
a `line' to be the union of the infinite proper class $C$ (rather than a
separate mathematical object as in classical logic) is also entirely
natural and consistent with the axiomatic nature of NAFL truth. Indeed, we
do not have any mental picture (i.e., a `construction') of a \emph{completed}
line as a separate mathematical object; we can only conceive of an infinitely
long line as obtained by the process represented in $C$ of extending
a (finite) line segment infinitely many times. This is an inherent
human limitation of being unable to conceive of the infinite and is
intuitively another reason why $N$ and $C$ must be proper classes in NAFL;
the classical position that infinite entities nevertheless `exist' as
mathematical objects (i.e., as sets) independently of human limitations
violates the axiomatic nature of NAFL truth and leads to Platonism and
inconsistency (from the NAFL viewpoint).
In conclusion, we have demonstrated in this section the importance of
diagrams as formal objects in EG(NAFL) and that the provability of
Euclid's fifth postulate in EG(NAFL) (as demanded by consistency) poses
severe restrictions on the concept of `line', which cannot be treated as
a mathematical object; consequently, quantification over `lines' is banned
in EG(NAFL). It was stated in \cite{srneg} that a problem for
future research is to figure out how the classical continuum of
real numbers can be handled in EG(NAFL). Indeed, not only `lines', but
even `points' and `line segments' must be \emph{analytically} represented
in any continuum theory by (collections of) real numbers, which are also
infinite proper classes in NAFL. So the question of how points and/or line
segments may be treated \emph{analytically} (rather than diagrammatically)
as mathematical objects in EG(NAFL) must be resolved. The author believes
that \emph{physically meaningful} statements about (sets/classes of)
real numbers, as represented by diagrams, could possibly be treated
analytically in NAFL by some sort of `translation' procedure into
Peano Arithmetic (or equivalently, finite set theory); such a procedure
will have to differ radically from its classical equivalents.
The resulting NAFL theory, if accomplished, should
satisfactorily resolve classical paradoxes, such as, the Banach-Tarski
paradox or Zeno's paradoxes of motion, without having to deny the
existence of precise positions in space or instants in time as done
by Lynds~\cite{lynds1, lynds2}. Indeed, it is already clear that
`points'~(real numbers) must exist in NAFL theories as infinite proper
classes and these can certainly represent precise positions in space
or instants in time. It is only quantification over sets/classes of
real numbers that is problematic in NAFL, for the real numbers
constitute neither a set nor a class (being themselves proper classes);
here justification of any formal translation procedure of statements
that involve quantification over real numbers into finite set theory
must necessarily come from diagrams as formal objects. Thus the formal
existence of diagrams would still play an essential and indispensable
role in justifying such an analytical treatment of geometry/analysis.
The author also believes that any continuum theory of space, time and
matter, even if possible to formulate consistently in NAFL, is
nevertheless an approximation to reality and will fail at quantum scales.
As noted in the concluding remarks of \cite{nafl2}, the ultimate NAFL
theory that describes reality must be one in which \emph{everything},
including space, time and matter, is discrete~(quantized).
\section{Relativistic determinism -- the clash with logic}
As in \cite{srneg}, we consider the theory of special relativity~(SR)
formalized in classical first-order predicate logic~(FOPL).
The context for this section is best explained by the following
extensive quote from Sec.~1 of \cite{srneg}:
\begin{quote}
``Let A and B be relatively moving inertial observers who happen to
coincide in space at a given instant defined by $t=0$ in A's frame and
$t^{\prime} = 0$ in B's frame. Let $C$ be an instantaneous
event that is localized in space and distant to both A and B.
Let $U(IBC)$ define a non-trivial universe of
material objects with certain well-posed
initial-boundary conditions~$IBC$. Define the proposition $P$
as ``From A's point of view, $C$ occurs in
$U(IBC)$ when A's local clock reads $t=0$'' and the
proposition $Q$ as ``From B's point of view,
$C$ occurs in $U(IBC)$ when B's local clock reads
$t^{\prime}=T$''. Here $T>0$ is a constant obtained from the
Lorentz transformations as applied to the event $C$ in A's and B's
inertial frames. Relativistic determinism asserts that if $P$ is
true then $Q$ must be true (or $P \Rightarrow Q$); in other words,
B's future at time $t^{\prime}=0$ is determined by the fact
that A has observed $C$ at precisely that instant (when A and B
coincided) and so B must necessarily observe $C$ at $t^{\prime}=T$.''
``In order to obtain a logical contradiction from the above scenario,
let us further stipulate that the proposition
``Event $C$ occurs in $U(IBC)$'' is undecidable in SR, i.e.,
in particular, neither A nor B can either prove or refute this proposition.
Such undecidability could occur in many ways, for example, as a result of
G\"odel's incompleteness theorems; alternatively, $C$ could be a
probabilistic event, such as, the outcome of a coin toss experiment
or some quantum phenomenon; or else, $C$ could be completely
unpredictable as a result of being decided by the instantaneous free
will of a human being. It immediately follows that $P$ and $Q$ are
undecidable in SR; see the ensuing paragraph for the definition of
such undecidability. Note however, that SR requires
$P \Leftrightarrow Q$ to be a theorem despite the undecidability of $P$
and $Q$; this fact immediately makes SR inconsistent in
the non-Aristotelian finitary logic~(NAFL) proposed by the author
in~[4] and [5] (in particular, see Remark~5 of
[4] and Section~2.2 of [5]). This argument for
inconsistency of SR in NAFL is simpler than the one given using inertial
frames in~[6]. It follows that the philosophy of formalism
as embodied by NAFL~[5] immediately rejects relativistic
determinism. The goal of this paper is to show
that an inconsistency can be deduced in SR even
within FOPL, if one insists on formalism.''
``Henceforth, whenever we refer to A~(B), it is to be
understood that our argument may apply equally well to any observer in
A's~(B's) set of inertial frames. Note that we require the following
restrictions regarding propositions involving $P$ and $Q$. The
truth of $P$~($Q$) can be \emph{asserted} (via an
observation, for example) or \emph{deduced} in SR \emph{only} by A~(B).
However, B~(A) can consider and either accept or refute in SR any
assertion/deduction of the truth of $P$~($Q$) made by A~(B);
but B~(A) cannot \emph{assert} or \emph{deduce} the truth of
$P$~($Q$). The undecidability of $P$~($Q$) in SR means that A~(B)
can neither prove nor refute $P$~($Q$) in SR. $P \Rightarrow Q$
is a theorem in B's (and not A's) frame; in other words, only B
has the right to deduce $Q$ in SR from an assertion of $P$ made
by A (if B happens to agree with A's assertion). Similarly,
$Q \Rightarrow P$ is a theorem in A's (and not B's) frame.
In fact \\ $P \Rightarrow Q$ and $Q \Rightarrow P$ are illegitimate
propositions in A's and B's frames respectively. The idea behind
these restrictions is to allow A~(B) to consider the truth of
$Q$~($P$) without undermining the Lorentz transformations.''
``In particular, $Q$ is undecidable in SR, which
means, as noted above, that B can neither prove nor refute $Q$ in SR.
The question we wish to consider is as follows. Given that A has
asserted the truth of $P$, and given that $P \Rightarrow Q$ is a
theorem of SR in B's frame, can B accept A's assertion and conclude
$Q$? In the metatheorem that follows, we argue that B in fact has a
formal refutation of A's assertion; i.e., B has a
proof of $\neg P$ in SR and hence B has no way to
conclude $Q$ despite A's assertion of $P$. However,
B does not have the right to use $Q \Rightarrow P$ along with
the said proof of $\neg P$ to deduce $\neg Q$, because, as noted
above, $Q \Rightarrow P$ is a theorem of A's (and not B's) frame.
Hence $Q$ continues to remain undecided in SR (in B's frame)
despite A's assertion of $P$. See Remark~6 below for
further clarifications.''
``Before proceeding to the main result in the metatheorem below,
we observe that an additional restriction is necessary, as follows.
A and B accept each other's observations/theorems as true/valid
if and only if there is no disagreement with (or a refutation of) the
observations or any step used in the proof of the said theorems,
including the theorems themselves. As an example, suppose A
asserts $\neg P$ and concludes $\neg Q$ from
the theorem $Q \Rightarrow P$ of A's frame. Then B accepts
A's assertion $\neg P$ as true and A's inference
$\neg P \Rightarrow \neg Q$ as valid despite that fact that
such an inference is illegal in B's frame. Thus B accepts A's
conclusion $\neg Q$ as true; i.e., B does not insist that
because of the illegality of the inference
$\neg P \Rightarrow \neg Q$ in B's frame, there must exist a
model for SR in which A asserts $\neg P$ and B asserts $Q$.''
\end{quote}
As explained in the above quote, we considered in~\cite{srneg} the
situation when A asserts $P$ and we concluded that B cannot accept this
assertion because B has a formal refutation of $P$; consequently,
B concludes that $Q$ continues to remain undecided in SR
despite A's assertion, as noted in the metatheorem of~\cite{srneg}.
Conversely, we argued in Remark~3 and Remark~4 of~\cite{srneg}
that A cannot likewise refute an assertion of $Q$ made by B; hence A
accepts that $P$ follows from the theorem $Q \Rightarrow P$ of A's frame.
In the present note, we wish to focus on the misleading claim made in
the final paragraph of the above quote (and also in Remark~6
of~\cite{srneg}) that B will automatically accept any assertion of
$\neg P$ made by A, because B has a formal proof of $\neg P$ in SR.
The problem with this claim is that B's concept of the negation
of $P$ is not necessarily the same as that of A and so the said
claim leads to the incorrect consequence that B must necessarily accept
A's conclusion of $\neg Q$ from $\neg P$. This is best illustrated
with the following example.
Let the event~$C$ in the above quote denote
the outcome `heads' in an instantaneous coin toss experiment~$E$ that
is distant to both A and B; the definitions of $P$ and $Q$ follow.
Further, let $R$ denote ``From A's point of view, the coin toss $E$
occurs in $U(IBC)$ when A's local clock reads $t=0$'', and let $S$ denote
``From B's point of view, the coin toss $E$ occurs in $U(IBC)$ when
B's local clock reads $t^{\prime}=T$''; here $T$ is the same positive
constant obtained from the Lorentz transformations as in the definition
of $Q$. For clarity, information about the spatial locations of events
is suppressed from all propositions defined here and in~\cite{srneg},
although strictly speaking, such information must be considered to be tacitly
present. Finally, let $U$ denote ``From A's point of view, the outcome
of the coin toss $E$ in $U(IBC)$ is `tails' when A's local clock
reads $t=0$'', and let $V$ denote ``From B's point of view, the
outcome of the coin toss $E$ in $U(IBC)$ is `tails' when B's local
clock reads $t^{\prime}=T$ ''. It follows that $P$ and $Q$ are
undecidable in SR (in the sense noted in the above quote) and so
are $U$ and $V$; further, $U$ and $V$ must satisfy the same
restrictions as $P$ and $Q$ respectively, as quoted above, and may
be substituted for $P$ and $Q$ in the metatheorem of~\cite{srneg}.
\begin{remark}\label{rm1}
In the first instance, suppose that A can prove $R$ in SR;
consistency demands that $IBC$ must be such that B must also be able
to prove $S$ in SR. Note that B does not have to rely on A's claim of
$R$ in order to prove $S$; indeed $B$ has a refutation of $R$ by
the metatheorem of~\cite{srneg}. Hence $R \Leftrightarrow S$ is
a theorem of SR and decidable propositions by themselves do not pose
any problem for relativistic determinism. It immediately follows
that
\begin{equation}\label{eq1}
U \Leftrightarrow \neg P
\end{equation}
is a theorem of SR in A's frame, and
\begin{equation}\label{eq2}
V \Leftrightarrow \neg Q
\end{equation}
is a theorem of SR in B's frame. Next suppose A claims $U$; from A's
point of view, (\ref{eq1}) implies that a claim of $U$ is equivalent
to a claim of $\neg P$. Since \\ $\neg P \Rightarrow \neg Q$ is a theorem
of A's frame, A concludes $\neg Q$. However, by the metatheorem of
\cite{srneg}, B has a refutation of $U$ \emph{and} a proof of $\neg P$
in SR. Clearly, B does not accept the equivalence in (\ref{eq1}). Since
B has a refutation of $R$ in SR (again, by the metatheorem
of \cite{srneg}), B only accepts $\neg P$ in the sense
that the coin toss did not happen at all when A and B
coincided at $(t=0, \: t^{\prime}=T)$. Hence B concludes that A has
\emph{deduced} $\neg P$ from a false assertion of $U$, and consequently
\emph{does not accept} the validity of A's conclusion $\neg Q$.
So from B's point of view there must still exist a model for SR in
which $Q$ is the case, despite A's claim of $U$; equation~(\ref{eq2})
shows that this is in agreement with the metatheorem of~\cite{srneg},
which asserts that from B's point of view, there must exist a model for
SR in which $\neg V$ is the case despite A's claim of $U$. Conversely,
suppose B claims $V$, or equivalently, from (\ref{eq2}), $\neg Q$. A
accepts this claim and concludes $U$ from the theorem $V \Rightarrow U$
of A's frame. A also accepts the validity of B's conclusion of
$\neg P$ from the theorem $\neg Q \Rightarrow \neg P$ of B's frame,
despite its illegality in A's frame, and one sees that A is acting
consistently with (\ref{eq1}). There is no clash with the metatheorem
of~\cite{srneg} because, as noted in Remark~3 and
Remark~4 of~\cite{srneg}, A cannot refute any claim of $Q$ or
$V$ made by B. It follows that A accepts the equivalence in
(\ref{eq2}) and one again sees the asymmetry noted in
Remark~3 and Remark~4 of~\cite{srneg}.
\end{remark}
\begin{remark}
Secondly, suppose that A does not have a proof or refutation
of $R$ in SR. Consistency demands that B likewise does not have a proof
or refutation of $S$ in SR. By our definition of undecidability,
$R$ and $S$ are undecidable in SR. Next suppose that A claims
$\neg P$; it is easy to see that, by our definitions, B will accept
this claim \emph{if and only if} it is also accompanied by a claim
of $\neg R$ by A. Consequently, B accepts the truth of A's
conclusion $\neg Q$ in this instance, despite the illegality of A's
inference $\neg P \Rightarrow \neg Q$ in B's frame. But if
A claims $R \& \neg P$, the conclusions of Remark~\ref{rm1}
will again hold. Conversely, A will accept an assertion of $\neg Q$ by
B irrespective of the status of B's claim with respect to $S$;
consequently, A unconditionally accepts B's conclusion of $\neg P$ from
the theorem $\neg Q \Rightarrow \neg P$ of B's frame, despite its
illegality in A's frame.
\end{remark}
In summary, we have established here and in \cite{srneg} that the
concepts of negation and implication for undecidable propositions
in SR are problematic in FOPL and lead to inconsistency from the point
of view of formalism. The non-Aristotelian finitary
logic~(NAFL)~\cite{nafl, nafl2} proposed by the author also demands that
these concepts be handled in a different manner for undecidable
propositions; the problems with FOPL should encourage further
investigation of these concepts in NAFL, despite that fact that
NAFL refutes SR as noted in \cite{srneg}.
\section*{Dedication}
The author dedicates this research to his son R.~Anand and wife
R.~Jayanti.
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