Neutrosophy and Neutrosophic Logic
Dr. Florentin Smarandache
University of New Mexico
Gallup, NM 87301, USA
smarand@unm.edu
Neutrosophy studies the origin, nature, and scope of neutralities, as well as
their interactions with different ideational spectra. It considers that every
idea <A> tends to be neutralized, balanced by <Non-A> ideas - as a
state of equilibrium.
Neutrosophy is the base of neutrosophic
logic, neutrosophic set theory which
generalizes the fuzzy set, and of neutrosophic
probability theory and neutrosophic
statistics, which generalize the classical and imprecise probability and
statistics respectively. Neutrosophic
Logic is a multiple-valued logic in which each proposition is estimated to
have the percentages of truth, indeterminacy, and falsity in T, I, and F
respectively, where T, I, F are standard or non-standard subsets included in
the non-standard unit interval ] -0,
1+[. It is an
extension of fuzzy, intuitionistic, paraconsistent logics.
1.0 Neutrosophy
(Smarandache, 2002)
1.1. Etymology
Neutro-sophy [French neutre
< Latin neuter, neutral, and Greek
sophia, skill/wisdom] means knowledge
of neutral thought.
1.2. Definition
Neutrosophy is a branch of philosophy that studies the origin,
nature, and scope of neutralities, as well as their interactions with different
ideational spectra.
1.3. Characteristics
This mode of thinking:
- proposes new philosophical theses, principles, laws, methods,
formulas, movements;
- reveals that world is full of indeterminacy;
- interprets the uninterpretable dealing with paradoxes (Le, 1996)
and paradoxism (Popescu, 2002) };
- regards, from many different angles, old concepts, systems,
showing that an idea, which is true in a given referential system,
may be false in another one, and vice versa;
- attempts to make peace in the war of ideas and to make war among
ideas which are at peace;
- measures the stability of unstable systems and the instability of
stable systems.
1.4. Methods of Neutrosophic
Study
The methods of neutrosophic study include mathematization
(neutrosophic logic, neutrosophic probability and statistics, duality),
generalization, the identification of complementarities, contradictions,
paradoxes, tautologies, analogies, and the reinterpretation of old ideas in new
contexts, including their combination and mutual interference with each other.
Its approach is aphoristic, linguistic, and transdisciplinary.
1.5. Introduction to Non-Standard Analysis
In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of
mathematical logic, that rigorously defines the infinitesimals. Informally, an
infinitesimal is an infinitely small number.
Formally, x is said to be infinitesimal if and only if for all positive
integers n one has xxx < 1/n. Let &>0 be a such infinitesimal
number. The hyper-real number set is an extension of the real number set, which
includes classes of infinite numbers and classes of infinitesimal numbers. Let’s consider the non-standard finite
numbers 1+ = 1+&, where
“1” is its standard part and “&” its
non-standard part, and –0 = 0-&, where “0” is its standard part and “&” its non-standard part. Then, we call ]-a, b+[
a non-standard unit interval.
Obviously, 0 and 1, and analogously non-standard numbers infinitely
small but less than 0 or infinitely small but greater than 1, belong to the
non-standard unit interval. Actually,
by “-a” one signifies a monad, i.e. a set of hyper-real numbers in
non-standard analysis:
(-a)= {a-x:
xcR*, x is infinitesimal},
and similarly “b+” is a monad:
(b+)= {b+x:
xcR*, x is infinitesimal}.
Generally, the left and right borders of a non-standard interval ]-a,
b+[ are vague, imprecise, themselves being non-standard (sub)sets (-a)
and (b+) as defined above.
Combining the two before mentioned definitions one gets, what we
would call, a binad of “-c+”:
(-c+)= {c-x: xcR*, x is infinitesimal} 4 {c+x: xcR*, x is infinitesimal},
which is a collection of open punctured neighborhoods (balls) of c.
Of course, –a < a, and b+ > b. No order is between –c+ and
c.
1.6. Neutrosophic components
Let T, I, F be standard or non-standard real subsets of ]-a,
b+[. These T, I, F are not
necessarily intervals, but may be any real sub-unitary subsets: discrete or continuous; single-element,
finite, or (countably or uncountably) infinite; union or intersection of
various subsets; etc. They may also
overlap. The real subsets could
represent the relative errors in determining t, i, f (in the case when the
subsets T, I, F are reduced to points).
In this article, T, I, F, called neutrosophic components, will represent the truth value,
indeterminacy value, and falsehood value respectively referring to neutrosophy,
neutrosophic logic, neutrosophic set, neutrosophic probability, neutrosophic
statistics.
This representation is closer to the way in which the human mind
actually reasons than other models. It
characterizes/catches the imprecision
of knowledge or linguistic inexactitude received by various observers (that’s
why T, I, F are subsets - not necessarily single-elements), uncertainty due to incomplete knowledge
or acquisition errors or stochasticity (that’s why the subset I exists), and vagueness due to lack of clear contours
or limits (that’s why T, I, F are subsets and I exists; in particular for the
appurtenance to the neutrosophic sets).
1.7. Formalization
Let's note by <A> an idea, or proposition, theory, event,
concept, entity, by <Non-A> what is not <A>, and by <Anti-A>
the opposite of <A>. Also,
<Neut-A> means what is neither <A> nor <Anti-A>, i.e.
neutrality in between the two extremes.
And <A'> a version of <A>.
<Non-A> is
different from <Anti-A>.
For example, if <A> = white, then <Anti-A> = black
(antonym), but <Non-A> = green, red, blue, yellow, black, etc. (any
color, except white), while <Neut-A> = green, red, blue, yellow, etc.
(any color, except white and black), and <A'> = dark white, etc. (any
shade of white).
<Neut-A> h <Neut-(Anti-A)>, neutralities of
<A> are identical with neutralities of <Anti-A>.
<Non-A> q <Anti-A>, and <Non-A> q <Neut-A> as well,
also
<A> 3 <Anti-A> = Â,
<A> 3 <Non-A> = Â.
<A>,
<Neut-A>, and <Anti-A> are disjoint two by two.
<Non-A> is
the completeness of <A> with respect to the universal set.
1.8. Main Principle
Between an idea <A> and its opposite <Anti-A>, there is
a continuum-power spectrum of neutralities <Neut-A>.
1.9. Fundamental Thesis
Any idea <A> is T% true, I% indeterminate, and F% false,
where T, I, F _ ]-0,
1+ [.
1.10. Main Laws
Let <"> be an attribute, and (T, I, F) _ ]-0, 1+ [3. Then:
- There is a proposition <P> and a referential system {R},
such that <P> is T% <">, I% indeterminate or <Neut-">, and F% <Anti-">.
- For any proposition <P>, there is a referential system {R},
such that <P> is T% <">, I% indeterminate or <Neut-">, and F% <Anti-">.
- <"> is at some degree <Anti-">, while <Anti-"> is at some degree <">.
Therefore:
For each proposition <P> there are referential systems {R1},
{R2}, ..., so that <P> looks differently in each of them -
getting all possible states from <P> to <Non-P> until
<Anti-P>. And, as a consequence, for any two propositions <M> and
<N>, there exist two referential systems {RM} and {RN}
respectively, such that <M> and <N> look the same. The referential
systems are like mirrors of various curvatures reflecting the propositions.
1.11. Mottos
- All is possible, the impossible too!
- Nothing is perfect, not even the perfect!
1.12. Fundamental Theory
Every idea <A> tends to be neutralized, diminished, balanced
by <Non-A> ideas (which includes, besides Hegel’s <Anti-A>, the
<Neut-A> too) - as a state of equilibrium. In between <A> and <Anti-A> there are infinitely many
<Neut-A> ideas, which may balance <A> without necessarily
<Anti-A> versions. To neuter an idea one must discover all its three
sides: of sense (truth), of nonsense (falsity), and of undecidability
(indeterminacy) - then reverse/combine them.
Afterwards, the idea will be classified as neutrality.
1.13. Delimitation from Other Philosophical Concepts and Theories
a) Neutrosophy is based not only on analysis of oppositional
propositions, as dialectic does, but
on analysis of neutralities in between them as well.
b) While epistemology
studies the limits of knowledge and justification, neutrosophy passes these
limits and takes under magnifying glass not only defining features and
substantive conditions of an entity <E> - but the whole <E'>
derivative spectrum in connection with <Neut-E>. Epistemology studies philosophical
contraries, e.g. <E> versus <Anti-E>, neutrosophy studies
<Neut-E> versus <E> and versus <Anti-E> which means logic
based on neutralities.
c-d) Neutral monism
asserts that ultimate reality is neither physical nor mental. Neutrosophy considers a more than
pluralistic viewpoint: infinitely many separate and ultimate substances making
up the world.
e) Hermeneutics is the
art or science of interpretation, while neutrosophy also creates new ideas and
analyzes a wide range ideational field by balancing instable systems and
unbalancing stable systems.
f) Philosophia Perennis
tells the common truth of contradictory viewpoints, neutrosophy combines with
the truth of neutral ones as well.
g) Fallibilism attributes
uncertainty to every class of beliefs or propositions, while neutrosophy
accepts 100% true assertions, and 100% false assertions as well - moreover, checks
in what referential systems the percent of uncertainty approaches zero or 100.
1.14. Evolution of an Idea
<A> in the world is not cyclic (as Marx said), but
discontinuous, knotted, boundless:
<Neut-A> = existing ideational background, before arising
<A>;
<Pre-A> = a pre-idea, a forerunner of <A>;
<Pre-A'> = spectrum of <Pre-A> versions;
<A> = the idea itself, which implicitly gives birth to
<Non-A> = what is outer <A>;
<A'> = spectrum of <A> versions after
(mis)interpretations (mis)understanding by different people, schools, cultures;
<A/Neut-A> = spectrum of <A> derivatives/deviations,
because <A> partially mixes/melts first with neuter ideas;
<Anti-A> = the straight opposite of <A>, developed
inside of <Non-A>;
<Anti-A'> = spectrum of <Anti-A> versions after
(mis)interpretations (mis)understanding by different people, schools, cultures;
<Anti-A/Neut-A> = spectrum of <Anti-A>
derivatives/deviations, which means partial <Anti-A> and partial <Neut-A> combined in
various percentage;
<A'/Anti-A'> = spectrum of derivatives/deviations after
mixing <A'> and <Anti-A'>
spectra;
<Post-A> = after <A>, a post-idea, a conclusiveness;
<Post-A'> = spectrum of <Post-A> versions;
<Neo-A> = <A> retaken in a new way, at a different
level, in new conditions, as in a non-regular curve with inflection points, in evolute and involute periods, in
a recurrent mode; the life of <A>
restarts.
Marx's 'spiral' of evolution is replaced by a more complex
differential curve with ups-and-downs, with knots - because evolution means
cycles of involution too.
This is dynaphilosophy =
the study of infinite road of an idea.
<Neo-A> has a larger sphere (including, besides parts of old
<A>, parts of <Neut-A> resulted from previous combinations), more
characteristics, is more heterogeneous (after combinations with various
<Non-A> ideas). But,
<Neo-A>, as a whole in itself, has the tendency to homogenize its
content, and then to de-homogenize by mixture with other ideas. And so on, until the previous <A> gets
to a point where it paradoxically incorporates the entire <Non-A>, being
indistinct of the whole. And this is
the point where the idea dies, cannot be distinguished from others. The Whole breaks down, because the motion is
characteristic to it, in a plurality of new ideas (some of them containing
grains of the original <A>), which begin their life in a similar
way.
Thus, in time, <A> arrives to mix with <Neut-A> and
<Anti-A>.
2. Neutrosophic Logic (Smarandache, 2002)
As an alternative to the existing logics we propose a non-classical
one, which represents a mathematical model of uncertainty, vagueness,
ambiguity, imprecision, undefined, unknown, incompleteness, inconsistency, redundancy,
contradiction.
2.1. Definition
A logic in which each proposition is estimated to have the
percentage of truth in a subset T, the percentage of indeterminacy in a subset
I, and the percentage of falsity in a subset F, where T, I, F are defined
above, is called Neutrosophic Logic.
We use a subset of truth (or indeterminacy, or falsity), instead of
a number only, because in many cases we are not able to exactly determine the
percentages of truth and of falsity but to approximate them: for example a
proposition is between 30-40% true and between 60-70% false, even worst:
between 30-40% or 45-50% true (according to various analyzers), and 60% or
between 66-70% false.
The subsets are not necessary intervals, but any sets (discrete,
continuous, open or closed or half-open/half-closed interval, intersections or
unions of the previous sets, etc.) in accordance with the given proposition.
A subset may have one element only in special cases of this logic.
Constants: (T, I, F) truth-values, where T, I, F are standard or
non-standard subsets of the non-standard interval ]-0, 1+ [,
where ninf = inf T + inf I + inf F m -0, and nsup =
sup T + sup I + sup F [ 3+.
Atomic formulas: a, b, c, … .
Arbitrary formulas: A, B, C, …
Neutrosophic logic is a formal frame trying to measure the truth,
indeterminacy, and falsehood. There are
many neutrosophic rules of inference (Dezert, 2002).
2.2. History
Classical Logic, also called Bivalent Logic for taking only two
values {0, 1}, or Boolean Logic from British mathematician George Boole
(1815-64), was named by the philosopher Quine (1981) “sweet simplicity.”
Peirce, before 1910, developed a semantics for three-valued logic
in an unpublished note, but Emil Post’s dissertation (1920s) is cited for
originating the three-valued logic.
Here “1” is used for truth, “1/2” for indeterminacy, and “0” for
falsehood. Also, Reichenbach, leader of
the logical empiricism, studied it.
Three-valued logic was employed by Halldén (1949), Korner (1960),
Tye (1994) to solve Sorites Paradoxes.
They used truth tables, such as Kleene’s, but everything depended on the
definition of validity. A three-valued paraconsistent system (LP) has the
values: ‘true’, ‘false’, and ‘both true and false’. The ancient Indian metaphysics considered four possible values of
a statement: ‘true (only)’, ‘false (only)’, ‘both true and false’, and ‘neither
true nor false’; J. M. Dunn (1976) formalized this in a four-valued
paraconsistent system as his First Degree Entailment semantics.
The Buddhist logic added a fifth value to the previous ones, ‘none
of these’ (called catushkoti).
The {0, a1, ..., an, 1} Multi-Valued, or
Plurivalent, Logic was developed by ºukasiewicz, while Post originated
the m‑valued calculus.
Many-valued logic was replaced by Goguen (1969) and Zadeh (1975)
with an Infinite-Valued Logic (of continuum power, as in the classical
mathematical analysis and classical probability) called Fuzzy Logic, where the
truth-value can be any number in the closed unit interval [0, 1]. The Fuzzy Set was introduced by Zadeh in
1975.
Therefore, we finally generalize fuzzy logic to a transcendental
logic, called “neutrosophic logic”: where the interval [0, 1] is exceeded, i.e.
, the percentages of truth, indeterminacy, and falsity are approximated by
non-standard subsets – not by single numbers, and these subsets may overlap and
exceed the unit interval in the sense of the non-standard analysis; also the
superior sums and inferior sum, nsup = sup T + sup I + sup F c ]-0, 3+[, may be
as high as 3 or 3+, while ninf = inf T + inf I + inf F c ]-0, 3+[, may be
as low as 0 or –0.
The idea of tripartition (truth, falsehood, indeterminacy) appeared
in 1764 when J. H. Lambert investigated the credibility of one witness affected
by the contrary testimony of another.
He generalized Hooper’s rule of combination of evidence (1680s), which
was a Non-Bayesian approach to find a probabilistic model. Koopman in 1940s introduced the notions of
lower and upper probability, followed by Good, and Dempster (1967) gave a rule
of combining two arguments. Shafer
(1976) extended it to the Dempster-Shafer Theory of Belief Functions by
defining the Belief and Plausibility functions and using the rule of inference
of Dempster for combining two evidences proceeding from two different sources.
Belief function is a connection between fuzzy reasoning and probability. The
Dempster-Shafer Theory of Belief Functions is a generalization of the Bayesian
Probability (Bayes 1760s, Laplace 1780s); this uses the mathematical
probability in a more general way, and is based on probabilistic combination of
evidence in artificial intelligence.
In Lambert “there is a chance p that the witness will be faithful
and accurate, a chance q that he will be mendacious, and a chance 1-p-q that he
will simply be careless” [apud Shafer (1986)].
Therefore three components: accurate, mendacious, careless, which add up
to 1.
Van Fraassen introduced supervaluation semantics in his attempt to
solve the sorites paradoxes, followed by Dummett (1975) and Fine (1975). They all tripartitioned, considering a vague
predicate which, having border cases, is undefined for these border cases. Van Fraassen took the vague predicate ‘heap’
and extended it positively to those objects to which the predicate definitively
applies and negatively to those objects to which it definitively doesn’t
apply. The remaining object’s border
was called its penumbra. A sharp
boundary between these two extensions does not exist for a soritical
predicate. Inductive reasoning is no
longer valid too; if S is a Sorites predicate, the proposition “½n(San&¾San+1)” is false. Thus, the predicate Heap (positive
extension) = true, Heap (negative extension) = false, Heap (penumbra) =
indeterminate.
Narinyani (1980) used the tripartition to define what he called the
“indefinite set”, and Atanassov (1982) continued on tripartition and gave five
generalizations of the fuzzy set, studied their properties and applications to
the neural networks in medicine:
a) Intuitionistic Fuzzy Set (IFS): Given an universe E, an IFS A
over E is a set of ordered triples <universe_element,
degree_of_membership_to_A(M), degree_of_non-membership_to_A(N)> such that
M+N [ 1 and M, N c [0, 1]. When M + N = 1 one obtains the fuzzy set, and if M + N < 1
there is an indeterminacy I = 1-M-N.
b) Intuitionistic L-Fuzzy Set (ILFS): This set is similar to IFS,
but M and N belong to a fixed lattice L.
c) Interval-valued Intuitionistic Fuzzy Set (IVIFS): This is set
similar to the IFS, but M and N are subsets of [0, 1] and sup M + sup N [ 1.
d) Intuitionistic Fuzzy Set of Second Type (IFS2): This is similar
to IFS, but M2 + N2 [ 1. M and N are
inside of the upper right quarter of unit circle.
e) Temporal IFS: This set similar to the IFS, but M and N are
functions of the time-moment too.
This neutrosophic logic is an attempt to unify many logics in a
single field. Yet, a too large generalization may sometimes have no
practical impact. Such unification
attempts have been done in the history of sciences:
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