- Email: [email protected]
LIMITATIONS OF CLASSICAL LOGIC
209
and L makes a logical matrix in a sense "conscious" of itself. The scalar calculations for the truth-values are carried over directly to the operator calculation and vice versa. The matrix polynomial can be rearranged to give L n = . klL n'l. k2Ln'2. ...- kn.lL - knl which expresses any power L n as a linear combination of the powers L n'l L n'2,..., L , I. For the nonsingular L, we can premultiply the matrix polynomial by the inverse L "l which, divided by k n ;e 0, gives the template
9
for computing the inverse as a linear combination of powers of L up to L n-l" L -1 = . (Ln'l+ kiLn'2+ ... + kn.ll) / k n A strong cognitive element, a built-in 'self-knowing', distinguishes the knowledge operators which transform every normed logical state into an eigenstate. The logical momentum M forms an eigenstate directly by acting on an arbitrary state: M Iq> = IS_> M IS.> = ~ IS.>, ~ = O. These and other intelligent properties of operator description can be used to overcome many of the limitations of classical logic. But first what are these limitations?
LIMITATIONS OF CLASSICAL LOGIC To appreciate fully the need for the operator reformulation of logic inspired by quantum thinking, we must examine the limitations of classical logical theory commonly in use. Logic is a window on the mind. Just as the wavefunction enables us to think quantum-mechanically, logic has been the only tool, at least until now, which enables us to think consciously. The
210
algebraic formalization of logic has opened up new opportunities for the conversion of human thought into precise mathematical formulas. The formalization of logic, however, is not achieved without a price. Classical logic is blind to the original and extraordinary. Basically it is a logic of fools and computers, reminding us forcefully that intelligent thinking should not be taken for granted. When we hold a conversation, dial a phone or decide to take an umbrella, expecting rain, the underlying thought processes may seem trivial. Its exceptional complexities, however, quickly become apparent when one attempts to simulate even elementary mental activity on a computer. Finding an algorithm even for simple intelligent behaviour turns out to be a horrendous undertaking. To form an intelligent thought is to form right connections between abstract concepts which are either given or must first be developed. As soon as the necessary concepts are developed and adequate connections are formed, the deduction process follows naturally. However, finding the proper connections is not always easy. Often we are swept away by the multiplicity of language. A simple conjunction can take many different forms, such as 'and', 'also', 'but also', 'but', 'while', 'not only ... but also', 'although', 'as well as' and so on, which all may serve as valid operations of conjunction. Valid logical thoughts take the shape of so-called well-formed formulas (wff). In such formulas logical concepts, denoted by variables x, y, z, ... are joined together by the unambiguously defined logical connectives AND, OR, NOT, IMPLY, etc. If the connectives are correct and the priorities of logical operations are respected, the truth or falsity of the formula for a given set of logical variables can be determined algorithmically, according to well-defined rules. The classical logic we commonly employ conforms to the rules of Boolean algebra, and is often called simply Boolean logic. Certain formulas - tautologies and contradictions - are universal, being universally true or universally false under the arbitrary interpretation of the logical variables. Such are, for example, the fundamental syllogism law" ((X IMPLY y)AND(y IMPLY z)) IMPLY (x IMPLY z) - TRUE, and the Modus Ponens inference rule: ((X AND (x IMPLY y)) IMPLY y
-
TRUE.
In the framework of brain science the universal logical formulas must be treated as cognitive constants which hold no matter what the conditions of the world. They represent the fundamental coding units of the thought process, a fact sometimes poorly grasped even by professional logicians. Any logical inference rule we employ is necessarily a tautology, and the intelligence code is a tautology of tautologies, a global tautology built up from elementary tautologies of the lower levels. Tautologies are not just a universal code of communication, but only tautologies are covariant and only true results of
211
intelligence are tautologies [Ref 89]. Because much of the puzzle of consciousness can be traced to the fundamental covariant construct of tautology, the study of the laws governing tautologies is instrumental in the deciphering of the intelligence code. Although classical logic excels in proving deterministic tautologies, it is yet to learn how to formulate tautologies in randomized statistical universes with probabilistic correlations. Classical logic was devised to articulate and communicate logical truths. The use of symbolic notation and the development of Boolean logic algebra has brought logic closer to the realm of mathematics and the exact sciences. However, in spite of being able to capture important formal aspects of deductive reasoning, logic still lacks the decisive features that would allow it to function fully as an instrument of calculation. Different formalizations enjoy different degrees of success. Mathematical logic continues to be developed, to a great extent under the influence of problems arising in applications, and it is here where many limitations of classical logic become apparent. Propositional Boolean logic restricts itself to the study of true and false logic expressions, with true and false representing the so-called binary alphabet of truth-values. A logic designed over such an alphabet is called a two-valued or binary logic. The convenient simplicity of this alphabet of truthvalues, however, imposes severe limitations on the objects of logical inquiry, since it disregards those expressions which cannot be defined in terms of true and false, such as those which are undecided, undetermined, unknown, meaningless or superposed. Investigation of these and other intelligent properties of logical discourse is attempted in multivalued and modal logics, which deal with the truths that lie beyond the grasp of Boolean 'black-orwhite' logic. Multivalued or k-valued logical systems introduce, albeit arbitrarily, other truth-values beside true and false. This prompts the question: what is the fundamental value of k? The existing multivalued logics are not able to answer this question from first principles. In spite of engendering much excitement and interest the multivalued approach has so far offered no fundamental breakthrough or prediction. Mathematical logic is often understood as a quest for a certainty which will enable us to place often the vague intuitive structures of the thought process into the definite framework of Boolean algebra. Boolean logic thrives on precision and its whole purpose is to exclude multiplicity. However, precision, useful in some situations, can be detrimental in others, constraining the ability of classical logic to accommodate the complexities and ambiguities of thinking. Such features of the thought process as indeterminism and uncertainty, spontaneous but nonetheless correct insights and decisions, unpredictability, interference and intuition, etc. are not formalizable in classical logic. Even though classical logic is an important component of the thought process, Boolean logic does not encompass the entire spectrum of thinking operations, due to a number of fundamental limitations. Before we start to seek ways to overcome these limitations we must first clearly identify them.
212
TRUTH-TABLES The development of logical theory follows closely in the footsteps of mathematics and the use of symbolic notation has brought it closer to the realm of mathematics. However, classical logic is more mathematical in form than in substance. Mathematics cannot be properly dealt with without the techniques that afford certain and rigorous computation, and the lack of such an overall computing capability is precisely what separates the domain of logic from that of mathematics. Although the boundary between the two is somewhat vague and overlapping, it is nevertheless real and significant. A major leap forward in the mathematization of logic occurred in the 19th century with the development by August DeMorgan and George Boole of what is known at present as Boolean logic. In scientific folklore no discovery is ever attributed to the right person. Indeed DeMorgan, whose name is immortalized in the famous duality rules" xny= uy, xuy=~ny, preceded Boole, if not by much, in realizing the possibility of an algebraic formulation of logic. This formulation become possible upon the realization that logical statements, if represented by binary numbers, can be written as algebraic equations and treated purely computationally. A standard way of describing a logical operation is by defining its truthtable. For two Boolean variables x and y, one can write down 16 different truth-tables corresponding to 16 different logical operations. Although we can achieve a great deal with 16 logical operations, and one needs three, two or even one operation to design logical calculus, the thought process clearly involves much more. We know how to write a truth-table for connectives like 'neither x nor y' or for 'absolutely not x' or 'necessarily x', for 'unless', for 'of course', 'x but not y' and a number of others. But classical logic does not have a truth-table for 'create', for 'retrace', for 'abort', for 'come back' or 'cancel' and many others. It is poorly understood that the limitations and many drawbacks in the study of the actual thought process stem from making use of just 16 binary logical operations. This number is increased dramatically in the formalism of matrix logic which takes the full step into the maths. The logical connectives are given the mathematical status of matrix operators, replacing the Boolean operations with one universal procedure of matrix multiplication. In classical logic there is a strict division into unary and binary connectives. It seems quite natural to consider, in addition to the binary operations, the unary operations of logical operators. In doing so, one ought to recognize that in matrix logic, as in conventional logic, a division into unary and binary operations does exist, but a division into unary and binary connectives or operators ceases to have similar implications. Indeed, in contrast to ordinary connectives, the same logical operator is able to act either as a binary operator between two logical vectors, or as a unary operator upon a single logical vector. This implies, furthermore, that the set of 4 unary operations of Boolean logic is enlarged to a set of 16 unary operations, making the number
213
of binary and unary operations in matrix logic equal, in total, 32 standard operations.
DEALING WITH ABSURDS The key notions of classical logic are those of a logical variable and of a logical connective. In Boolean logic logical variables are defined in the binary set E2- {0,1} where the values 0 and 1 correspond to false and true respectively. As we mentioned above, there are unary connectives, like NOT, which act on a single atomic variable, and binary connectives like AND, OR, I M P L Y , IF, etc., which act on a pair of variables, joining them into a compound formula. Each logical connective is defined by a corresponding truth-table, which tells us exactly the result of the logical operation, depending on the value of the input variables. With the notable exception of the implications IF and I M P L Y , Boolean connectives have a clear intuitive meaning. Although implication is in common use and we master this logical operation easily, the controversy surrounding implication remains unresolved. Because the implication syllogism lies at the heart of the matrix principle formulated earlier, we must look more closely at the truth-table for implication, which is a common source of confusion:
IMPLY
y=0
x=0
TRUE
x=l
FALSE
i
y---I
HI
i
ii
TRUE
TRUE
On the one hand, implication completely rules out the possibility of a true antecedent x = 1 implying a false consequent y -- 0, which fully agrees with our intuition. But on the other hand, implication is valid for the false x = 0 implying the true y = 1. This can lead to valid but absurd and meaningless situations, and one has to be reminded that a logical implication represents only a relationship between the truth status of x and y and not a relationship between their meanings. Among the various limitations of classical logic this is probably the best known. Boolean implication enables the truth of y to be inferred from the truth of x, and nothing in particular to be inferred from the falsity of x. The difficulty arises when a false antecedent implies a true consequent and one cannot interpret logical implication in terms of cause and effect in a physical sense.
214
The failure to provide a clear and satisfactory explanation of the bizarre but mathematically correct and combinatorially necessary properties of implication is a major weakness of classical logic. One understands implication as a conditional relation between two statements x and y, x implies y, when the relation is true. However, implication does not necessarily represent the logical force of conditional statements. If x is false or y is true, the implication is true. Therefore classical logic will accept as valid implications like "If liars have integrity, cats can walk on water". Two lies can form a true implication! These paradoxes of implication have led to the search for definitions of strict implication: (x .-.> y) where the modal connective necessarily is introduced. In various texts we are told that solves the paradox. It doesn't. It simply axiomatically imposes a constraint to suit the logician's need to live in a perfectly meaningful world. The world, however, is not perfect at all. Since the truth-value of ( x ~ y ) does not depend wholly on the truth-values of x and y, we have here a modal and not a truth-functional link. But it is the truth-functional complexity of implication which is the source of the problem. The resolution of the paradox of logical implication is not simply beyond the grasp of classical logic, it lies outside logic, or at least what we have understood by logic until now.
NEGATION An important fact commonly overlooked or perhaps not realized by many logicians is that Boolean logic is a theory of the fixed point: X2 ----X.
Put another way, the essence of classical logic, as well as its limitations, stems from the assumption that logical values are idempotent. Clearly, this idempotency axiom immediately restricts the possible values of logical variables to just two values 0 and 1, corresponding to false and true respectively. Classical logic is two-valued. The existence of the fixed point entails that logical negation is computationally a binary complement, N O T x =:~ 1-x. Then the classical Law of Contradiction is deduced naturally: X-X2-X(I-
x)=O
215
According to the Law of Contradiction no statement can be true and false simultaneously" x g - 0 , which, employing the DeMorgan duality rule, turns into the classical Law of Excluded Middle, the tautology O R x -1. Making use of the fixed point, one builds up Boolean logic from first principles, which is advantageous, but this first principle has fundamental limitations. Whereas the thought process frequently involves subtle, partial, local or incomplete forms of negation, Boolean negation is a total and nondivisible logical operation, unintelligently and globally swapping true and false. Classical negation has no parts, but there are many. This question is intimately linked to the question of logical reversibility. The negation of negation is identity: X =X,
which tells us that Boolean NOT is a self-inverse, in fact the nonsingular operation in Boolean logic. Classical logic improperly, that only the negation operation can be reversed logical operations do not have inverses and are not inve~ible. fundamental limitation and a major blunder of classical theory.
only nontgivial insists, quite while all other This is another
AN UNNOTICED OPERATION IN BOOLEAN LOGIC The development of mathematical logic became possible upon the realization that if the propositions of logic could be represented by precise symbols, then the relation between them could be read as precisely as algebraic equations. From the fixed point axiom, we determine that N O T can be formalized as a subtraction from 1. Further, the logical AND and OR are identified as binary algebraic operations, product and addition respectively" (XANDy)
(xORy)
=:~ x . y
x+y
To make this representation fully legitimate another serious limitation has to be imposed. The sum of two truths must be a truth. Logical addition follows the unnatural involution rule" 1+1-1, responsible for the awkward fact that a given logical expression may correspond to different algebraic expressions.
216
Given a set of functions (operations), we are typically interested in determining the minimal irreducible basis set. The three logical operations NOT, AND and OR form a functionally complete basis set. The textbooks on logic usually tell us that these operations are the only operations which can be represented as algebraic operations over binary numbers: subtraction, product and addition respectively. Allegedly there are not enough algebraic operations in Boolean algebra to allow every logical connective to be associated with a particular algebraic operation. In fact, there is a fourth, and a very important logical operation, implication, which can also be expressed as an algebraic operation over binary numbers. For that we explore the powers yX, where x, y ~ E2. Then, setting 0~ 1, we can represent the implication as follows:
(x implies y) - yx Taking the power x in yX to be the antecedent and the base y to be the consequent, one arrives at the following truth table:
0 0
0
0
0-1
I-1
Oi=O
11=1
Since any value raised to the power of null is unity:
(anything)~
1,
if the consequent is true, the implication is true" 1~ and I t=l, and is not dependent, as required, on the value of the antecedent. As unravelled by the matrix principle, implication is central to thinking operations. What Boole and his followers either did not pay attention to or did not know is that implication can be modelled by a binary exponent, fact that went unnoticed for a long time. However, important as it is, the algebraic formalisation of NOT, AND and OR, and now also IF, is restricted to only these three (four) connectives, which is another fundamental limitation of classical logic. Whenever one wishes to bring the 13 (12) other Boolean operations into algebraic form, one can do this only indirectly by expressing them in the basis set of connectives NOT, AND and OR, (IF). Consider for example implication expressed in the minor basis sets" x IMPLYy =(NOTx) ORy=(l-x)+y=l-x+y or
x I M P L Y y = NOT(x 9 (NOT y)) = 1-(x (1- y)) = 1 - x + xy.
217
The logical connectives in Boolean logic are operations, and there exist only a limited number of them in ordinary algebra, a natural limitation classical logic has no means to overcome. Abstract thought requires many, not just a few operations, just as one cannot create a life form with just a few genes but need thousands of them. Not surprisingly, classical logic fails dramatically to explain induction and creative high-level intelligence. Abstract thought manipulates operations, not just variables as classical logic does. But there is nothing in classical theory which could enable one to gain direct control over the operations. The strict division into operators and operands exist in classical logic, in contrast to consciousness, where the distinction between them is fuzzy or dissolves altogether. When one moves up the ladder of abstraction, from the basic level of data to the top level of consciousness, one realizes that classical logical theory is confined to just the few first levels of data and information processing, modestly probing the next level of the knowledge processing. Although logic forms a core of high-level intelligence, our knowledge of high-level algorithms is poor. The only algorithms we fully understand and manage confidently are the algorithms of computation, and these alone in their present form are insufficient to handle high-level cognitive functions such as abstraction, induction, intuition and creativity.
CONSCIOUSNESS ,
,
,,,
INTELLIGENCE KNOWLEDGE INFORMATION DATA . _
Fig. 21 The complexity levels of consciousness
218
As far as intelligence and ultimately consciousness are concerned, logical theory has a long way to go before it can be meaningfully and effectively applied there. To come to grips with the problem of consciousness one needs a deeper exploration of logic.
LOGIC INTERPOLATES, MIND EXTRAPOLATES Mathematical and physical theories are commonly idealizations and approximations, which often disregard actual physical features. A ball is not a perfect sphere, there is no pointlike mass and no absolutely smooth surface without friction. In the realm of idealizations logic is no exception, a fact which has both its advantages and its drawbacks. But besides the flaws that are due to idealization, such as two-valuedness of classical logic, there are defects in logical theory that are fundamental in nature. This fact becomes clear when one attempts to employ classical logic to describe the intelligent processes in the brain. One soon discovers that while logic interpolates, mind extrapolates"
e o
o Q
Q Q
emuooaeammle0in~ connectives
INTERPOLATION
Q o
EXTRAPOLATION
Our fascination with the elegance and effectiveness of logic in the realm of deduction is combined with our frustration at its impotence in the realm of induction.
219
What are logical interpolation and extrapolation? When, given two logical statements, we connect them by means of connectives, this in some abstract topological space amounts to interpolation between two logical points. Technically speaking we establish validity of an expression by computing some formula which must be well-formed. Logic doesn't extrapolate. Although the usage of the modal connectives possibly 0 and necessarily can be useful, they are not genuine computational connectives, and provide little support, if any, to the attempt to formulate and compute the unknown. In cognitive logic, circumstances are fundamentally different. A thought very often must run forward, into unknown and uncharted territory. The final destination point is not given but has to be found or created. The fundamental extrapolation mechanism enters the thought process, indicating that all attempts to solve the question of artificial intelligence in the framework of a logic which only interpolates are doomed to failure.
TIMELES S LOGIC Another limitation of classical logic is its inability to account for dynamical aspect of thinking. Logic is static. It does not tell us about time. Among the various limitations of classical logic the most serious one concems its timeless character. The dynamical aspect of intelligent thinking is totally unaccounted for, even though some attempts were made in tense logic. The introduction of time parameters into logic brought no significant results, and is basically philosophical not computational. In this study the question of logical dynamics is of paramount importance, and will be treated by making use of the differential calculus, a tool unfamiliar in mathematical logic. The question of time in logic is closely connected to the question of commutation. A major defect of classical logic is that it often commutes in situations where it shouldn't. The truth-value of the expression J F K was assassinated AND buried and the expression JFK was buried AND assassinated, is according to the rule of Boolean logic the same, although the second expression is a meaningless statement. Likewise, the conjunction: 'a girl got pregnant AND gave birth to a child' is truth-functionally equivalent to a conjunction 'a girl gave birth to a child AND got pregnant', which underscores the disregard for time and causality in classical logic. Boolean variables are Abelian and commute under conjunction. The inability of classical logic to distinguish between Abelian and non-Abelian systems not only offends common sense, it reveals the inadequacy of classical logic to deal with high-level intelligence. Noncommutation is a fundamental feature of logic which naturally leads us to the development of non-Abelian matrix logic in the framework of the matrix principle.
and L makes a logical matrix in a sense "conscious" of itself. The scalar calculations for the truth-values are carried over directly to the operator calculation and vice versa. The matrix polynomial can be rearranged to give L n = . klL n'l. k2Ln'2. ...- kn.lL - knl which expresses any power L n as a linear combination of the powers L n'l L n'2,..., L , I. For the nonsingular L, we can premultiply the matrix polynomial by the inverse L "l which, divided by k n ;e 0, gives the template
9
for computing the inverse as a linear combination of powers of L up to L n-l" L -1 = . (Ln'l+ kiLn'2+ ... + kn.ll) / k n A strong cognitive element, a built-in 'self-knowing', distinguishes the knowledge operators which transform every normed logical state into an eigenstate. The logical momentum M forms an eigenstate directly by acting on an arbitrary state: M Iq> = IS_> M IS.> = ~ IS.>, ~ = O. These and other intelligent properties of operator description can be used to overcome many of the limitations of classical logic. But first what are these limitations?
LIMITATIONS OF CLASSICAL LOGIC To appreciate fully the need for the operator reformulation of logic inspired by quantum thinking, we must examine the limitations of classical logical theory commonly in use. Logic is a window on the mind. Just as the wavefunction enables us to think quantum-mechanically, logic has been the only tool, at least until now, which enables us to think consciously. The
210
algebraic formalization of logic has opened up new opportunities for the conversion of human thought into precise mathematical formulas. The formalization of logic, however, is not achieved without a price. Classical logic is blind to the original and extraordinary. Basically it is a logic of fools and computers, reminding us forcefully that intelligent thinking should not be taken for granted. When we hold a conversation, dial a phone or decide to take an umbrella, expecting rain, the underlying thought processes may seem trivial. Its exceptional complexities, however, quickly become apparent when one attempts to simulate even elementary mental activity on a computer. Finding an algorithm even for simple intelligent behaviour turns out to be a horrendous undertaking. To form an intelligent thought is to form right connections between abstract concepts which are either given or must first be developed. As soon as the necessary concepts are developed and adequate connections are formed, the deduction process follows naturally. However, finding the proper connections is not always easy. Often we are swept away by the multiplicity of language. A simple conjunction can take many different forms, such as 'and', 'also', 'but also', 'but', 'while', 'not only ... but also', 'although', 'as well as' and so on, which all may serve as valid operations of conjunction. Valid logical thoughts take the shape of so-called well-formed formulas (wff). In such formulas logical concepts, denoted by variables x, y, z, ... are joined together by the unambiguously defined logical connectives AND, OR, NOT, IMPLY, etc. If the connectives are correct and the priorities of logical operations are respected, the truth or falsity of the formula for a given set of logical variables can be determined algorithmically, according to well-defined rules. The classical logic we commonly employ conforms to the rules of Boolean algebra, and is often called simply Boolean logic. Certain formulas - tautologies and contradictions - are universal, being universally true or universally false under the arbitrary interpretation of the logical variables. Such are, for example, the fundamental syllogism law" ((X IMPLY y)AND(y IMPLY z)) IMPLY (x IMPLY z) - TRUE, and the Modus Ponens inference rule: ((X AND (x IMPLY y)) IMPLY y
-
TRUE.
In the framework of brain science the universal logical formulas must be treated as cognitive constants which hold no matter what the conditions of the world. They represent the fundamental coding units of the thought process, a fact sometimes poorly grasped even by professional logicians. Any logical inference rule we employ is necessarily a tautology, and the intelligence code is a tautology of tautologies, a global tautology built up from elementary tautologies of the lower levels. Tautologies are not just a universal code of communication, but only tautologies are covariant and only true results of
211
intelligence are tautologies [Ref 89]. Because much of the puzzle of consciousness can be traced to the fundamental covariant construct of tautology, the study of the laws governing tautologies is instrumental in the deciphering of the intelligence code. Although classical logic excels in proving deterministic tautologies, it is yet to learn how to formulate tautologies in randomized statistical universes with probabilistic correlations. Classical logic was devised to articulate and communicate logical truths. The use of symbolic notation and the development of Boolean logic algebra has brought logic closer to the realm of mathematics and the exact sciences. However, in spite of being able to capture important formal aspects of deductive reasoning, logic still lacks the decisive features that would allow it to function fully as an instrument of calculation. Different formalizations enjoy different degrees of success. Mathematical logic continues to be developed, to a great extent under the influence of problems arising in applications, and it is here where many limitations of classical logic become apparent. Propositional Boolean logic restricts itself to the study of true and false logic expressions, with true and false representing the so-called binary alphabet of truth-values. A logic designed over such an alphabet is called a two-valued or binary logic. The convenient simplicity of this alphabet of truthvalues, however, imposes severe limitations on the objects of logical inquiry, since it disregards those expressions which cannot be defined in terms of true and false, such as those which are undecided, undetermined, unknown, meaningless or superposed. Investigation of these and other intelligent properties of logical discourse is attempted in multivalued and modal logics, which deal with the truths that lie beyond the grasp of Boolean 'black-orwhite' logic. Multivalued or k-valued logical systems introduce, albeit arbitrarily, other truth-values beside true and false. This prompts the question: what is the fundamental value of k? The existing multivalued logics are not able to answer this question from first principles. In spite of engendering much excitement and interest the multivalued approach has so far offered no fundamental breakthrough or prediction. Mathematical logic is often understood as a quest for a certainty which will enable us to place often the vague intuitive structures of the thought process into the definite framework of Boolean algebra. Boolean logic thrives on precision and its whole purpose is to exclude multiplicity. However, precision, useful in some situations, can be detrimental in others, constraining the ability of classical logic to accommodate the complexities and ambiguities of thinking. Such features of the thought process as indeterminism and uncertainty, spontaneous but nonetheless correct insights and decisions, unpredictability, interference and intuition, etc. are not formalizable in classical logic. Even though classical logic is an important component of the thought process, Boolean logic does not encompass the entire spectrum of thinking operations, due to a number of fundamental limitations. Before we start to seek ways to overcome these limitations we must first clearly identify them.
212
TRUTH-TABLES The development of logical theory follows closely in the footsteps of mathematics and the use of symbolic notation has brought it closer to the realm of mathematics. However, classical logic is more mathematical in form than in substance. Mathematics cannot be properly dealt with without the techniques that afford certain and rigorous computation, and the lack of such an overall computing capability is precisely what separates the domain of logic from that of mathematics. Although the boundary between the two is somewhat vague and overlapping, it is nevertheless real and significant. A major leap forward in the mathematization of logic occurred in the 19th century with the development by August DeMorgan and George Boole of what is known at present as Boolean logic. In scientific folklore no discovery is ever attributed to the right person. Indeed DeMorgan, whose name is immortalized in the famous duality rules" xny= uy, xuy=~ny, preceded Boole, if not by much, in realizing the possibility of an algebraic formulation of logic. This formulation become possible upon the realization that logical statements, if represented by binary numbers, can be written as algebraic equations and treated purely computationally. A standard way of describing a logical operation is by defining its truthtable. For two Boolean variables x and y, one can write down 16 different truth-tables corresponding to 16 different logical operations. Although we can achieve a great deal with 16 logical operations, and one needs three, two or even one operation to design logical calculus, the thought process clearly involves much more. We know how to write a truth-table for connectives like 'neither x nor y' or for 'absolutely not x' or 'necessarily x', for 'unless', for 'of course', 'x but not y' and a number of others. But classical logic does not have a truth-table for 'create', for 'retrace', for 'abort', for 'come back' or 'cancel' and many others. It is poorly understood that the limitations and many drawbacks in the study of the actual thought process stem from making use of just 16 binary logical operations. This number is increased dramatically in the formalism of matrix logic which takes the full step into the maths. The logical connectives are given the mathematical status of matrix operators, replacing the Boolean operations with one universal procedure of matrix multiplication. In classical logic there is a strict division into unary and binary connectives. It seems quite natural to consider, in addition to the binary operations, the unary operations of logical operators. In doing so, one ought to recognize that in matrix logic, as in conventional logic, a division into unary and binary operations does exist, but a division into unary and binary connectives or operators ceases to have similar implications. Indeed, in contrast to ordinary connectives, the same logical operator is able to act either as a binary operator between two logical vectors, or as a unary operator upon a single logical vector. This implies, furthermore, that the set of 4 unary operations of Boolean logic is enlarged to a set of 16 unary operations, making the number
213
of binary and unary operations in matrix logic equal, in total, 32 standard operations.
DEALING WITH ABSURDS The key notions of classical logic are those of a logical variable and of a logical connective. In Boolean logic logical variables are defined in the binary set E2- {0,1} where the values 0 and 1 correspond to false and true respectively. As we mentioned above, there are unary connectives, like NOT, which act on a single atomic variable, and binary connectives like AND, OR, I M P L Y , IF, etc., which act on a pair of variables, joining them into a compound formula. Each logical connective is defined by a corresponding truth-table, which tells us exactly the result of the logical operation, depending on the value of the input variables. With the notable exception of the implications IF and I M P L Y , Boolean connectives have a clear intuitive meaning. Although implication is in common use and we master this logical operation easily, the controversy surrounding implication remains unresolved. Because the implication syllogism lies at the heart of the matrix principle formulated earlier, we must look more closely at the truth-table for implication, which is a common source of confusion:
IMPLY
y=0
x=0
TRUE
x=l
FALSE
i
y---I
HI
i
ii
TRUE
TRUE
On the one hand, implication completely rules out the possibility of a true antecedent x = 1 implying a false consequent y -- 0, which fully agrees with our intuition. But on the other hand, implication is valid for the false x = 0 implying the true y = 1. This can lead to valid but absurd and meaningless situations, and one has to be reminded that a logical implication represents only a relationship between the truth status of x and y and not a relationship between their meanings. Among the various limitations of classical logic this is probably the best known. Boolean implication enables the truth of y to be inferred from the truth of x, and nothing in particular to be inferred from the falsity of x. The difficulty arises when a false antecedent implies a true consequent and one cannot interpret logical implication in terms of cause and effect in a physical sense.
214
The failure to provide a clear and satisfactory explanation of the bizarre but mathematically correct and combinatorially necessary properties of implication is a major weakness of classical logic. One understands implication as a conditional relation between two statements x and y, x implies y, when the relation is true. However, implication does not necessarily represent the logical force of conditional statements. If x is false or y is true, the implication is true. Therefore classical logic will accept as valid implications like "If liars have integrity, cats can walk on water". Two lies can form a true implication! These paradoxes of implication have led to the search for definitions of strict implication: (x .-.> y) where the modal connective necessarily is introduced. In various texts we are told that solves the paradox. It doesn't. It simply axiomatically imposes a constraint to suit the logician's need to live in a perfectly meaningful world. The world, however, is not perfect at all. Since the truth-value of ( x ~ y ) does not depend wholly on the truth-values of x and y, we have here a modal and not a truth-functional link. But it is the truth-functional complexity of implication which is the source of the problem. The resolution of the paradox of logical implication is not simply beyond the grasp of classical logic, it lies outside logic, or at least what we have understood by logic until now.
NEGATION An important fact commonly overlooked or perhaps not realized by many logicians is that Boolean logic is a theory of the fixed point: X2 ----X.
Put another way, the essence of classical logic, as well as its limitations, stems from the assumption that logical values are idempotent. Clearly, this idempotency axiom immediately restricts the possible values of logical variables to just two values 0 and 1, corresponding to false and true respectively. Classical logic is two-valued. The existence of the fixed point entails that logical negation is computationally a binary complement, N O T x =:~ 1-x. Then the classical Law of Contradiction is deduced naturally: X-X2-X(I-
x)=O
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According to the Law of Contradiction no statement can be true and false simultaneously" x g - 0 , which, employing the DeMorgan duality rule, turns into the classical Law of Excluded Middle, the tautology O R x -1. Making use of the fixed point, one builds up Boolean logic from first principles, which is advantageous, but this first principle has fundamental limitations. Whereas the thought process frequently involves subtle, partial, local or incomplete forms of negation, Boolean negation is a total and nondivisible logical operation, unintelligently and globally swapping true and false. Classical negation has no parts, but there are many. This question is intimately linked to the question of logical reversibility. The negation of negation is identity: X =X,
which tells us that Boolean NOT is a self-inverse, in fact the nonsingular operation in Boolean logic. Classical logic improperly, that only the negation operation can be reversed logical operations do not have inverses and are not inve~ible. fundamental limitation and a major blunder of classical theory.
only nontgivial insists, quite while all other This is another
AN UNNOTICED OPERATION IN BOOLEAN LOGIC The development of mathematical logic became possible upon the realization that if the propositions of logic could be represented by precise symbols, then the relation between them could be read as precisely as algebraic equations. From the fixed point axiom, we determine that N O T can be formalized as a subtraction from 1. Further, the logical AND and OR are identified as binary algebraic operations, product and addition respectively" (XANDy)
(xORy)
=:~ x . y
x+y
To make this representation fully legitimate another serious limitation has to be imposed. The sum of two truths must be a truth. Logical addition follows the unnatural involution rule" 1+1-1, responsible for the awkward fact that a given logical expression may correspond to different algebraic expressions.
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Given a set of functions (operations), we are typically interested in determining the minimal irreducible basis set. The three logical operations NOT, AND and OR form a functionally complete basis set. The textbooks on logic usually tell us that these operations are the only operations which can be represented as algebraic operations over binary numbers: subtraction, product and addition respectively. Allegedly there are not enough algebraic operations in Boolean algebra to allow every logical connective to be associated with a particular algebraic operation. In fact, there is a fourth, and a very important logical operation, implication, which can also be expressed as an algebraic operation over binary numbers. For that we explore the powers yX, where x, y ~ E2. Then, setting 0~ 1, we can represent the implication as follows:
(x implies y) - yx Taking the power x in yX to be the antecedent and the base y to be the consequent, one arrives at the following truth table:
0 0
0
0
0-1
I-1
Oi=O
11=1
Since any value raised to the power of null is unity:
(anything)~
1,
if the consequent is true, the implication is true" 1~ and I t=l, and is not dependent, as required, on the value of the antecedent. As unravelled by the matrix principle, implication is central to thinking operations. What Boole and his followers either did not pay attention to or did not know is that implication can be modelled by a binary exponent, fact that went unnoticed for a long time. However, important as it is, the algebraic formalisation of NOT, AND and OR, and now also IF, is restricted to only these three (four) connectives, which is another fundamental limitation of classical logic. Whenever one wishes to bring the 13 (12) other Boolean operations into algebraic form, one can do this only indirectly by expressing them in the basis set of connectives NOT, AND and OR, (IF). Consider for example implication expressed in the minor basis sets" x IMPLYy =(NOTx) ORy=(l-x)+y=l-x+y or
x I M P L Y y = NOT(x 9 (NOT y)) = 1-(x (1- y)) = 1 - x + xy.
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The logical connectives in Boolean logic are operations, and there exist only a limited number of them in ordinary algebra, a natural limitation classical logic has no means to overcome. Abstract thought requires many, not just a few operations, just as one cannot create a life form with just a few genes but need thousands of them. Not surprisingly, classical logic fails dramatically to explain induction and creative high-level intelligence. Abstract thought manipulates operations, not just variables as classical logic does. But there is nothing in classical theory which could enable one to gain direct control over the operations. The strict division into operators and operands exist in classical logic, in contrast to consciousness, where the distinction between them is fuzzy or dissolves altogether. When one moves up the ladder of abstraction, from the basic level of data to the top level of consciousness, one realizes that classical logical theory is confined to just the few first levels of data and information processing, modestly probing the next level of the knowledge processing. Although logic forms a core of high-level intelligence, our knowledge of high-level algorithms is poor. The only algorithms we fully understand and manage confidently are the algorithms of computation, and these alone in their present form are insufficient to handle high-level cognitive functions such as abstraction, induction, intuition and creativity.
CONSCIOUSNESS ,
,
,,,
INTELLIGENCE KNOWLEDGE INFORMATION DATA . _
Fig. 21 The complexity levels of consciousness
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As far as intelligence and ultimately consciousness are concerned, logical theory has a long way to go before it can be meaningfully and effectively applied there. To come to grips with the problem of consciousness one needs a deeper exploration of logic.
LOGIC INTERPOLATES, MIND EXTRAPOLATES Mathematical and physical theories are commonly idealizations and approximations, which often disregard actual physical features. A ball is not a perfect sphere, there is no pointlike mass and no absolutely smooth surface without friction. In the realm of idealizations logic is no exception, a fact which has both its advantages and its drawbacks. But besides the flaws that are due to idealization, such as two-valuedness of classical logic, there are defects in logical theory that are fundamental in nature. This fact becomes clear when one attempts to employ classical logic to describe the intelligent processes in the brain. One soon discovers that while logic interpolates, mind extrapolates"
e o
o Q
Q Q
emuooaeammle0in~ connectives
INTERPOLATION
Q o
EXTRAPOLATION
Our fascination with the elegance and effectiveness of logic in the realm of deduction is combined with our frustration at its impotence in the realm of induction.
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What are logical interpolation and extrapolation? When, given two logical statements, we connect them by means of connectives, this in some abstract topological space amounts to interpolation between two logical points. Technically speaking we establish validity of an expression by computing some formula which must be well-formed. Logic doesn't extrapolate. Although the usage of the modal connectives possibly 0 and necessarily can be useful, they are not genuine computational connectives, and provide little support, if any, to the attempt to formulate and compute the unknown. In cognitive logic, circumstances are fundamentally different. A thought very often must run forward, into unknown and uncharted territory. The final destination point is not given but has to be found or created. The fundamental extrapolation mechanism enters the thought process, indicating that all attempts to solve the question of artificial intelligence in the framework of a logic which only interpolates are doomed to failure.
TIMELES S LOGIC Another limitation of classical logic is its inability to account for dynamical aspect of thinking. Logic is static. It does not tell us about time. Among the various limitations of classical logic the most serious one concems its timeless character. The dynamical aspect of intelligent thinking is totally unaccounted for, even though some attempts were made in tense logic. The introduction of time parameters into logic brought no significant results, and is basically philosophical not computational. In this study the question of logical dynamics is of paramount importance, and will be treated by making use of the differential calculus, a tool unfamiliar in mathematical logic. The question of time in logic is closely connected to the question of commutation. A major defect of classical logic is that it often commutes in situations where it shouldn't. The truth-value of the expression J F K was assassinated AND buried and the expression JFK was buried AND assassinated, is according to the rule of Boolean logic the same, although the second expression is a meaningless statement. Likewise, the conjunction: 'a girl got pregnant AND gave birth to a child' is truth-functionally equivalent to a conjunction 'a girl gave birth to a child AND got pregnant', which underscores the disregard for time and causality in classical logic. Boolean variables are Abelian and commute under conjunction. The inability of classical logic to distinguish between Abelian and non-Abelian systems not only offends common sense, it reveals the inadequacy of classical logic to deal with high-level intelligence. Noncommutation is a fundamental feature of logic which naturally leads us to the development of non-Abelian matrix logic in the framework of the matrix principle.