Re: T-relevant logic

On Jan 1, 8:03 am, Newberry <newberr...@xxxxxxxxx> wrote:
We need to replace the classical logic with truth-relevant logic (see
below.) Here is why.
Let us consider the sentence "There are no round squares":

~(Ex)(Sx & Rx)        (1)

If there are no round squares, to what are we then attributing non-
existence? Do round squares subsist in order to enable us to talk
about them even though they do not exist? Certainly not. The sentence
above can be interpreted as " 'round square' does not have a
denotatum."

We can also interpret (1) as "all squares are non-round."

(x)(Sx --> ~Rx)        (2)

But 'round square' still does not denote anything.

What about "All round squares are large"?

(x)((Sx & Rx) --> Lx)       (3)

Since 'round square' does not denote anything, (3) has exactly the
same meaning as "All  ghudalbirgs are large", that is, it has no
meaning.

Can (3) be interpreted as

~(Ex)((Sx & Rx) & ~Lx)        (4)  ?

That is, can we say then that " 'small round square' does not have a
denotatum"? Perhaps,  but (3) can also be interpreted as attempting to
attribute a property to round squares.

We need a logic where

~(Ex)(Px & ~Px)        (5)

is derivable, but

(x)((Px & ~Px) --> Qx)        (6)

is not. What kind of logic will accomplish this? We will not be able
to derive (6) if

P & ~P --> Q        (7)

is not derivable. (7) is indeed notoriously counterintuitive - it is
the paradox of material implication. This is a further indication that
we are proceeding in the right direction.


In the logic NAFL, P&~P-->Q is theorem of a consistent theory T in
which either P or ~P is provable. If P is undecidable in T (i.e.,
neither P nor ~P is provable in T), then P&~P-->Q is not provable in
T. This makes perfect sense in NAFL, where P&~P-->Q is equivalent to
Pv~PvQ, and Pv~P is provable in T when either P or ~P is provable (and
not otherwise). Note that there is absolutely nothing counter-
intuitive about this.

Since P&~P-->O is not provable in a consistent NAFL theory T in which
P is undecidable, there must exist a model for T in which P&~P is the
case, but an arbitrary proposition will not turn out true in that
model. At first signt the truth of P&~P in the said non-classical
model appears counter-intuitive, but it has a perfectly valid
explanantion. See the sci.logic thread "FOL/Inuitionistic logic versus
NAFL. Part 1. Failure of non-contradiction" for explanations:

http://groups.google.co.bw/group/sci.logic/browse_thread/thread/48894ac0f1d11787/f34781b18deff9c4?#f34781b18deff9c4



We have two options:
1)  P --> Q  #  ~P v Q  =  ~(P & ~Q)
2)  P --> Q  =  ~P v Q  =  ~(P & ~Q)
Several logics were proposed along the lines of 1)
* Lewis's system of strict implication
* Ackermann's system of strenge Implikation
* Church's system of weak implication
* Various systems of Anderson and Bellnap

There are two systems of logic in the second category
* truth-relevant logic
* occurrence-relevant logic
There are minor differences between the two.

None of the system in the first category achieved a general
acceptance. Nobody was able to provide a satisfactory interpretation
of any of them. Richard Diaz developed t-relevant and o-relevant
logics as a result of his criticism of the systems of Anderson and
Belnap.  We will use them for a different purpose.

In t-relevant logic NONE of the formulas below are derivable

(P & ~P) --> Q        (8)
(~P v P) v Q        (9)
~((P & ~P) & ~Q)        (10)

and in our interpretation they are all meaningless. By generalization
we also find that NONE of

(x)((Px & ~Px) --> Qx)        (11)
(x)((~Px v Px) v Qx)        (12)
~(Ex)((Px & ~Px) & ~Qx)        (13)

are derivable and they are all meaningless.

Let Pxy stand for x is the proof of y, let Qx be satisfied by only one
x = m. And let the Goedel number of

~(Ex)(Ey)(Pxy & Qy)        (14)

be m. Then (14) is a Goedel formula. We then observe that if

~(Ex)Pxm       (15)

then (14) is meaningless analogically to (13). This is in accordance
with our expectations; we came to a similar conclusion in another
thread.http://groups.google.com/group/sci.logic/browse_frm/thread/e295044b4d...
This gives us the basic insight that in all likelihood t-relevant
logic is the way to go.


The way forward is NAFL. I will open a long-pending thread on NAFL now
that I have cleared off my year-end commitments at IBM. I can always
hope that in this new year, someone somewhere in the academic
community will recognize NAFL's worth and be honest enough and
straightforward enough to acknowledge it.

Regards, RS

.

Relevant Pages

  • T-relevant logic
    ... If there are no round squares, to what are we then attributing non- ... Several logics were proposed along the lines of 1) ... Nobody was able to provide a satisfactory interpretation ...
    (sci.logic)
  • Re: T-relevant logic
    ... But 'round square' still does not denote anything. ... Several logics were proposed along the lines of 1) ... Nobody was able to provide a satisfactory interpretation ... and in our interpretation they are all meaningless. ...
    (sci.logic)
  • Re: Why Some Sentences Lack Truth Values
    ... whether or not you consider "All round squares ... are green" meaningless or not, it is perfectly possible to *reason* ... vacuous sentences does not make a system "semantically" complete. ...
    (sci.logic)
  • Re: Why Some Sentences Lack Truth Values
    ... of affairs. ... "There are no round squares" is ... but its negation is meaningless. ...
    (sci.logic)
  • Re: Why Some Sentences Lack Truth Values
    ... Yeah, yeah. ... "There are no round squares" is ... but its negation is meaningless. ... it is not a picture of any possible state of affairs. ...
    (sci.logic)