Re: The logic of atheism

Paul Holbach <paulholbachDELETETHENAME@xxxxxxxxxx> wrote:
prabbit1@xxxxxxxxxxxxxxx wrote:
Paul Holbach <paulholbachDELETETHENAME@xxxxxxxxxx> wrote:
Paul Holbach wrote:

Remark:
This definition [x is omniscient =def Ap((p & <>Kp) -> Kxp)]
also covers possible cases in which "p" is neither true
nor false or both true and false:

Of course, classical logicians, who adhere to the law of excluded
middle, consider such cases impossible.

Actually, that's not the law of excluded middle.

What is not the law of excluded middle?

From

http://en.wikipedia.org/wiki/Principle_of_bivalence

"In logic, the principle of bivalence states that for any proposition P,
either P is true or P is false.

This is not to be confused with the law of excluded middle and the law of
noncontradiction. See bivalence and related laws for a summary of the
differences.

In classical logic, the principle of bivalence is equivalent to the result
that there are no propositions that are neither true nor false. A
proposition P that is neither true nor false is undecidable. In
intuitionistic logic, sometimes the truth-value of a proposition P cannot be
determined (i.e. P cannot be proved nor disproved). In such a case, P simply
does not have a truth-value. Other logics, e.g. multi-valued logic, may
assign P an indeterminate truth-value."

Okay, I see what your point is.

The classical law of excluded middle (LEM), p v ~p, is formulated as an
inclusive disjunction with the following truth table:

0 v 0 => 0
0 v 1 => 1
1 v 0 => 1
1 v 1 => 1

That is, LEM alone is not violated in case both p and ~p are true. It
merely demands that /at least one/ proposition of the pair <p, ~p> be
true. So LEM guarantees that it is impossible that neither p nor ~p is
true.
Of course, the law of non-contradiction (LNC) is also classically
valid, guaranteeing that it is impossible that both p and ~p are true;
and it's the combination of LEM and LNC that guarantees that of the
propositional pair <p, ~p> /at least one/ proposition is true and /at
most one/ proposition is true:

((p v ~p) & ~(p & ~p)) <-> ~(p <-> ~p)

Actually, I just read this again and find myself still disagreeing (although
I started to agree in a previous post.) The principle of bivalence is the
one that requires that at least one proposition of the pair is true. The law
of excluded middle doesn't really address the truth value of the individual
parts ("p" and "~p") but simply says that the compound statement of "p or
~P" must be true. One example is in a future event: "P will happen tomorrow
or P will not happen tomorrow" is true based on the LEM (i.e. there's no
other "middle" possibility) but yet it can be argued that both "P will
happen tomorrow" and "P will not happen tomorrow" are, at best,
indeterminate currently or that either, taken alone, doesn't even have a
truth-value at all. So the principle of bivalence (p is either true or
false) may not apply here at all. The differences ARE admittedly very subtle
and it took me some time to understand them but I found a good article at
http://en.wikipedia.org/wiki/Bivalence_and_related_laws that shows the
differences pretty well (as well as how they tie in together.)


--
Mike

-------------------------------
"Our enemies are innovative and resourceful, and so are we. They never stop
thinking about new ways to harm our country and our people, and neither do
we," George W. "Shrub" Bush Aug 5, 2004
.

Relevant Pages

  • Re: The logic of atheism
    ... that's not the law of excluded middle. ... "In logic, the principle of bivalence states that for any proposition P, ... sometimes the truth-value of a proposition P cannot be ... Other logics, e.g. multi-valued logic, may ...
    (talk.atheism)
  • about law of excluded middle
    ... i have a question about how the law of excluded middle (LEM) is to be ... understood in classical logic compared with other logics. ... the law of bivalence states that for every formula p, ...
    (sci.logic)
  • Re: The logic of atheism
    ... "In logic, the principle of bivalence states that for any proposition P, ... This is not to be confused with the law of excluded middle and the law of ... sometimes the truth-value of a proposition P cannot be ... Other logics, e.g. multi-valued logic, may ...
    (talk.atheism)
  • Re: about law of excluded middle
    ... understood in classical logic compared with other logics. ... the law of bivalence states that for every formula p, ... addition to LEM? ...
    (sci.logic)
  • Re: The logic of atheism
    ... What is not the law of excluded middle? ... sometimes the truth-value of a proposition P cannot be ... Other logics, e.g. multi-valued logic, may ... LEM alone is not violated in case both p and ~p are true. ...
    (talk.atheism)