[extropy-chat] Non-classic logics

[Ian Goddard]

--- Technotranscendence <neptune at superlink.net> wrote:
 
> Modal operators might be an example where
> commutativity fails.  L~P is not the same as ~LP -- 
> where L stands for the "It is necessarily the
> case that..."  Ditto for M~P and ~MP -- where M
> stands for "It is possibly the case that..."


 The same holds for the quantifiers in predicate
logic. For the existential quantifier (Ex) it's the
case that -ExPx (where '-' denotes NOT) is not the
same as Ex-Px. -ExPx is the same as the universally
quantified statement Ax-Px. And the same holds if we
invert each Ex to Ax and the Ax to Ex in the last two
sentences.

 That modal-predicate analogy exists because the modal
operator [] ("It is necessary that") is analogues to
the universal quantifier Ax ("For all x") since []P is
true in some world w iff P is true in all other worlds
accessible to w. And the modal operator <> ("It is
possible that") is analogous to the existential
quantifier Ex ("For at least one x") since <>P is true
in some world w iff P is true in at least one other
world accessible to w. Gamut covers this analogy in
Vol II (p.23). [*] 

(Btw, with [] I try to emulate the box used for
necessity and with <> to emulate the diamond used for
possibility. This would also help differentiate these
operators from predicates letters.)

[*]
http://www.press.uchicago.edu/cgi-bin/hfs.cgi/00/7088.ctl



> With classical propositional logic (cpl), wouldn't
> the same be true for
> ~ and distribution?  ~(P & Q) is not the same as
> (~P) & (~Q) -- think of
> the case where the truth value of P does not equal
> the truth value of Q,
> assuming bivalent logic.  In that case, the former
> statement is true
> while the latter is false.


 I guess DeMorgan's rule is a special distribution
property for NOT that affects the operator (OR or AND)
of the statement that NOT is distributed over such
that OR becomes AND or AND becomes OR. 

~(P & Q)  ::  (~P v ~Q)

~(P v Q)  ::  (~P & ~Q)

 When students would say to the professor in my first
logic course that something in logic is like
distribution in math, he'd be like, "Well, not
exactly." Although Copi's symbolic logic does refer to
some replacement rules I cited as distribution rules.
If there's one thing one notices in logic it's how
many different symbolic conventions and views there
can be from one source to another. It's not like that
in mathematics proper. 

~Ian

http://iangoddard.net


		
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