-.- A comparative microstudy of 3-valued logics CopyRight (C) 1986-2007 , Jan Hajek, NL, version 3.11 of 2007-5-15 Keywords : 3-valued logic , many-valued logic , multi-valued logics implication Summary : The implication (and its negation) is the most challenging logical operation. No wonder that it is the most controversial one, especially in multi-valued logics. It is the only non-commutative function of both input variables. Its nontrivial asymmetry makes it the prime candidate for logical inference, and it has some potential for causal inference, simply because we do not have much other choice. Therefore it is here compared for 5 three-valued logics by 9 thinkers. The results : - The most rational logic is that by S.C. Kleene and by Jan Hajek ; - the least rational logic is by T. Kool . Mottos: CERTAINTY CAN ARISE NEITHER FROM UNCERTAINTY ALONE, NOR FROM AMBIGUITY { Jan Hajek's rationale } [ Fuzzy set ] theory cannot be either right or wrong. It is applicable math tested by its uses. However, the rationale behind it, the systemic principles involved can be right or wrong. { Brian Gaines, 1985 } A meaning arises from an intention, over distinction, to functionality. A meaning of a human invention (eg a logic) is mainly determined by its functionality. { Jan Hajek } Even a Berkeley computer science and electrical engineering professor can get implication wrong. { Eric Hehner , University of Toronto, p8 } Never run after a bus, a (wo)man, or a multi-valued logic, because there will be another one in ten minutes. { my politically correct paraphrase of a Yale professor who thus spoke about a bus, a woman or a cosmological theory } These Copenhagen people are so clever in their use of language that, even after they have answered your question, you still dont know whether the answer was "yes" or "no"! { Eugene Wigner, a Hungarian-American physicist & Nobel Prize winner has worked on the Manhattan project } ... there's a third possible logical term equal to yes and no which is capable of expanding our understanding in unrecognized direction. We don't even have term for it, so I'll have to use the Japanese mu. Mu means "no thing". Like "Quality" it points outside the process of dualistic discrimination. Mu simply says, "No class; not one, not zero, not yes, not no". It states that the CONTEXT OF THE QUESTION is such that a yes or no answer is in error and should not be given. "Unask the question" is what it says. { Robert Pirsig, Zen and the Art of Motorcycle Maintenance, p.314 in Bantam edition, April 1975 } { JH: Japanese mu = wu in Chinese, as eg in wu-wei } -.- +Caution: NO part of this document may be published, implemented, programmed, copied or communicated by any means without an explicit & full reference to this author + the full title + the website www.humintel.com/hajek for the freshest version + the CopyRight note in texts and in ALL references to this. An implicit, incomplete, indirect, disconnected or unlinked reference (in your text and/or on www) does NOT suffice. All based on 1st-hand experience. All rights reserved. -.- +Contents: +Warm up with examples of (non)probabilistic (non)implications x -> y : +Three valued logics (is the main topic, hence at the top) +Examples of x -> y +Basics of Boolean logic (is the support, hence below) -.- +Warm up with examples of (non)probabilistic (non)implications x -> y : Ex0: IF c THEN b ; is not the same as IF not c THEN not b ; See +Basics of Boolean logic (far below). Ex1: (x -> y) == (~y -> ~x) is implication's contrapositive property : "IF Fire x THEN probably smoke y" makes sense, and so does "IF No smoke ~y THEN probably no fire ~x" which works ok in daily life. Alas, "Fire x CAUSES smoke y" makes sense, but "No smoke ~y CAUSES no fire ~x" is nonsense. Hence it is implication's contrapositivity which makes it less suitable as an indicator or as a measure of causation. Alas, we do not have much other choice, so we have to use implication as an indicator of possible causation tendency. Ex2: Probability P(c|b) does not say that b implies c, although the converse is true: IF b implies c, THEN P(c|b) = 1. Therefore high P(c|b) is just an indication of a possible implication b -> c, ie that "b is SUFicient for c". From the set theory follows that P(c|b) is also an indication of "c is NECessary for b". Ex3a: Independence implies uncorrelatedness but not necessarily vice versa, ie uncorrelatedness does not necessarily imply independence, ie uncorrelated may nevertheless be independent. However, 2 jointly Gaussian random vectors or r.v.'s which are uncorrelated are also independent. But Gaussian r.v.'s need not to be jointly Gaussian. Very low correlation is just an indication of a possible independence. Very high correlation is just an indication of a possible causation. Ex3b: A linear relationship implies a correlation coefficient near to 1 or to -1, but not necessarily vice versa. High correlation is not equivalent (= a 2-way implication) to "most probably a linear relationship". The correct reasoning holds the opposite 1-way only : IF there is a linear relation between X and E[Y|X] , THEN the value of the correlation coefficient will be close to an extreme, and this will be due to a small mean squared error MSE . Very high correlation is just an indication of a possibly linear relationship. Ex3c: IF at least one of 2 r.v.'s has a ZERO-mean , eg Mo = 0 , THEN orthogonality & uncorrelatedness mutually imply each other, ( 2-way implication ie equivalence ) hence : IF none of both r.v.'s has a ZERO-mean , THEN they cannot be orthogonal & uncorrelated simultaneously. -.- +Three valued logics compared via implication : Alas, too many logics were formulated already. Rather than yet another new logic we need to know their strengths & weaknesses, when (not) & how (not) to apply which logic. What matters most is not some formal "beauty" like eg some formalistic consistency. It is their opeRational interpretation & clarity & robustness ie ease of use by as ordinary as possible humans, that's what matters most. Like eg fiber optics, a logic is of little use if u need a crew of PhD's from Harvard, MIT, Oxbridge or Stanford to install, connect and use it. The 1st Amendment to the US Constitution guarantees freedom of expression. Hence everybody is free to polute other peoples mails & minds with her/his nonsense and/or with her/his private logic. Theoretically, everyone's logic should be equal before the law, but even in daily legal practice, not every reasoning is considered valid. In Orwell's Animal Farm, some Animals were more equal than the other. At university all people are equal, but professors are more equal than students and janitors. To say that all logics happily co-exist in The Universe is MEANINGLESS for everybody except for extreme Platonists. As long as we doNt identify distinct pros & cons of each logic, and doNt spell them out in intelligible terms, nobody (but few) will know what is who talking about, and we should not use such logic(s). For serious purposes (not for private intellectual mental games) every tool must be DECONSTRUCTED ie destructively stress-tested. Hence this evaluative comparative microstudy. Various 3-valued logics have been proposed by various thinkers, eg by : (a leading + means "his logic is identical with his predecessor's") - Lotfi Zadeh (in 1965 became the father of fuzzy logic; his precursor with vague logic was Max Black in 1930ies, also born in Baku), + Ronald Yager (published a lot on fuzzy logics since early 1980ies); - S.C. Kleene (in 1952; in 1956 he fathered regular expressions for automata theory, since 1968 used eg in string pattern matching), + Jan Hajek (in 1980ies designed and implemented expert systems based on his 3-valued logic extended with new unique, because partially !!! self-checking, operators: alt , Salt , altS , Xalt used thus : x alt y ie x is an alternative to y , alt is commutative x Salt y ie x is a Superalternative to y , is non-commutative x altS y ie x is a subalternative to y , is non-commutative x Xalt y ie x and y are mutually exclusive alternatives , is comm; all of these OR-like connectives derive from the set theory and can automatically intercept ie to detect & report many, though obviously not all, SEMANTIC ERRORS in the inference rules expressed with these "ALT-OPS" supplied by fallible human beings. New AND-like connectives are: TrueThen , FalseThen , MayThen. All operators are designed so as to provide safe MAXIMUM of CONDITIONAL EVALUATIONS requiring MINIMUM of inputs/questions/ interrogations/data to reach a conclusion. Therefore Hajek's expert system was called QUIXPERT written in Turbo Pascal ); - Jan Lukasiewicz (in 1920ies; 1954 fathered "Polish notation" used in virtually all compilers and calculators), + Hans Reichenbach (an important philosopher of science); - Kurt Goedel aka Kurt Godel (top theoretical logician of the past century; he went nuts and starved himself to death), + A. Heyting (in 1930ies); - T. Kool in 2007 . 3-valued logics with identical truth-tables (a + indicates identity): Z+Y is fuzzy logic by Zadeh , and by Yager ; K+J is by Kleene , and by Jan Hajek , which is the most reasonable one, for the obvious reasons spelled out in Q1 / A1 below ; L+R is by Lukasiewicz , and by Reichenbach ; G+H is by Goedel / Godel , and by Heyting; Kool is a loner, not identical with anyone, therefore his logic is an anomaly. Could it be an ingenious anomaly ? Let's check & see. If it's not checked, it's wrong. Seeing is believing. Let's denote : Words: our in this epaper means "those considered here", ie here our does not mean my ie Jan Hajek's. Operands : 0 is "certainly False" ; 1 is "certainly True". ? is 1/2 , 0.5 , uncertain, doNt know, maybe (as in human affairs), indeterminate (eg as in quantum mechanics), vague, intermediate (eg as in fuzzy logic) ; Operators : == is equivalent , synonymous , 2-way implication , if and only if. Everybody here defines negation as ~z == 1 - z , eg ~? == ? ; (x or y) is numerically MAX(x;y) in all our logics (x,y) ie (x & y) is numerically Min(x;y) in all our logics <> means unequal => isNt used here, is meaningless, forbidden, no op here . (x >= y) means (x greater than y) ; (x <= y) means (x smaller than y) , (x entails y) , (x -> y) (x -> y) means (x implies y) , numerical formulae & results differ depending on the formulas for our (x -> y)'s which work not only with the values 0, 0.5, 1, but also with any real numbers 0 <= x,y <= 1 thus as implications in multi-valued logics : Now is the right spot to sum up the most important tautologies, ie equivalences which are allways true, for the Boolean implication : (x -> y) == ~(x,~y) == (~x or y) , the 2nd == is by DeMorgan's rule == [(x or y)==y] == (~y -> ~x) a " contrapositive " property undesirable for causation . Expressing these various ==s by means of Min(x;y) and MAX(x;y) , various formulae for (x -> y) obtain : ZY is fuzzy logic intended for > 3 values, so its (x -> y) exists in various forms, and I am showing just one of them, intentionally chosen to differ from K+J , see Q7 / A7 : [~x or (x,y)] == (~x or y) ZY := Max( 1-x ; Min(x;y) ); is from [~x or (x,y)] == ~[x,~(x,y)] KJ := Max( 1-x ; y ); is from (~x or y) == ~(x,~y) LR := Min( 1-x + y; 1 ); if x <= y then GH :=1 else GH := y if x <= y then Kool:=1 else Kool:= 0 From the inputs 0, ?, 1, the just shown formulas always produce 0 , 1/2 ie 0.5 ie ?, 1. There exist other formulas for many-valued logics which produce reals in the range from 0 to 1, but with (many) other values in between, eg: (x & y) == x.y ie a product, hence (x or y)== x + y - x.y = x + (1-x).y = y + (1-y).x . I designed the following table so as to be comparative ie to highlight the differences between (x -> y)'s in different logics, hence my ='s : Table for x implies y : x is the antecedent , y is the consequent Inputs | x -> y out : similar logics are in neigboring columns x y | Z+Y K+J L+R G+H Kool here = means "like K+J" : -------|---------------------|----table hand-crafted by robots ---- 0 0 | = 1 = = = | 0 ? | = 1 = = = | 0 1 | = 1 = = = | ? 0 | = ? = 0 0 | ? ? | = ? 1 1 1 |... on crazy irrational 1's see Q4 ? 1 | ? 1 = = = |... this Z+Y's ? is anomalous, see Q7 1 0 | = 0 = = = | 1 ? | = ? = = 0 |... Kool's 0 is anomalous, see Q3 Q5 1 1 | = 1 = = = | -------|------------------------- Kool's (x -> y) never yields a ? Problems with 3-valued logics : The very fact that there exist several 3-valued logics is an indication of problems. To have several logics is a sad fact; a single logic, however not ideal, would be better for standard reasoning. The standard voltage of 230 V is not ideal for all apps, but it is much better than 5 or more voltages. Note that all logics here are identical for the 4 inputs when x, y are 0, 1. In fact they all are identical for one more input (0 -> ?). If we change Z+Y to another arithmetic formula used by many fuzzy logicians, then Z+Y will be identical with K+J. Various logicians had various motives for and various methods how to derive their various logics. I have derived my no-nonsense 3-valued logic independently of Kleene. Q1 / A1 tells my rationale on how I derived it : Q1: Why is K+J logic the most rational one ? A1: Because it is based on my most rational reasoning for its derivation. I (not Kleene) express it in opeRational terms for any logical expression thus : IF some operands are ? and we replace them successively with ALL possible combinations of 0s and 1s, and AT LEAST ONE pair of those evaluations yield different outputs THEN the value of the logical expression must be THE LEAST CERTAIN value obtained ( my algorithmic rationale works also for multi-valued logics where 0, 1 are equally certain ) Pragmatics: P1: op1(?) == ? , (? op2 ?) == ? , ie No certainty out of uncertainty; P2: (? op certainty) must not yield contradictory results ; (certainty op ?) must not yield contradictory results . P3: My algorithm above applies to expressions with any number of operands and operators, monadic, dyadic, anyadic, prefix, infix, postfix, anyfix :-) P4: My "AT LEAST ONE pair" has the practical consequence that we can stop checking an expression at the 1st contradiction (after at least 2 different evaluations), when one of the resulting values is the MOST UNCERTAIN one, or when the logic has only 3 values (0, 1, ?). P5: Conditional evaluation ie partial evaluation is allowed (and encouraged to minimize questioning/interrogation) for those (sub)expressions for which it is clear (a priori ie before they are fully evaluated ie before all their operands were even assigned a value) that the remaining evaluation cannot change the result anymore. That's my PRINCIPLE OF LOGICAL INVARIANCE . { Kleene 1952, p.335 } calls Lukasiewicz's logic "irregular". Although Kleene's justification of regularity is unclear to me in terms of real-world semantics, his detection of (ir)regularity is remarkably clear, although I made it even more clear on more places thus : A given column (row) contains 1 in the ? row (column), only if the given column (row) consists entirely of 1s ; and likewise for 0. I add: and this must hold for all columns and all rows, otherwise a 3-valued logic is not regular in Kleene-sense. Kleene's truth-tables for (x -> y) are these square tables : K+J is regular L+R is irregular y: | 1 0 ? y: | 1 0 ? -----|------- -----|------- x: 1 | 1 0 ? x: 1 | 1 0 ? 0 | 1 1 1 0 | 1 1 1 ? | 1 ? ? ? | 1 ? 1...makes L+R irregular Kleene was no regular Joe since he had some obsession with regularity, as I said above. In fact I admit that I am not sure if his regularity and my common sense (also expressed as an easily PROGRAMMABLE RULE) are equivalent. The fact is that Kleene and I have both obtained the same truth-table for a 3-valued logic, and that my rationale/justification/ derivation is understandable & communicable, while Kleene's is not. Feel free to test on your spouse if (s)he better understands Hajek or Kleene :-) Unlike Kleene's metamath which is a tough read, my rationality is a communicable common sense rooted in living reality , not in a dead formalism. Finally it is the opeRational interpretation which matters more than some formal consistency. Q2: The law of contradiction (z & ~z) == 0 doesNt hold, eg (? & ~?) == ? The law of excluded middle (z or~z) == 1 doesNt hold, eg (? or~?) == ? Is it bad ? A2: Not too bad, as in practice we normally doNt do such ops with a single operand. Moreover, the problem of no tautologies (including the just mentioned non-laws) has been resolved in { John Grant , 1974 } for Kleene's 3-valued logic, hence also for Jan Hajek's 3-valued logic . Q3: Why has Kool (1 -> ?) == 0 ?? Because he wishes (x -> y) == (x <= y) hence (1 <= 0.5) == 0. Is it a reasonable wish ? A3: Only seemingly, but NOT really, because ( see A1 ) : (1 -> 0) == 0 , (1 -> 1) == 1 in his and in all other logics, hence the result depends on the actual value of the 2nd operand, hence the result must be a ? (as it is in all other logics). !!! Koole's ordering is 0 < 0.5 < 1 where math doesNt capture the meaning; !!! K+J ordering is ? < 0 < 1 where math does capture the meaning: !!! CERTAINTY CANNOT ARISE FROM AMBIGUITY . Q4: Is (? -> ?) == 1 reasonable in L+R , G+H , Kool ? A4: Not if we reason as rationally as I do in A1 and A3 . !!! CERTAINTY CANNOT ARISE FROM UNCERTAINTY ALONE. Q5: What are other weaknesses of Kool's logic in addition to the sad fact that his (x -> y) always creates a certainty out of uncertainty ? ? A5: (x -> y) == (~x or y) == ~(x,~y) , the 2nd == is by DeMorgan's rule ; the 1st == holds for K+J and for L+R (with some outputs different). The 1st == is needed eg for multiobjective decisions a la Yager. Alas, the 1st == does NOT hold for Kool's logic : (1 -> ?) == 0 by Kool, but his ~(1,~?) == (0 or ?) == ? differs . Q6: What about equivalence (x == y) in Kool ? A6: (x == y) == [(x -> y) & (y -> x)] in any decent logic. Since his (x -> y) canNOT yield a ? , his (x == y) produces certainty (be it 1, or be it 0) out of uncertainty. This certainly is no good. Q7: What makes this this particular Z+Y to differ from K+J ? A7: This Z+Y's (? -> 1) == ? versus 1 by K+J is caused by the fact that Max(1-x ; Min(x;y)) <> Max(1-x; y) for x=0.5 and y=1 ; this numerical inequality despite formally correct translation [~x or (x,y)] == (~x or y) from Boolean logic to 3-valued numerics unveils a fundamental problem with this particular Z+Y. Q8: Which logic would you choose for controling the flight of your plane, or Star Wars anti-rock sys, or your salary sys ? A8: I gave my preference in the Abstract already. How about you ? -.- +Basics of Boolean logic : There exist only 4^2 = 16 Boolean operators aka Boolean functions ie binary logical operations (~ is non ie a complement) : Inputs | function's output : x op y | f0 f1 f2 f3 f4 f5 f6 f7 f8 f9 10 11 12 13 14 15 are f0 up to f15 -------|--c--c--------------c--c--c--c--------------c--c-|--- c = commutative 0 op 0 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | DeMorgan's laws: 0 op 1 | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | ~(x or y)=(~x & ~y) 1 op 0 | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | ~(x & y)=(~x or ~y) 1 op 1 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | -------|--c--c--------------c--c--c--c--------------c--c-|--- c = commutative No fun=0 2 2 y 2 x 2 2 2 2 ~x 2 ~y 2 2 0 | a 2 = fun(x , y) f4 = x > y hence ~f4 = f11 = (x <= y) = ~(x,~y) = (x entails y) = (x -> y) f2 = y > x hence ~f2 = f13 = (y <= x) = ~(y,~x) = (y entails x) = (y -> x) f2 = ~f13 = ~~(y,~x) = (y,~x) = ( y unlessBlockedBy x) == (~x,y) = (~x unlessBLockedBy ~y) , formally ; Isomorphy between Boolean logic and set theory and probability theory allows us to directly write formulas in probabilistic logic : eg: fire z := burning y unlessBlockedBy the use of extinguisher x P(z) = Py - Pxy; fire z := burning y unlessBlockedBy the ~x=no oxygen P(z) = Py - P(y,~x) = Py - ( Py - Pxy ) = Pxy is no good; clearly ! do not use complements if you don't want simplistic results. f2: if ~x then z:=y else z:=0 ; f13: if ~x then z:=~y else z:= 1; f2: if x then z:=0 else z:=y ; f13: if x then z:= 1 else z:=~y; f2 is better understood than f13 , since the result z is just z:=y unless x f2: P(y,~x) = Py - Pxy, = Py if Pxy = 0 ie disjoint = 0 if Pxy = Py ie P(x|y) = 1, needs Py <= Px > 0 if Pxy < Py ie P(x|y) < 1. 4+2 = 6 = 4 functions of one variable only & 2 non-functions of x, or of y. 4*2 = 8 complementary pairs of f#'s (note that 0+15 = 1+14 = .. = 7+8 = 15 ) : 4*2 = 8 are commutative (marked by c ) ie symmetric wrt x, y, hence UNsuitable as indicators of causation which is asymmetric wrt x, y : c | no fun ~f0 = f15 no fun c | ~(x & y) = ~f1 = f14 = (x nand y) = Sheffer function ! | (~y or x) = ~f2 = f13 = (y implies x) = (y->x) = ~(y & ~x) = (~x->~y) = | f13 = [(y or x)==x] | ~(y) = ~f3 = f12 = ~y ! | (~x or y) = ~f4 = f11 = (x implies y) = (x->y) = ~(x & ~y) = (~y->~x) = | f11 = [(x or y)==y] | ~(x) = ~f5 = f10 = ~x c | ~(x Xor y) = ~f6 = f9 = (x eqv y) = (x == y) = equivalence of x, y c | ~(x or y) = ~f7 = f8 = (x nor y) = Pierce function Any (not only those listed here) valid rule (ie an lhs = rhs ) has its DUAL rule (the DUALITY is mutual ie a symmetric 2-way relation), obtained thus: change ANDs into ORs, ORs into ANDs, False ie 0 into True ie 1, 1 into 0, but don't change the parentheses and negations (ie non, not, ~ ). Examples: 0 & 1 = 0 is dual with 1 or 0 = 1 ~(x,~y) = (~x or y) is dual with ~(x or ~y) = (~x,y) Only 2 pairs of f's are non-commutative functions of both variables x, y ; these 2 pairs of complementary functions are ( f2 , f13 ) and ( f4 , f11 ). Only these might provide the necessary asymmetry for a candidate measure of causation obtainable from probabilistic logic which follows from the isomorphism between logic and measures on sets (actually true metrics m ). Alas, we shall see that the following property of the Boolean implication (x implies y) == (~y implies ~x) aka CONTRAPOSITIVE property is UNdesirable since a causation measure requires M(x:y) <> M(~y:~x) because eg: "fire x causes smoke y" makes sense, while "no smoke ~y causes no fire~x" is NONSENSE . Such examples and the fact that only the implications and their complements are the only asymmetric functions of both variables x, y finish ! my PROOF that we cannot use any purely Boolean function as an M(:). Also a single probability P(.) or P(.|.) should not be used even as a corroboration as is shown in great detail in { Popper , appendix IX , pp.387-398 }, and as discussed by Sir Karl Popper vs Leblanc in { The British Journal for the Philosophy of Science, vol.X, 1960, pp.313-318 }. Here come 2 visualisations of the same basic law for any elementary measure m which derives from (hence is isomorphic with) Boolean logic and set theory (which are isomorphic). Just imagine 2 rectangularly cut pizzas or pancakes Px and Py placed upon each other so that they (don't)(partially) overlap : == is equivalent , synonymous ; = is equal ; ~ non, negation, complement 0 <= m(.) is a measure of a set , eg a number n(.) of elements, N = m(All) eg P(.) is a measure of a set , P(.) = n(.)/N , is & is a joint ie an overlap ; U is "or" is a union (w/o an overlap ) Venn diagram enhanced : More expressive one fits 2x2 table : ___ Universe = m(All) ________ ___ 1verse = P(All) = 1 ______ | | | | | | __m(y)___________ | | P(x,y) | P(x,~y) P(x) | | ________|__m(x)__ | | == Pxy | = Px - Pxy | | | | | | | |-----------|------------------| | | m(y-x) | m(x,y) | m(x-y) | | | | | | |________|________| | | | P(~x,y) | P(~x,~y) P(~x) = 1-Px | |_________________| | | = Py -Pxy | = P(~(x or y) | | | | | | | m(~x,~y) = m(~(x U y)) | | | | |______________________________| |___ P(y) __|__ P(~y) = 1-Py __| ---yyyyyyyy,,,,,,,,,,xxxxxxxx--- in 1-dimension where ,,, are joint m(x,y) From 3 axioms m(empty set) = 0 ; m(x) >= 0 for any set x ; m(x or y) = m(x) + m(y) + 0 if m(x,y)=0 ie if x, y are disjoint sets ; all what comes next follows : m(x-y) == m(x,~y) = m(x) - m(x,y) P(x,~y) = Px - Pxy == P(x-y) m(x-y) +m(y-x) +m(x,y) = m(x or y) P(x,~y)+P(~x,y)+Pxy = P(x or y) m(x-y) +m(y-x) +m(x,y) +m(~x,~y) = m(All) P(x,~y)+P(~x,y)+Pxy+P(~x,~y) = 1 m(x or y) +m(~x,~y) = m(All) P(x or y) + P(~x,~y) = 1 because m(x or y) m(~x,~y) are disjoint since they can never overlap ; m(x or y) = m(x) + m(y) +0 iff m(x,y) = 0 ie if no overlap ; basic Bayes : m(y|x).m(x) = m(x,y) = m(y).m(x|y) ; P(y|x).P(x) = P(x,y) = P(y).P(x|y) basic bounds ( Bonferroni inequality is on lhs ) : Max[ 0, m(x) + m(y) - m(All) ] <= m(x,y) <= min[ m(x), m(y) ] Max[ 0, Px + Py - 1 ] <= Pxy <= min[ Px , Py ] Max[ m(x), m(y) ] <= m(x U y) = m(x)+m(y)-m(x,y) <= min[m(x)+m(y), m(All) ] Max[ Px, Py ] <= P(x or y) = Px + Py - Pxy <= min[ Px + Py , 1 ] so if Px + Py > 1 we are not completely free to choose any Pxy <= min[Px, Py] in tests or examples; for more find Bonferroni here and in { Hajek www }. Disjoint x, y have Pxy = 0 hence x, y are dependent since the condition for independence ie Px.Py = 0 cannot hold if Px > 0 and Py > 0. m(~x & ~y) = m(~(x or y)) , P(~x , ~y) = P(~(x or y)) by DeMorgan's law m(~x or ~y) = m(~(x & y)) , P(~x or ~y) = P(~(x,y)) = 1-Pxy via De Morgan d(x,y) = m(x or y) - m(x,y) = m(x) + m(y) - 2.m(x,y) = m(x-y) + m(y-x) = symmetrical distance between x, y = sum of asymmetrical distances = is a metric distance since as it holds : d(x,y) = d(y,x) >= d(x,x) = 0 = d(y,y) d(x,y) + d(y,z) >= d(x,z) is the triangle inequality For a better interpretability of numerical values, we often want a measure m(.) NORMALIZED to the scale [0..1] or [0..100%]. Eg: Q: How should we normalize the overlap Pxy ? A: m(x,y)/[ m(x) + m(y) ] has in fact the range of only [0..1/2] , so 2.m(x,y)/[ m(x) + m(y) ] might work (in analogy to harmonic average ). [0..0.5] is due to MAXimal possible overlap iff m(y) = m(x,y) = m(x). Hence a SHARPER normed overlap is m(x,y)/[m(x) + m(y) - m(x,y)] , which in fact is the normed equivalence m(x==y), in some applications interpretable as SIMILARITY or PROXIMITY. These meanings become clear when we derive normed DISSIMILARITY as non-equivalence ie XOR ie symmetrical distance by taking the complement of m(x==y) : m(x <> y) ie m(x =/= y) = ~[ m(x==y) ] = m(All) - m(x==y) . For probabilities P(x <> y) ie P(x =/= y) = ~[ P(x==y) ] = P(All) - P(x==y) = 1 - P(x==y) = = 1 - Pxy/( Px + Py - Pxy ) = non-proximity = distance ! = ( Px + Py - 2.Pxy )/( Px + Py - Pxy ) = [ P(x or y) - Pxy ]/P(x or y) ! = [(Px-Pxy)+(Py-Pxy)]/( Px + Py - Pxy ) = [ P(x,~y) + P(y,~x) ]/P(x or y) is the normed probabilistic DISTANCE , isomorphic with the d(x,y)/m(x or y). It can be visually appreciated in the pizzagram aka Venn diagram above. It has useful applications as a measure of DISSIMILARITY. For more on probabilistic logic see my other epapers at : !!! www.humintel.com/hajek -.- +References : John Grant : A non-truth-functional 3-valued logic ; Mathematics Magazine Sept-Oct 1974, pp.221-223. He points out that "in Kleene's system there are no tautologies" and then shows how to obtain them. Jan Hajek : A Knowledge Engineering Logic for Smarter, Safer and Faster (Expert) Systems, February 1988, 54 pages x 53 lines/page in A4. !!! www.humintel.com/hajek has epapers on probabilistic causation aka probabilistic causality and on probabilistic logic Eric Hehner : From Boolean algebra to unified algebra ; The Mathematical Intelligencer, 26, 2004/2, 3-11, see p8 Stephen Kleene : Introduction to Metamathematics, 1952; see pp. 334-335 T. Kool : doesNt want to be referenced Popper Karl: The Logic of Scientific Discovery, 6th impression (revised), March 1972; with new appendices, on corroboration is Appendix IX to his old Logik der Forschung, 1935, where in his Index: Gehalt, Mass des Gehalts = Measure of content (find SIC ). His oldfashioned p(x,y) is modern P(x|y), and his confirmation = corroboration C(x,y) is my C(y:x) Ronald R. Yager : A new methodology for ordinal multiobjective decisions based on fuzzy logic; Decision Sciences 12 (1981), 589-600 -.-