1) NEUTROSOPHIC LOGIC (OR SMARANDACHE LOGIC as named in the
Denis Howe's On-Line Dictionary of Computing) is:
A logic in which each proposition is t% true, i% indeterminate, and
f% false, with no restriction on t,i,f nor on their sum n=t+i+f.
Hence, the neutrosophic logic generalizes:
- the intuitionistic logic, which supports incomplete theories (for
0<n<100 and i=0, 0<=t,i,f<=100);
- the fuzzy logic (for n=100 and i=0, and 0<=t,i,f<=100);
- the Boolean logic (for n=100 and i=0, with t,f either 0 or
100);
- the multi-valued logic (for 0<=t,i,f<=100);
- the paraconsistent logic (for n>100 and i=0, with both
t,f<100);
- the dialetheism, which says that some contradictions are true
(for t=f=100 and i=0; some paradoxes can be denoted this way).
Compared with all other logics, the neutrosophic logic
introduces a percentage of "indeterminacy" - due to unexpected parameters
hidden in some propositions, and let each component t,i,f be even boiling
over 100 or freezing under 0.
For example: in some tautologies t>100, called "overtrue".
Similarly, a proposition may be "overindeterminate" (for i>100, in some
paradoxes), "overfalse" (for f>100, in some unconditionally false
propositions); or "undertrue" (for t<0, in some unconditionally false
propositions), "underindeterminate" (for i<0, in some unconditionally true
or false propositions), "underfalse" (for f<0, in some unconditionally true
propositions).
This is because we should make a distinction between
unconditionally true (t>100, and f<0 or i<0) and conditionally true
propositions (t<=100, and f<=100 or i<=100).
While in classical true/false logic it is possible to define
precisely 2m different m-ary operators for each m>0 (Charles D. Ashbacher),
the neutrosophic p-ary operators may be defined in uncountably infinite
different ways. The good operatorial selection would lead to applications
in neural networks, automated reasoning, and probabilistic models.
In the paraconsistent logic one cannot derive all statements
from a contradiction, ex contradictione quodlibet fails. In the
neutrosophic logic from a given contradiction one can derive a specific
statement only, depending on the neutrosophic operator used and the given
contradiction.
In the dialetheism it works the metaphysical thesis that some
contradictions are true. In the neutrosophic logic there are
contradictions denoted by t=f=100, which means 100% true and 100% false in
the same time; even more, it is possible to have a proposition which is,
say, 70% true and 60% false, while in the fuzzy logic it is not - because
the components should sum to 100, i.e. 70% true and 30% false.
The neutrosophic logic unifies many logics; it is like Felix
Klein's program in geometry, or Einstein's unified field in physics.
2) NEUTROSOPHIC SET:
An element x(t,i,f) belongs to a set M in the following way: it is
t% true in the set, i% indeterminate in the set, and f% false, with no
restriction on t,i,f nor on their sum n=t+i+f.
From the intuitionistic logic, paraconsistent logic,
dialetheism, paradoxes, and tautologies we transfer the "adjectives" to
the sets, i.e. to intuitionistic set, paraconsistent set, dialetheist set,
paradoxist set, and tautologic set respectively.
Hence, the neutrosophic set generalizes:
- the intuitionistic set, which supports incomplete set
theories (for 0<n<100 and i=0, 0<=t,i,f<=100);
- the fuzzy set (for n=100 and i=0, and 0<=t,i,f<=100);
- the classical set (for n=100 and i=0, with t,f either 0 or
100);
- the paraconsistent set (for n>100 and i=0, with both
t,f<100);
- the dialetheist set, which says that the intersection of some
disjoint sets is not empty (for t=f=100 and i=0; some paradoxist sets can
be denoted this way).
Compared with all other types of sets, the neutrosophic set
introduces a percentage of "indeterminacy" - due to unexpected parameters
hidden in some sets, and let each component t,i,f be even boiling over 100
or freezing under 0.
For example: an element in some tautological sets may have
t>100, called "overincluded". Similarly, an element in a set may be
"overindeterminate" (for i>100, in some paradoxal sets), "overexcluded"
(for f>100, in some unconditionally false propositions); or "undertrue"
(for t<0, in some unconditionally false propositions), "underindeterminate"
(for i<0, in some unconditionally true or false propositions), "underfalse"
(for f<0, in some unconditionally true propositions).
This is because we should make a distinction between
unconditionally true (t>100, and f<0 or i<0) and conditionally true
propositions (t<=100, and f<=100 or i<=100).
3) NEUTROSOPHIC PROBABILITY:
A generalization of the classical probability in which the
chance that an event occurs is t% true, i% indeterminate, and f% false,
with no restriction on t,i,f nor on their sum n=t+i+f.
An interesting particular case is for n=100, with
0<=t,i,f<=100, which is closer to the classical probability.
For n=100 and i=0, with 0<=t,f<=100, one obtains the classical
probability.
4) NEUTROSOPHIC STATISTICS:
Analysis of the events described by the neutrosophic
probability.
[Abstract extracted from "A Unifying Field in Logic. Neutrosophy:
Neutrosphic Probability, Set, and Logic", second edition, by F.
Smarandache, American Research Press, 1999.]
Papers on this new unifying logic used in quantum physics and
computer programming are welcome, and they will be published in the
Proceedings of Neutrosophic Logic and Their Applications. The authors are
invited to submit them on the address:
JoAnne McGray, secretary
American Research Press
Rehoboth, Box 141, NM 87322, USA
E-Mail: M_L_Perez@yahoo.com
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