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Dear logicians,
with this post I propose to discuss some topics of the logic of being and
existence.
From a logical point of view the clear distinction of being and existence
was done by Russell in the &427 of its Principles of Mathematics: a
countable thing is a thing that have the being; to talk about something is
to admitt that such a thing is or have the being. On the other hand, the
existence is a property that someone of the thing that has the being have.
Existing objects are a subset of the univers of the being.
The rest of this post is dedicate to the development of these ideas at the
light of free logics.
Let us consider the univers of speech, named U, of all the things that have
the being, and M, the world of existing things. The complementary set of M,
named -M, is the set of all things, if any, that have the being but don't
exist.
An atomic phormula of the form ?Pa? is true in M, accordingly to the canonic
rules, if and only if the existent thing denotates by ?a? belongs to the
set of things denotates by ?P?. But when Pa is true in -M?
Suppose that Father Christmas don't exist. Than what is of the proposition
(1) ?Father Christmas has a white beard??
In M (1) is false, because Father Christmas is supposed do be inexistent,
but we can wish that (1) is true in -M.
In our logic we can say that the perfect triangle of Euclidean geometry
doesn't exist in nature and, contemporary, that it is true that its
internal angles are equal to two right ones.
Propositions that are true in M and in -M are Axioms.
If we abbreviate ?x belongs to U? with ?E!x?, we have the following axiom:
(2) for all x, E!x
But we can easily notate that some of the theorems of first order logic are
not theorems of our logic.
As an istance we consider the propositions
(3) Exist an x such that (x=y)
It is true in M, according with the ordinary semantic rules, but it is false
in -M. In fact doesn't exist a thing that is identical with Father
Christmas, because Father Christmas doesn't exist.
As another istance consider a therorem of firs order logic that is invalid
in our logic:
(4) Pa -> exist an x such that (Px)
It is easy to see that (4) is true in M but is false in -M. In fact if I
exist, you can deduce from ?I have a white beard? that ?there exist white
beared things?, but if Father Christmas doesn't exist, you can not deduce
that ?there exist white beared things? from ?Father Christmas has a white
beard?.
Another point of interest is that our logic seems to be more general of the
logic of Principia. As Russell admitted, the logic of Principia involves
that the univers of speech is not empty. You can demonstrate in the logic
of Principia that at leat an individual have to exist:
(5) (a=a) -> (a=a)
(6) (a=a) -> exist an x such that (x=x)
(7) (a=a)
(8) exist an x such that (x=x)
and therefore U would be different from the empty set.
But in our perspective, (6) is false, and can not be derived from (5), that
is true.
In our logic we must have different rules from the treatment of quantifiers.
I hope someone would like to work with me about these rules or give some
suggestions to me,
greetings
--
Roberto Vescarelli
http://www.faberbox.com/roby/
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