NEUTROSOPHIC LOGIC, A GENERALIZATION OF FUZZY LOGIC --------------------------------------------------- by Florentin Smarandache - Fifth Part - A) Introduction: One passes from the classical {0, 1} Bivalent Logic of George Boole, to the Three-Valued Logic of Reichenbach (leader of the logical empiricism), then to the {0, a1, ..., an, 1} Plurivalent one of ţukasiewicz (and Post's m-valued calculus), and finally to the [0, 1] Infinite Logic as in classical mathematical analysis and probability: a Transcendental Logic (with values of the power of continuum), or Fuzzy Logic. Falsehood is infinite, and truthhood quite alike; in between, at different degrees, indeterminacy as well. In the neutrosophic theory between being and nothingness existence and nonexistence geniality and mediocrity certainty and uncertainty value and nonvalue and generally speaking and there are infinitely many transcendental states. An infinitude of infinitudes. They are degrees of neutralities combined with and . In fact there also are steps between being and being existence and existence geniality and geniality possible and possible certainty and certainty value and value and generally speaking between and . The notions, in a pure form, last in themselves only, but outside they have an interfusion form. Infinitude of shades and degrees of differentiation: between white and black there exists a unbounded palette of colors resulted from thousands of combinations among them. All is alternative: progress alternates with setback, development with stagnation and underdevelopment. In between objective and subjective there is a plurality of shades. In between good and bad... In between positive and negative... In between possible and impossible In between true and false... In between "A" and "Anti-A"... As a neutrosophic circle: | | Good | Neighbourhood . . | . . | . . | . . | . Indeterminate . | . Indeterminate --------------.------------|------------.-------------- Neighbourhood . | . Neighbourhood . | . . | . . | . . | . . Bad | Neighbourhood | | Everything is G% good, I% indeterminate, and B% bad, where G + I + B = 100. Besides Diderot's dialectics on good and bad ("Rameau's Nephew", 1772), any act has its percentage of "good", "indeterminate", and of "bad" as well incorporated. Rodolph Carnap said: "Metaphysical propositions are neither true nor false, because they assert nothing, they contain neither knowledge nor error (...)". Hence, there are infinitely many statuses in between "Good" and "Bad", and generally speaking in between "A" and "Anti-A", like on the real number segment: [0, 1] False True Bad Good Non-sense Sense Anti-A A 0 is the absolute falsity, 1 the absolute truth. In between each oppositing pair, normally in a vicinity of 0.5, are being set up the neutralities. There exist as many states in between "True" and "False" as in between "Good" and "Bad". Irrational and transcendental standpoints belong to this interval. Even if an act apparently looks to be only good, or only bad, the other haded side should be sought. The ratios Anti-A Non-A --------, ------- A A vary indefinitely. They are transfinite. If a statement is 30%T (true) and 60%I (indeterminate), then it is 10%F (false). This is somehow alethic, meaning pertaining to truthhood and falsehood in the same time. B) Definition of Neutrosophic Logic: This is a generalization (for the case of null indeterminacy) of the fuzzy logic. In opposition to Fuzzy Logic, if a proposition is t% true, doesn't necessarily mean it is (100-t)% false. There should also be a percent of indeterminacy on the values of . A better approach is t% true, f% false, and i% indeterminate, where t+i+f = 100 and t,i,f belong to [0, 100], called neutrosophic logical value of , and noted by n(A) = (t,i,f). Neutrosophic Logic means the study of neutrosophic logical values of the propositions. There exist, for each individual one, PRO parameters, CONTRA parameters, and NEUTER parameters which influence the above values. Indeterminacy results from any hazard which may occur, from unknown parameters, or from new arising conditions. This resulted from practice. C) Applications: Neutrosophic logic is useful in the real-world systems for designing control logic, and may work in quantum mechanics. ţ The candidate C, who runs for election in a metropolis M of p people with right to vote, will win. This proposition is, say, 25% true (percent of people voting for him), 35% false (percent of people voting against him), and 40% indeterminate (percent of people not coming to the ballot box, or giving a blank vote - not selecting anyone, or giving a negative vote - cutting all candidates on the list). ţ Tomorrow it will rain. This proposition is, say, 50% true according to meteorologists who have investigated the past years' weather, 30% false according to today's very sunny and droughty summer, and 20% undecided. ţ This is a heap. As an application to the sorites paradoxes, we may now say that this proposition is t% true, f% false, and i% indeterminate (the neutrality comes for we don't know exactly where is the difference between a heap and a non-heap; and, if we approximate the border, our 'accuracy' is subjective). We are not able to distinguish the difference between yellow and red as well if a continuum spectrum of colors is painted on a wall imperceptibly changing from one into another. D) Definition of (Truth-)Neutrosophic Logical Connectors: One uses the definitions of neutrosophic probability and neutrosophic set (operations). Let 0 <= t1, t2 <= 1 represent the truth-probabilities, 0 <= i1, i2 <= 1 the indeterminacy-probabilities, and 0 <= f1, f2 <= 1 the falsity-probabilities of two propositions A1 and A2 respectively, where t1 + i1 + f1 = 1 and t2 + i2 + f2 = 1. One notes the neutrosophic logical values of A1 and A2 by n(A1) = (t1, i1, f1) and n(A2) = (t2, i2, f2). _ Also, for any 0 <= x <= 1 one notes 1-x = x. Let t = (t1, t2), i = (i1, i2), f = (f1, f2). Let W(a,b,c) = (1-a)/(b+c) and W(R) = W(R(t),R(i),R(f)) for any tridimensional vector R = ( R(t),R(i),R(f) ). Negation: _ Let N(x) = 1-x = x. Then: n(ŞA1) = ( N(t1), N(i1)W(N), N(f1)W(N) ). Conjunction: Let C(x,y) = xy, and C(z1,z2) = C(z) for any bidimensional vector z = (z1, z2). Then: n(A1 /\ A2) = ( C(t), C(i)W(C), C(f)W(C) ). (And, in a similar way, generalized for n propositions.) Weak or inclusive disjunction: _ _ Let D1(x,y) = x+y-xy = x+xy = y+xy, and D1(z1,z2) = D1(z) for any bidimensional vector z = (z1, z2). Then: n(A1 \/ A2) = ( D1(t), D1(i)W(D1), D1(f)W(D1) ). (And, in a similar way, generalized for n propositions.) Strong or exclusive disjunction: _ _ __ Let D2(x,y) = x(1-y)+y(1-x)-xy(1-x)(1-y) = xy+xy-xyxy, and D2(z1,z2) = D2(z) for any bidimensional vector z = (z1, z2). Then: n(A1 \_/ A2) = ( D2(t), D2(i)W(D2), D2(f)W(D2) ). (And, in a similar way, generalized for n propositions.) Material conditional (implication): _ _ Let I(x,y) = 1-x+xy = x+xy = 1-xy, and I(z1,z2) = I(z) for any bidimensional vector z = (z1, z2). Then: n(A1->A2) = ( I(t), I(i)W(I), I(f)W(I) ). Material biconditional (equivalence): _ _ _ _ Let E(x,y) = (1-x+xy)(1-y+xy) = (x+xy)(y+xy) = (1-xy)(1-xy), and E(z1,z2) = E(z) for any bidimensional vector z = (z1, z2). n(A1A2) = ( E(t), E(i)W(E), E(f)W(E) ). Sheffer's connector: Let S(x,y) = 1-xy, and S(z1,z2) = S(z) for any bidimensional vector z = (z1, z2). n(A1 | A2) = n(ŞA1 \/ ŞA2) = ( S(t), S(i)W(S), S(f)W(S) ). Peirce's connector: __ Let P(x,y) = (1-x)(1-y) = xy, and P(z1,z2) = P(z) for any bidimensional vector z = (z1, z2). n(A1A2) = n(ŞA1 /\ ŞA2) = ( P(t), P(i)W(P), P(f)W(P) ). E) Properties of Neutrosophic Logical Connectors: Let's note by t(A) the truth-component of the neutrosophic value n(A), and t(A) = a, t(B) = b. Also f(A) and i(A) represent the falsity-component and indeterminacy-component of the neutrosophic value n(A) = (t(A),i(A),f(A)). a) Negation: lim t(ŞA) = 1 t(A)->0 lim t(ŞA) = 0 t(A)->1 b) Conjunction: t(A /\ B) = t(B /\ A) ó min {a, b}, equality occurs when t(A) or t(B) is 1. lim t(A /\ B) = b t(A)->1 lim t(A /\ B) = 0 t(A)->0 ě t( /\ A) = 0 if t(A) is different from 1. k=1 m m t( /\ Ak) = Product t(Ak). k=1 k=1 c) Weak Disjunction: t(A \/ B) = t(B \/ A) ň max {a, b}, equality occurs when: at least one of t(A), t(B) is 1, or both t(A) = t(B) = 0. lim t(A \/ B) = 1 t(A)->1 lim t(A \/ B) = b t(A)->0 ě t( \/ A) = 1 if t(A) is different from 0. k=1 By mathematical induction one finds: m m k t( \/ Ak) = Sigma (-1)^(k+1) Sigma Product t(Aj ). k=1 k=1 1ój1<...1 lim t(A \_/ B) = b t(A)->0 lim t(A \_/ B) = 0 t(A)->0 t(B)->0 lim t(A \_/ B) = 0 t(A)->1 t(B)->1 lim t(A \_/ B) = 1 t(A)->0 t(B)->1 With a TI-82 graphing calculator, defining a function y1 = 2x(1-x)-x^2(1-x)^2 and then composing it with itself many times for different truth- values, t(A) in between (0, 1), one finds that ě t( \_/ A) = 0.430159709... = ksd = strong disjunction constant. k=1 The cases t(A) = 0 or 1 are trivial. A recurrent sequence results, defined as follows: c(1) = c in (0, 1), and c(m+1) = y1(c(m)) for m ň 1. If c < ksd, then cm is strictly increasing, if c = ksd, then cm is constant, if c > ksd, then cm is strictly decreasing. The graph of the function y1, for 0 ó x ó 1, is framed by [0, 1], therefore cm is lower and upper bounded. Thus it is convergent, and the limit l belongs to [0, 1]. Then, computing lim c(m+1) = lim y1(c(m)), or y1(l) - l = 0. m->ě m->ě Graphing again, with the calculator, the function y2(x) = y1(x) - x and, looking for the nonzero root of the equation y2(x) = 0 in the interval [0, 1], one obtains that l = ksd and the first nine decimal places of the strong disjunction constant ksd are those listed above. See, as an example, the following two repeating Strong Disjunction Tables: y1(0.1), y1(2)(0.1) = y1(y1(0.1)), and so on: 0.1 \_/ 0.1 = 0.1719 0.1719 \_/ 0.1719 = 0.2644 and so on: 0.3512, 0.4038, 0.4235, 0.4287, 0.4298, 0.43009, 0.43014, 0.43015, 0.4301590896, 0.4301595784, 0.4301596815, 0.4301597032, 0.4301597078, 0.4301597087, 0.4301597089, 0.430159709, ... . And similarly y1(0.5), y1(2)(0.5) = y1(y1(0.5)), and so on: 0.5 \_/ 0.5 = 0.4375 0.4375 \_/ 0.4375 = 0.4316 and so on: 0.4305, 0.4302, 0.43017, 0.43016, 0.4301598362, 0.4301597358, 0.4301597147, 0.4301597102, 0.4301597093, 0.4301597091, ... . Let b be different from 0,1 and constant, and a (b) = (b^2-3b+1)/(2b^2-2b). max One has: max t(A \_/ B) occurs for: 0ót(A)ó1 a = a (b) if a (b) belongs to [0, 1], max max or a = 0 if a (b) < 0, max or a = 1 if a (b) > 1, max because the equivalence connector is described by a parabola of equation d2 (a) = (b-b^2)a^2 + (b^2-3b+1)a + b, b which is concave up. Another Strong Disjunction Table: (0.2 \_/ 0.2 = 0.2944 should be read as: if t(A) = 0.2 and t(B) = 0.2, then t(A \_/ B) = 0.2944): 0.0 \_/ 0.0 = 0.0000 0.1 \_/ 0.1 = 0.1719 0.2 \_/ 0.2 = 0.2944 0.3 \_/ 0.3 = 0.3759 0.4 \_/ 0.4 = 0.4224 0.5 \_/ 0.5 = 0.4375 0.6 \_/ 0.6 = 0.4224 0.7 \_/ 0.7 = 0.3759 0.8 \_/ 0.8 = 0.2944 0.9 \_/ 0.9 = 0.1719 1.0 \_/ 1.0 = 0.0000 e) Implication: lim t(A->B) = 1 t(A)->0 lim t(A->B) = b t(A)->1 lim t(A->B) = 1-a t(B)->0 lim t(A->B) = 1 t(B)->1 t(A->A) = 1 if t(A) = 0 or 1, and > a otherwise. With a TI-82 graphing calculator, defining a function y1 = 1-x+x^2 and then composing it with itself many times for different truth- values t(A) in between [0, 1], one find that ě t(A -> A) = 1, which means infinite repetition, k=1 for example: A1 đ (A->A), A2 đ (A1->A1), A3 đ (A2->A2), etc. See, as an example, these three Implication Tables (0.2->0.6 = 0.92 should be read as: if t(A) = 0.2 and t(B) = 0.6, then t(A->B) = 0.92): 0.0->0.6 = 1.00 0.1->0.6 = 0.96 0.2->0.6 = 0.92 0.3->0.6 = 0.88 0.4->0.6 = 0.84 0.5->0.6 = 0.80 0.6->0.6 = 0.76 0.7->0.6 = 0.72 0.8->0.6 = 0.68 0.9->0.6 = 0.64 1.0->0.6 = 0.60 0.6->0.0 = 0.40 0.6->0.1 = 0.46 0.6->0.2 = 0.52 0.6->0.3 = 0.58 0.6->0.4 = 0.64 0.6->0.5 = 0.70 0.6->0.6 = 0.76 0.6->0.7 = 0.82 0.6->0.8 = 0.88 0.6->0.9 = 0.94 0.6->1.0 = 1.00 0.0->0.0 = 1.00 0.1->0.1 = 0.91 0.2->0.2 = 0.84 0.3->0.3 = 0.79 0.4->0.4 = 0.76 0.5->0.5 = 0.75 0.6->0.6 = 0.76 0.7->0.7 = 0.79 0.8->0.8 = 0.84 0.9->0.9 = 0.91 1.0->1.0 = 1.00 f) Equivalence: t(AB) = t(BA) = t(ŞAŞB) lim t(A B) = 1 t(A)->0 t(B)->0 lim t(A B) = 1 t(A)->1 t(B)->1 lim t(A B) = 0 t(A)->0 t(B)->1 lim t(A B) = 1-b t(A)->0 lim t(A B) = b t(A)->1 Let b be different from 0,1 and constant, and a (b) = (b^2-3b+1)/(2b^2-2b). max One has: max t(A B) occurs for: 0<=t(A)<=1 a = a (b) if a (b) is in [0, 1], max max or a = 0 if a (b) < 0, max or a = 1 if a (b) > 1, max because the equivalence connector is described by a parabola of equation e (a) = (b^2-b)a^2 + (-b^2+3b-1)a + (1-b), b which is concave down. With a TI-82 graphing calculator, defining a function y1 = (1-x+x^2)^2 and then composing it with itself many times for different truth- values t(A) in between [0, 1], one finds that ě t(A A) = 1, which means infinite repetition, k=1 for example: A1 đ (A A), A2 đ (A1 A1), A3 đ (A2 A2), etc. See, as an example, this Equivalence Table: (0.2 0.6 = 0.92 x 0.52 = 0.4784 should be read as: if t(A) = 0.2 and t(B) = 0.6, then t(A B) = t(A->B)/\t(B->A) = 0.92 x 0.52 = 0.4784): 0.0 0.6 = 1.00 x 0.40 = 0.4000 0.1 0.6 = 0.96 x 0.46 = 0.4416 0.2 0.6 = 0.92 x 0.52 = 0.4784 0.3 0.6 = 0.88 x 0.58 = 0.5104 0.4 0.6 = 0.84 x 0.64 = 0.5376 0.5 0.6 = 0.80 x 0.70 = 0.5600 0.6 0.6 = 0.76 x 0.76 = 0.5776 0.7 0.6 = 0.72 x 0.82 = 0.5904 0.8 0.6 = 0.68 x 0.88 = 0.5984 0.9 0.6 = 0.64 x 0.94 = 0.6016 1.0 0.6 = 0.60 x 1.00 = 0.6000 g) Sheffer's Connector: t(A|B) = t(B|A) lim t(A|B) = 1 t(A)->0 lim t(A|B) = 1-b t(A)->1 With a TI-82 graphing calculator, defining a function y1 = 1-x^2 and then composing it with itself many times for different truth- values t(A) in between [0, 1], one find that ě t(A | A) = 0, 1 {two distinct subsequences converging to 0 and 1 k=1 respectively - hence the entire sequence is divergent}, which means infinite repetition, for example: A1 đ (A | A), A2 đ (A1 | A1), A3 đ (A2 | A2), etc. h) Peirce's Connector: t(AB) = t(BA) lim t(AB) = 1-b t(A)->0 lim t(AB) = 0 t(A)->1 With a TI-82 graphing calculator, defining a function y1 = (1-x)^2 and then composing it with itself many times for different truth- values t(A) in between [0, 1], one find that ě t(A A) = 0, 1 {two distinct subsequences converging to 0 and 1 k=1 respectively - hence the entire sequence is divergent}, which means infinite repetition, for example: A1 đ (A A), A2 đ (A1 A1), A3 đ (A2 A2), etc. F) Comparison between Fuzzy Logic and Neutrosophic Logic: The neutrosophic connectors have a better truth-value definition approach to the real-world systems than the fuzzy connectors. While the fuzzy negation is defined in the same way as the neutrosophic negation for the truth-value: v(ŞA1) = 1-t1, the fuzzy conjunction is different: v(A1/\A2) = min {t1, t2}, which is not very accurate, because no matter how bigger is, say, t2 than t1 < 1, min {t1, t2} = always to t1 (static result, not influenced by t2 anymore), but in the neutrosophic conjunction, where t(A1/\A2) = t1t2, the result varies directly proportional with both parameters t1 and t2. Similarly, the fuzzy weak or inclusive disjunction is different: v(A1\/A2) = max {t1, t2}, which is not very accurate, because no matter how smaller is, say, t1 than t2 > 0, max {t1, t2} = always to t2 (static result, not influenced by t1 anymore), but in the neutrosophic weak or inclusive disjunction, where t(A1\/A2) = t1+t2-t1t2, the result varies directly proportional with both parameters t1 and t2. If t1 decreases towards 0, the results decreases towards t2; if t1 increases towards t2, the result increases towards 2t2-t2^2 (> t2). Don't talk about the falsity-values of the fuzzy connectors and of the neutrosophic connectors respectively, which are totally distinct, even for the negation. G) Other Properties: In modal logic, the primitive operators /\ 'it is possible _ \/ that' and |_| 'it is necessary that' can be defined by: t(/\A) > 0, \/ _ and, because |_|A could be regarded as Ş(/\ŞA), _ \/ t(|_|A) < 1. The sufficient reason principle (Aristotle, Leibniz), which asserts that every statement has a grounding, partially works in this logic. Also, identity principle, that AA is true, partially works, because if say t(A) = 0.3 then t(AA) = 0.6241, the only cases when t(AA) = 1 are for t(A) = 0 or 1. Same thing for the principles of bivalence (a statement is either true or false), and of excluded middle (a statement with its negation is always true). The principle of noncontradiction (a statement and its negation may not both be true) functions only if t(A) is straight 0 or 1, otherwise 0 < t(A/\ŞA) < 1. Neutrosophy shows that a philosophical idea, no matter if proven true by ones or false by others, may get any truth-value in the interval [0, 1] depending on the referential system we are reporting it to. The conjunction is well-defined, associative, commutative, admits a unit element U with t(U) = 1, but no element whose truth-component is different from 1 is inversable. The conjunction is not absorbent, i.e. t( A/\(A/\B) ) /\ t(A), except for the cases when t(A) = 0, or t(A) = t(B) = 1. The disjunction is well-defined, associative, commutative, admits a unit element O with t(O) = 0, but no element whose truth-component is different from 0 is inversable. The disjunction is not absorbent, i.e. t( A\/(A\/B) ) \/ t(A), except for the cases when one of t(A) = 1, or t(A) = t(B) = 0. None of them is distributive with respect to the other. De Morgan laws do not apply either. Therefore (NL, /\, \/, Ş), where NL is the set of neutrosophic logical propositions, is not an algebra. Nor (P([0, 1]), ď, U, C), where P([0, 1]) is the set of all subsets of [0, 1], and C(A) is the neutrosophic complement of A. One names a set N, endowed by two associative unitary internal laws * and #, which are not inversable except for their unit elements respectively, and not distributive with respect to each other, Ninversity. If both laws are commutative, then N is called a Commutative Ninversity. For a better understanding of the neutrosophic logic one needs to study the commutative ninversity. H) Other Types of Neutrosophic Logical Connectors: There are situations when we have to more focus on the percent of falsity or of indeterminacy than on the percent of truth. Thus, we define in a similar way the logical connectors, but the main component will then be considered the last or second one respectively. An intriguing idea would be to take the arithmetic average of the corresponding components of the truth-, indeterminacy-, and false-neutrosophic connectors. Let's reconsider the previous notations. Definitions of Falsity-, Indeterminacy-, and Mean- Neutrosophic Logical Connectors: To be able to distinguish between Truth-, Falsity-, Indeterminacy-, and Mean-Neutrosophic Connectors one uses the nt(A) or n(A), nf(A), ni(A), nm(A) notations respectively. Let's again the neutrosophic probability of A1 be (t1, i1, f1) and of A2 be (t2, i2, f2). Let's Wt(a,b,c) = W(a,b,c) = (1-a)/(b+c), Wi(a,b,c) = (1-b)/(c+a), and Wf(a,b,c) = (1-c)/(a+b). Also, M(a,b,c) = ( M1(a), M2(b), M3(c) ) = ( (b+c)/3, (c+a)/3, (a+b)/3 ). Then one defines: Negation: nt(ŞA1) = ( N(t1), N(i1)Wt(N), N(f1)Wt(N) ), while ni(ŞA1) = ( N(t1)Wi(N), N(i1), N(f1)Wi(N) ), nf(ŞA1) = ( N(t1)Wf(N), N(i1)Wf(N), N(f1) ), nm(ŞA1) = ( N(t1)M1(Wt(N)), N(i1)M2(Wi(N)), N(f1)M3(Wf(N)) ). Conjunction: nt(A1/\A2) = ( C(t), C(i)Wt(C), C(f)Wt(C) ), while ni(A1/\A2) = ( C(t)Wi(C), C(i), C(f)Wi(C) ), nf(A1/\A2) = ( C(t)Wf(C), C(i)Wf(C), C(f) ), nm(A1/\A2) = ( C(t)M1(Wt(C)), C(i)M2(Wi(C)), C(f)M3(Wf(C)) ). Weak or inclusive disjunction: nt(A1\/A2) = ( D1(t), D1(i)Wt(D1), D1(f)Wt(D1) ), while ni(A1\/A2) = ( D1(t)Wi(D1), D1(i), D1(f)Wi(D1) ), nf(A1\/A2) = ( D1(t)Wf(D1), D1(i)Wf(D1), D1(f) ), nm(A1\/A2) = ( D1(t)M1(Wt(D1)), D1(i)M2(Wi(D1)), D1(f)M3(Wf(D1)) ). Strong or exclusive disjunction: nt(A1\_/A2) = ( D2(t), D2(i)Wt(D2), D2(f)Wt(D2) ), while ni(A1\_/A2) = ( D2(t)Wi(D2), D2(i), D2(f)Wi(D2) ), nf(A1\_/A2) = ( D2(t)Wf(D2), D2(i)Wf(D2), D2(f) ), nm(A1\_/A2) = ( D2(t)M1(Wt(D2)), D2(i)M2(Wi(D2)), D2(f)M3(Wf(D2)) ). Material conditional (implication): nt(A1->A2) = ( I(t), I(i)Wt(I), I(f)Wt(I) ), while ni(A1->A2) = ( I(t)Wi(I), I(i), I(f)Wi(I) ), nf(A1->A2) = ( I(t)Wf(I), I(i)Wf(I), I(f) ), nm(A1->A2) = ( I(t)M1(Wt(I)), I(i)M2(Wi(I)), I(f)M3(Wf(I)) ). Material biconditional (equivalence): nt(A1A2) = ( E(t), E(i)Wt(E), E(f)Wt(E) ), while ni(A1A2) = ( E(t)Wi(E), E(i), E(f)Wi(E) ), nf(A1A2) = ( E(t)Wf(E), E(i)Wf(E), E(f) ), nm(A1A2) = ( E(t)M1(Wt(E)), E(i)M2(Wi(E)), E(f)M3(Wf(E)) ). Sheffer's connector: nt(A1|A2) = ( S(t), S(i)Wt(S), S(f)Wt(S) ), while ni(A1|A2) = ( S(t)Wi(S), S(i), S(f)Wi(S) ), nf(A1|A2) = ( S(t)Wf(S), S(i)Wf(S), S(f) ), nm(A1|A2) = ( S(t)M1(Wt(S)), S(i)M2(Wi(S)), S(f)M3(Wf(S)) ). Peirce's connector: nt(A1A2) = ( P(t), P(i)Wt(P), P(f)Wt(P) ), while ni(A1A2) = ( P(t)Wi(P), P(i), P(f)Wi(P) ), nf(A1A2) = ( P(t)Wf(P), P(i)Wf(P), P(f) ), nm(A1A2) = ( P(t)M1(Wt(P)), P(i)M2(Wi(P)), P(f)M3(Wf(P)) ). I) Tables of Truth-, Indeterminacy-, Falsity-, and Mean- Neutrosophic Logic Connectors (rounded to the nearest last decimal place in each particular case): NEGATION Truth Indeterminacy Falsity t i f A 0.4 0.5 0.1 nt(ŞA) 0.60 0.14 0.26 ni(ŞA) 0.20 0.50 0.30 nf(ŞA) 0.055 0.045 0.900 nm(ŞA) 0.285 0.228 0.487 CONJUNCTION Truth Indeterminacy Falsity t i f A 0.4 0.5 0.1 B 0.7 0.1 0.2 nt(A/\B) 0.28 0.51 0.21 ni(A/\B) 0.89 0.05 0.06 nf(A/\B) 0.83 0.15 0.02 nm(A/\B) 0.67 0.24 0.09 WEAK DISJUNCTION Truth Indeterminacy Falsity t i f A 0.4 0.5 0.1 B 0.7 0.1 0.2 nt(A\/B) 0.82 0.12 0.06 ni(A\/B) 0.34 0.55 0.11 nf(A\/B) 0.43 0.29 0.28 nm(A\/B) 0.53 0.32 0.15 STRONG DISJUNCTION Truth Indeterminacy Falsity t i f A 0.4 0.5 0.1 B 0.7 0.1 0.2 nt(A\_/B) 0.4896 0.3370 0.1734 ni(A\_/B) 0.3480 0.4775 0.1745 nf(A\_/B) 0.3819 0.3725 0.2456 nm(A\_/B) 0.4065 0.3957 0.1978 IMPLICATION Truth Indeterminacy Falsity t i f A 0.4 0.5 0.1 B 0.7 0.1 0.2 nt(A->B) 0.88 0.045 0.075 ni(A->B) 0.22 0.55 0.23 nf(A->B) 0.049 0.031 0.92 nm(A->B) 0.383 0.209 0.408 EQUIVALENCE Truth Indeterminacy Falsity t i f A 0.4 0.5 0.1 B 0.7 0.1 0.2 nt(A<->B) 0.5104 0.2003 0.2893 ni(A<->B) 0.1927 0.5225 0.2848 nf(A<->B) 0.1214 0.1242 0.7544 nm(A<->B) 0.27483 0.28233 0.44284 SHEFFER'S CONNECTOR Truth Indeterminacy Falsity t i f A 0.4 0.5 0.1 B 0.7 0.1 0.2 nt(A|B) 0.72 0.138 0.142 ni(A|B) 0.0212 0.95 0.0288 nf(A|B) 0.00862 0.01138 0.98 nm(A|B) 0.24994 0.36646 0.38360 PEIRCE'S CONNECTOR Truth Indeterminacy Falsity t i f A 0.4 0.5 0.1 B 0.7 0.1 0.2 nt(AB) 0.18 0.3154 0.5046 ni(AB) 0.11 0.45 0.44 nf(AB) 0.08 0.20 0.72 nm(AB) 0.1233 0.3218 0.5549 6) NEUTROSOPHIC TOPOLOGY A) Definitions: Let's construct a Neutrosophic Topology on NT = [0, 1], considering the associated family of subsets (0, a), for 0 ó a ó 1, the whole set [0, 1], and the empty set = (0, 0), called open sets, which is closed under set union and finite intersection. The union is defined as (0, a) U (0, b) = (0, d), where d = a+b-ab, and the intersection as (0, a) ď (0, b) = (0, c), where c = ab. The interval NT, endowed with this topology, forms a neutrosophic topological space. 7) NEUTROSOPHIC MEASURE A) Definitions: The collection of all subsets of [0, 1], defined above, constitute a neutrosophic sigma-algebra (or neutrosophic ĺ- algebra), because the set itself, the empty set, the complements in the set of all members, and all countable unions of members belong to the power set P([0, 1]). The complementary of (0, a) is (0, r), where r = 1-a, which is a closed set. The interval NT, endowed with this sigma-algebra, forms a neutrosophic measurable space. One defines a neutrosophic non-negative measure nm((0,a)) = a, which is ĺ-additive because the only disjoint subsets are (0,0) and (0,a) and a = nm((0,a)) = nm((0,0)U(0,a)) = nm((0,0))+nm((0,a)) = 0+a. nm : P([0, 1]) ---> [0, 1]. Of course nm(AUB) = nm(A)+nm(B)-nm(AďB), nm(AďB) = nm(A)nm(B), and nm(C(A)) = 1-nm(A). The above measurable space, endowed with the non-negative measure, forms a neutrosophic measure space. B) Isomorphicity: Neutrosophic Probability Space, Neutrosophic Logical Space, Neutrosophic Topological Space, and Neutrosophic Measure Space are all isomorphic. References: [ 1] Albee, Ernest, "History of English Utilitarianism", Collier Books, Crowell-Collier Publ. Co., N.Y., 1962. [ 2] Ayer, A.J., "Logical Positivism", The Free Press of Glencoe, New York, 1958. [ 3] Bailey, Cyril, "The Greek Atomists and Epicurus", Russell & Russell, Inc., New York, 1964. [ 4] Berlin, Isaiah (ed.), "The Empiricists: John Locke, George Berkeley, David Hume", Dolphin Books, Doubleday & Company, Inc., Garden City, N.Y., 1961. [ 5] Bouvier, Alain, George, Michel, "Dictionnaire des Math‚matiques", sous la direction de Fran‡ois Le Lionnais, Presses Universitaire de France, Paris, 1979. [ 6] Bouwsma, W.J., "The Culture of Renaissance Humanism", American Historical Association, Washington, 1973. [ 7] Burnet, John, "Greek Philosophy: Thales to Plato", St. Martin's Press, Inc., New York, 1962. [ 8] Carr‚, M.H., "Realists and Nominalists", Oxford University Press, Fair Lawn, NJ, 1946. [ 9] Copleston, Frederick, "Arthur Schopenhauer, Philosopher of Pessimism", Barnes and Noble Books, New York, 1975. [10] Hassing, Richard F., "Final Causality in Nature and Human Affairs", The Catholic University of America Press, Baltimore, 282p., 1997. [11] Hegel, G.W.F., "The Phenomenology of Spirit", trans., A.V.Miller, Clarendon Press, Oxford, 1977. [12] Hobbes, Thomas, "Body, Man and Citizen", Collier Books, Crowell-Collier Publishing Co., New York, 1962. [13] Iorga, Nicolae, "Cugetţri", edited by Elisabeta Jurca-Pod, The Yellow Bird Publ., Chicago, 1991. [14] Jaspers, K., "Nietzsche: An Introduction to the Understanding of His Philosophical Activity", University of Arizona Press, Tucson, 1965. [15] Jaspers, Karl, "General Psychopathology", translated by J. Hoenig and Marian W. Hamilton, Introduction by Paul McHugh, The John Hopkins University Press, Baltimore, Vol. I and II. [16] Kant, Immanuel, "Critique of Pure Reason", St. Martin's Press, New York, 1965. [17] Kenny, A., "Aquinas", Hill and Wang, Inc., New York, 1980. [18] Kockelmans, J.L., "Phenomenology: The Philosophy of Edmund Husserl and Its Interpretation", Doubleday and Company, Inc., Garden City, N.Y., 1967. [19] Le, Charles T., "The Smarandache Class of Paradoxes", in , Bombay, India, No. 18, 53-55, 1996. [20] Leff, Gordon, "Medieval Thought from St. Augustine to Ockham", Penguin Books, Inc., Baltimore, 1962. [21] Marcel, Gabriel, "Man against Mass Society", Henry Regnery Co., Chicago, 1962. [22] Marcuse, Herbert, "Reason and Revolution: Hegel and the Rise of Social Theory", Beacon Press, Boston, 1960. [23] McKeon, Richard P., "An Introduction to Aristotle", Random House, Inc., New York, 1947. [24] McNeil, Martin, F., Thro, Ellen, "Fuzzy Logic / A Practical Approach", Foreword by Ronald R. Yager, Academic Press, 1994. [25] Mehta, J.L., "Martin Heidegger: The Way and the Vision", University of Hawaii Press, Honolulu, 1976. [26] Munshi, K.M., Diwakar, R.R. (gen. eds.), "Introduction to Vedanta", by P. Nagaraja Rao, Bhavan's Book University, Chowpatty, Bombay, India, 1966. [27] Peirce, C.S., "Essays in the Philosophy of Science", The Liberal Arts Press, Inc., New York, 1957. [28] Popa, Constantin M., "The Paradoxist Literary Movement", Xiquan Publ., Hse., Phoenix, 1992. [29] Popescu, Titu, "Estetica paradoxismului", Tempus Publ. Hse., Bucharest, 1995. [30] Rousseau, Jean-Jacques, "On the Social Contract", trans. Judith R. Masters, St. Martin's Press, Inc., New York, 1978. [31] Russell, Bertrand, "Introduction to Mathematical Philosophy", Dover Publications, Inc., New York, 1993. [32] Ryle, Gilbert, "The Concept of Mind", Barnes and Noble, Inc., New York, 1950. [33] Sartre, Jean-Paul, "Existentialism and Human Emotions", Philosophical Library, Inc., New York, 1957. [34] Scruton, Roger, "A Short History of Modern Philosophy / From Descartes to Wittgenstein", Routledge, London, 1992. [35] Smarandache, Florentin, "Collected Papers", Vol. II, University of Kishinev Press, Kishinev, 1997. [36] Smarandache, Florentin, "Distihuri paradoxiste", Dorul, Nţrresundby, 1998. [37] Smarandache, Florentin, "Linguistic Paradoxists and Tautologies", Libertas Mathematica, University of Texas at Arlington, to appear. [38] Soare, Ion, "Un Scriitor al Paradoxurilor: Florentin Smarandache", Almarom, Rm. Vlcea, 1994. [39] Stephens, J., "Francis Bacon and the Style of Science", University of Chicago Press, Chicago, 1975. [40] TeSelle, E., "Augustine the Theologian", Herder & Herder, Inc., 1970. [41] Vasiliu, Florin, "Paradoxism's Main Roots", Ed. Haiku, Bucharest, 1994. [42] Veatch, H.B., "A Contemporany Appreciation", Indiana University Press, Bloomington, 1974. [43] Vlastos, Gregory, "The Philosophy of Socrates", Anchor Books, Garden City, New York, 1971. [44] Wittgenstein, L., "Tractatus Logico-Philosophicus", Humanitas Press, New York, 1961. CONTENT AND INDEX OF NEW TERMS Preface ...................................................... 1) NEUTROSOPHY, A NEW BRANCH OF PHILOSOPHY A) Etymology ............................................ B) Definition ........................................... C) Characteristics ...................................... D) Methods of Neutrosophic Study ........................ E) Formalization ........................................ F) Main Principle ....................................... G) Fundamental Thesis ................................... H) Main Laws ............................................ I) Mottos ............................................... J) Fundamental Theory ................................... K) Delimitation from Other Philosophical Concepts and Theories ............................................. L) Philosophy's Limits .................................. Tautologism ........................................ Nihilism ........................................... Philosophism ....................................... M) Classification of Ideas .............................. N) Evolution of an Idea ................................. Dynaphilosophy ..................................... O) Philosophical Formulas ............................... a) Law of Equilibrium .............................. b) Law of Anti-Reflexivity ......................... c) Law of Complementarity .......................... d) Law of Inverse Effect ........................... e) Law of Reverse Identification ................... f) Law of Joined Disjointedness .................... g) Law of Identities' Disjointedness ............... h) Law of Compensation ............................. i) Law of Prescribed Condition ..................... j) Law of Particular Ideational Gravitation ........ k) Law of Universal Ideational Gravitation ......... P) Neutrosophic Studies and Interpretations of Known Theories, Modes, Views, Processes of Reason, Acts, Concepts in Philosophy ............................... Trialectic ......................................... Pluralectic ........................................ Transalectic (ě-alectic) ........................... Comparative Philosophy ............................. Quantum Philosophy ................................. Neutrosophism ...................................... Ignorantism ........................................ Triplementarity .................................... N-plementarity ..................................... Infinit-plementarity (ě-plementarity) .............. Philosophy of Philosophy ........................... Inconsistent Systems of Axioms ..................... Contradictory Theory ............................... Hermeneutics of Hermeneutics ....................... Social Theory ...................................... Social Paradox ..................................... The Sets' Paradox .................................. A Paradoxist Psychological C˘mplex ................. Paradoxist Psychological Behavior .................. Ceaseless Anxiety .................................. Inverse Desire ..................................... Mathematician's Paradox ............................ Divine Paradox (I) ................................. Divine Paradox (II) ................................ Expect the Unexpected .............................. The Ultimate Paradox ............................... The Invisible Paradox .............................. Fuzzy Concept ...................................... Paradoxist Existentialism .......................... Semantic Paradox (I) ............................... Semantic Paradox (II) .............................. Tautologies ........................................ Paradox of the Paradoxes ........................... Semi-utilitarism ................................... Neutrosophic Evolutionism .......................... Anti-philosophy .................................... Will to Mediocrity ................................. Midman ............................................. Will to Weakness ................................... Underman ........................................... Infinitdisciplinarity .............................. Global Discipline .................................. Pluripsism ......................................... Tetranitarism ...................................... Plurinitarism ...................................... Idon ............................................... Idonical Measurement ............................... Ultimate Idea ...................................... Neutrosopher ....................................... Civilization Paradox ............................... Philosophy of Neutralities ......................... Concreteness of the Abstractness ................... Mechanical Philosophy .............................. NonLogos ........................................... Unconditioned Totalitarism ......................... Conditioned Totalitarism ........................... Philosophy-poetry .................................. Displacement Towards Neutrality .................... Neutrosophic Notions ............................... Antispirits ........................................ DevGod ............................................. Paradoxist Determinism ............................. Definitive Judgement ............................... Neutrosology ....................................... Neutrosology of the Culture ........................ Neutrosophic Anthropology .......................... Supra-man .......................................... Infra-man .......................................... Null-man ........................................... Neut-ward .......................................... Neutrosophic Behaviorism ........................... Neutrosophic Sphere ................................ Neutrosophic Existentialism ........................ Anti-tautology ..................................... Negative Definition ................................ Maieutic Neutrosophy ............................... Naive Philosophy ................................... Neutrosophic Phenomenology ......................... Contemporary Neutrosophic Moral .................... Postneutrosophy .................................... Labelism ........................................... Superego ........................................... Underego ........................................... Spiritual Pathology ................................ Immaterial Matter .................................. Homo Neutrosophus .................................. Pluri-philosophy ................................... 2) TRANSDISCIPLINARITY, A NEUTROSOPHIC METHOD A) Definition ........................................... B) Multi-Structure and Multi-Space ...................... Multi-Group ........................................ Multi-Ring ......................................... Multi-Field ........................................ Multi-Lattice ...................................... Multi-Module ....................................... K-Structured Space ................................. Infinite-Structured Space .......................... K-Structured Group ................................. Infinite-Structured Group, etc. .................... C) Psychomathematics .................................... D) Mathematical Modeling of Psychological Processes a) Improvement of Weber's and Fechner's Laws ....... b) Synonymity Test ................................. c) Linguistic Neutrosophy .......................... E) Psychoneutrosophy .................................... F) Socioneutrosophy ..................................... G) Econoneutrosoph ...................................... H) New types of Philosophies a) Object Philosophy ............................... b) Concrete Philosophy ............................. c) Sonorous Philosophy ............................. d) Fuzzy Philosophy ................................ Trichotomy .................................... Plurichotomy .................................. Transchotomy (ě-chotomy) ...................... Fuzzy-chotomy ................................. Neutro-chotomy ................................ e) Applied Philosophy .............................. f) Experimental Philosophy ......................... g) Futurist Philosophy ............................. h) Nonphilosophy ................................... I) New types of Philosophical Movements a) Revisionism ..................................... b) Inspirationalism ................................ c) Recurrentism .................................... d) Sophisticalism .................................. e) Rejectivism ..................................... f) Paradoxism ...................................... J) Logical and Combinatory Modeling in Experimental Literature a) An Avant-garde Literary Movement, the Paradoxism - The Basic Thesis of the Paradoxism .......... - The Essence of the Paradoxism ............... - The Delimitation from Other Avant-gardes .... - The Directions of the Paradoxism ............ b) New Types of 'Mathematical' Poetry with Fixed Form - Paradoxist Distich .......................... - Tautological Distich ........................ - Dualist Distich ............................. - Paradoxist Tertian .......................... - Tautological Tertian ........................ - Paradoxist Quatrain ......................... - Tautological Quatrain ....................... - Fractal Poem ................................ c) New Types of Short Story - Syllogistic Short Story ..................... - Circular Short Story ........................ d) New Types of Drama - Neutrosophic Drama .......................... - Sophistic Drama ............................. - Combinatory Drama ........................... 3) NEUTROSOPHIC PROBABILITY, A GENERALIZATION OF CLASSICAL PROBABILITY A) Definition ........................................... B) Neutrosophic Statistics .............................. C) Neutrosophic Probability Space ....................... D) Applications ......................................... 4) NEUTROSOPHIC SET, A GENERALIZATION OF FUZZY SET A) Definition ........................................... B) Neutrosophic Set Operations .......................... Complement of A .................................... Intersection ....................................... Union .............................................. Difference ......................................... Cartesian Product .................................. C) Applications ......................................... 5) NEUTROSOPHIC LOGIC, A GENERALIZATION OF FUZZY LOGIC A) Introduction ......................................... Neutrosophic Circle ................................ B) Definitions .......................................... Neutrosophic Logical Value ......................... Neutrosophic Logic ................................. C) Applications ......................................... D) Definition of (Truth-)Neutrosophic Logical Connectors Negation ........................................... Conjunction ........................................ Weak or Inclusive Disjunction ...................... Strong or Exclusive Disjunction .................... Material Conditional (Implication) ................. Material Biconditional (Equivalence) ............... Sheffer's Connector ................................ Peirce's Connector ................................. E) Properties of Neutrosophic Logical Connectors a) Negation ........................................ b) Conjunction ..................................... c) Weak Disjunction ................................ d) Strong Disjunction .............................. e) Implication ..................................... f) Equivalence ..................................... g) Sheffer's Connector ............................. h) Peirce's Connector .............................. F) Comparison between Fuzzy Logic and Neutrosophic Logic G) Other Properties ..................................... Ninversity, a New Algebraic Structure .............. H) Other Types of Neutrosophic Logic Connectors. Definitions of Falsity-, Indeterminacy-, and Mean- Neutrosophic Logic Connectors ...................... I) Tables of Truth-, Indeterminacy-, Falsity-, and Mean- Neutrosophic Logic Connectors ........................ 6) NEUTROSOPHIC TOPOLOGY A) Definitions .......................................... Neutrosophic Topology .............................. Neutrosophic Topological Space ..................... 7) NEUTROSOPHIC MEASURE A) Definitions .......................................... Neutrosophic Sigma-Algebra ......................... Neutrosophic Measurable Space ...................... Neutrosophic Non-negative Measure .................. Neutrosophic Measure Space ......................... B) Isomorphicity ........................................ References ................................................... Content and Index to New Terms ...............................