Bruno Marchal wrote:
Hi George, [out-of-line message]Imagine a three port device such as an electrically controlled switch. Let's say that this device has three lines connected to it: an input connected to p, a control connected to q and an output that we'll call qBp.
perhaps you could try to motivate your "qBp == If q then p".
I don't see the relation with "if q is 1 then p is known, and and if q is 0
then p is unknown". How do you manage the "known" notion.
If the control sets the switch to OFF (ie. q=0) , the output is not connected to the input. Therefore for anyone observing the output, the value of p is unknown, i.e., qBp = x. The electronic value of x can be any arbitrary value except 0 and 1 which are reserved for the possible known binary values.
If the control sets the switch to ON (ie. q=1), the output is connected to the input. Therefore for anyone observing the output, the value of p is known. It is either 0 or 1 depending on what the input p is.
Giving any logic L1, it is always interesting to look if
there is no other (perhaps better known) logic L2 such
that you can interpret L1 in L2.
Now it can be shown that most modal logic cannot be
easily or directly represented by a multi-valued logic, so
I doubt your proposal could work.
You can always try, but let us be sure we agree on the
"intuitive" meaning of Bp, in the case of the Smullyan's "self-
rererential" interpretation of the "B".
So we have a machine M.
The machine M print propositions from time to time.
Bp means that the machine M print p.
The machine could print Bp. In that case the machine
prints the proposition that she prints p.
A machine is self-referentially correct (SRC) if her use of
B is correct.
The following machine (programs) is SRC:
print "B hello"
The following machine (programs) is not SRC:
print "B hello"
because the machine pretend that she print hello, but will never
Of course, we will add conditions; mainly that
the machine' set of proposition will be closed for modus
ponens, i.e. that if she print (one day, soon or later) X, and
if she prints X->Y, then she will print Y.
To sum up: Bp means "M asserts p".
Bp is true (resp. false) if and only if M asserts p (resp. does not assert p).