Natural Science Forum / Physics / Relativity / September 2005

Of course A does not follow from B.
You don't understand what the symbols represent.

B') The transformation between coordinates (x,t) and (x',t') as used
     in two inertial frames is linear.
     If all the events on a light signal are described by a world line in
     one coordinate system with the equation x = ct, then the equation
     transforms to the equation x' = ct' in another coordinate system
     and vice versa.

A') There is a number L such that the following equation holds
     not only for events on the light signal, but for all events:
          ( x - c t ) = L * ( x' - c t' )

Then A' follows from B'.

Proof:

Linearity of the transformation means that there are numbers
P, Q, R, S such that the transformation can be written as

   {  x' = P x + Q t
   {  t' = R x + S t .

Then the transformation of the world line
   x = c t
can be found by inserting this in the transformation, giving:
   {  x' = P c t + Q t
   {  t' = R c t + S t
resulting in
   {  x' = (c P + Q) t
   {  t' = (c R + S) t
which gives after eliminating t:
   x' / t' = (c P + Q) / (c R + S)
or equivalently
   x' - (c P + Q) / (c R + S) t' = 0

Demanding that this is equivalent with
   x' - c t' = 0
implies that
   (c P + Q) / (c R + S) = c
and thus that
   (c P + Q) = c (c R + S)
and, for later use in (*):
   Q - c S = - c (P - c R)

So now we take the transformation equations and find that
they satisfy
   x' - c t' = P x + Q t - c (R x + S t)
             = (P - c R) x + (Q - c S) t
             = (P - c R) x - c (P - c R) t       (*)
             = (P - c R) (x - c t)
             = L ( x - c t ) ,
provided we take L as
   L = P - c R.

-----------

So we have proven that demanding that the world line
   x - c t = 0
gets linearly transformed into
   x' - c t' = 0 ,
implies that there is a number L such that the transformation satisfies
   x' - c t' = L ( x - c t )

Can you find the flaw in the proof, Valev?

Dirk Vdm

Dear Pentcho Valev

This is the truth-table of the "if and only if" double implication. So your
physical conditional is equivalence between statements.

But double implication is definible in terms of "->" and "&": p<->q =
(p->q)&(q->p). So your request is that two statements A and B form a
physical conditional "A implies B" if and only if they have the same truth
value. But we have just a connective for this, i.e. connective for
equivalence, "<->". But "ex falso sequitur quodlubet" and so your
connective for physical implication is not adequate for any case of
implication.

If you have the problem of sign up the physical implication you can see C.I.
Lewis, A Survey of Symbolic logic. Lewis read Russell's Principia finding
unsatisfied and limited its description of material implication. So
proposed that A "strictly implies" B if and only if is not possible A&
not-B. This is a modal view of implication that speacks about the modal
quality of the link between the antecedent and the consequent of an
implication.

By,

Roby

False, because arbitrary modeling decisions go into laying a foundation
to a theory and they need not have any certainty at all: such as the
decision to model matter as continuous, when no one believes that it
is, but hydrodynamics doesn't care.

What truth table above is called material equivalence and it is
bi-directional.

The standard truth table for implication (material implication) is:

 A  B  A->B
 T  T   T
 F  T   T
 T  F   F
 F  F   T

By the latter truth evaluations modus ponens is a tautology. By
material equivalence it is not!

It makes sense that modus ponens should be a tautology and any
contrivance to make it so is fully justified. Modus ponens is the
(tautology):

[A & (A -> B)] -> B