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RELEVANT REASONING AS THE LOGICAL BASIS OF KNOWLEDGE ENGINEERING Jingde Cheng Department of Computer Science and Communication Engineering Kyushu University 6-10-1 Hakozaki, Fukuoka 812-8581, Japan Reprinted from Proceedings of the 4th World Congress on Expert Systems Mexico City, Mexico, March 16-20, 1998 RELEVANT REASONING AS THE LOGICAL BASIS OF KNOWLEDGE ENGINEERING Jingde Cheng Department of Computer Science and Communication Engineering Kyushu University 6-10-1 Hakozaki, Fukuoka 812-8581, Japan [email protected] conclusion, and concerns the strength of the relation between them. Therefore, the correctness of an argument depends on the connection between its premises and its conclusion, and neither on whether the premises are true or not, nor on the conclusion is true or not. Thus, we have a fundamental question: What is the criterion by which to decide whether the conclusion really does follow from the premises or not? ABSTRACT Almost all the current knowledge-based systems are directly or indirectly based on classical mathematical logic which gives no guarantee that the conclusion of a reasoning is necessarily relevant to its premises, even if the reasoning is valid in the sense of the classical mathematical logic. It is this problem that causes the ineffectiveness and inefficiency of reasoning and learning engines of large-scale knowledge-based systems. To make the current knowledge-based systems more powerful and flexible, we have to solve this problem from both theoretical and practical aspects. This paper proposes that relevant reasoning based on paradox-free relevant logics should be taken as the logical basis of knowledge engineering. The paper points out why a reasoning based on the classical mathematical logic may be irrelevant, shows that a reasoning based on the paradox-free relevant logics is relevant and that it is the notion of entailment that plays the fundamental role in relevant reasoning, and gives some examples to explain why one should take relevant reasoning as the logical basis of knowledge engineering. 1. A logically valid reasoning is a reasoning such that its arguments or inferences are justified based on some logical criterion in order to obtain correct conclusions. Therefore, a reasoning may be valid on a logical criterion but invalid on another. Automated reasoning is concerned with the execution of computer programs that assist in solving problems requiring reasoning. To perform an automated reasoning, knowledge used in the reasoning must be represented in some explicit and formal form. Knowledge Engineering is a discipline concerned with constructing and maintaining knowledge bases to store knowledge in various domains of the real world and using the knowledge by automated reasoning technique to solve problems in the domains that ordinarily require human logical reasoning. Therefore, two key issues in knowledge engineering are how to construct and maintain knowledge bases effectively and efficiently, and how to reason out new knowledge from known knowledge effectively and efficiently. INTRODUCTION Reasoning is the process of drawing new conclusions from some premises which are known facts and/or assumed hypotheses. In general, a reasoning consists a number of arguments or inferences. An argument (or inference) is a set of declarative sentences consisting of one or more premises, which contain the evidence, and a declarative sentence as conclusion. In an argument, a claim is being made that there is some sort of evidential relationship between its premises and its conclusion: the conclusion is supposed to follow from the premises, or, equivalently, the premises are supposed to entail the conclusion. The correctness of an argument is a matter of the connection between its premises and its Almost all the current knowledge-based systems are directly or indirectly based on classical mathematical logic (CML for short). However, a reasoning based on the CML may be irrelevant, i.e., the conclusion reasoned and/or deduced from the premises of that reasoning may be irrelevant at all, in the sense of meaning, to the premises. In the framework of the CML, there is no guarantee that the conclusion of a reasoning is necessarily relevant to its premises, even if the reasoning is valid in the sense of the CML. As a result, for a conclusion reasoned and/or deduced based on the CML, we have to investigate whether it is relevant 449 to its premises or not by ourselves. Obviously, the more we have information and/or knowledge, the more difficult we do this investigation task. It is this problem that causes the ineffectiveness and inefficiency of reasoning and learning engines of large-scale knowledge-based systems. To make current knowledgebased systems more powerful and flexible, we have to solve this problem from both theoretical and practical aspects. theorems, laws, and principles to connect a concept, fact, situation or conclusion and its sufficient conditions. Indeed, a major work of almost all scientists is to discover some conditional and/or causal relationships between various phenomena, data, and laws in their research fields. In logic, the notion abstracted from various conditionals is called “entailment.” In general, an entailment, for instance, “A entails B” or “if A then B,” must concern two parts which are connected by the connective “... entails ...” and called the antecedent and the consequent of that entailment, respectively. The truth-value and/or validity of an entailment depends not only on the truth-values of its antecedent and consequent but also more essentially on a necessarily relevant and conditional relation between its antecedent and its consequent. The notion of entailment plays the most essential role in human logical thinking because any reasoning must invoke it. Therefore, it is historically always the most important subject studied in logic and is regarded as the heart of logic [1]. This paper proposes that relevant reasoning based on paradox-free relevant logics should be taken as the logical basis of knowledge engineering. The paper points out why a reasoning based on the CML may be irrelevant, shows that a reasoning based on the paradoxfree relevant logics is relevant and that it is the notion of entailment that plays the fundamental role in relevant reasoning, and gives some examples to explain why one should take relevant reasoning as the logical basis of knowledge engineering. 2. LOGIC AND ENTAILMENT When we study and use logic, the notion of entailment may appear in both the object logic (i.e., the logic we are studying) and the meta-logic or observer’s logic (i.e., the logic we are using to study the object logic). First of all, we define the terminology used in this paper for discussing our subject clearly, unambiguously, and formally. What is logic? Logic deals with what entails what or what follows from what. Its aim is to determine which are the correct conclusions of a given set of premises, i.e., to determine which arguments are valid. Therefore, the most essential and central concept in logic is the logic consequence relation that relates a given set of premises to those conclusions which validly follow from the premises. As the systematic study of fundamental principles that underlie various valid reasoning in order to obtain correct conclusions from premises, logic must abstract the forms of the reasoning from their contents. From the viewpoint of logic, the validity of a reasoning depends on the connection between its premises and its conclusion, and neither on whether the premises are true or not, nor on the conclusion is true or not. In the object logic, there usually is a connective, such as the material implication in CML, the relevant implication in relevant logic R, the necessary and relevant implication in modal relevant logic E, and so on, to represent the notion of entailment. On the other hand, in the meta-logic, the notion of entailment is usually used to represent a valid logical consequence relation. In general, for an object logic L, any set Γ of formulas, and any formula C, we say that Γ semantically or model-theoretically entails C, or that C semantically or modeltheoretically follows from Γ, or that C is a semantic or model-theoretical consequence of Γ, written as Γ |=L C (the relation |=L is called the semantic or model-theoretical consequence relation of L), if and only if C is interpreted to be true for any model of Γ; we say that Γ syntactically o r proof-theoretically entails C, or C syntactically or proof-theoretically f o l l o w s from Γ, or C is a syntactic or proof-theoretical consequence of Γ, written as Γ |−L C (the relation |− L is called the syntactic or proof-theoretical consequence relation of L), if and only if C is deducible from Γ. It is probably difficult, if not impossible, to find a sentence form in various natural and social scientific publications which is more generally used to describe various definitions, propositions, theorems, laws, and principles than the sentence form of “if ... then ... .” A sentence of the form “if ... then ...” is usually called a conditional which states that there exists a conditional and/or causal relationship between the “if” part and the “then” part of the sentence. Natural and social scientists always use conditionals in their descriptions of various definitions, propositions, 450 and any element of TLe(P) is called an empirical theorem of the formal theory. For a logic, an entailment formula (i.e., a formula in the form of A⇒B where “⇒“ is the connective used to represent the notion of entailment in the logic) is called an empirical entailment of the logic if its truth-value, in the sense of that logic, depends on the contents of its antecedent and consequent (i.e., from the viewpoint of the logic, the relevant relation between the antecedent and the consequent of that entailment is regarded to be empirical); an entailment formula is called a logical entailment of the logic if its truthvalue, in the sense of that logic, depends only on its abstract form but not on the contents of its antecedent and consequent, and therefore, it is considered to be universally true or false (i.e., from the viewpoint of the logic, the relevant relation between the antecedent and the consequent of that entailment is regarded to be logical). Indeed, the most intrinsic difference between some different logic systems is to regard what class of entailments as logical entailments. A formal theory TL(P) is said to be directly inconsistent if and only if there exists a formula A of L such that both A∈P and ¬A∈P hold. A formal theory T L (P) is said to be indirectly inconsistent if and only if there exists a formula A of L such that both A∈TL(P) and ¬A∈TL(P) but not both A∈P and ¬A∈P. A formal theory TL(P) is said to be consistent if and only if it is neither directly inconsistent nor indirectly inconsistent. In general, a formal theory constructed as a purely deductive science (e.g., CML and its various extensions) is consistent. However, almost all, if not all, formal theories constructed based on empirical and/or experimental sciences are generally indirectly inconsistent. A formal theory TL(P) is said to be meaningless or e x p l o s i v e if and only if A∈TL(P) for arbitrary formula A∈F(L). A meaningless or explosive formal theory is not useful at all. Obviously, if a formal logic system L is explosive, then any directly or indirectly inconsistent formal theory TL(P) must also be explosive. A formal logic system L is a triplet (F(L), |− L , Th(L)) where F(L) is the set of all well formed formulas of L, |− L is the logical consequence relation of L such that for P⊆F(L) and C∈F(L), P |− L C means that within the framework of L taking P as premises we can obtain C as a valid conclusion, and Th(L) is the set of logical theorems of L such that φ |− L t holds for any t∈Th(L). According to the representation of the logical consequence relation of a logic, the logic can be represented as a Hilbert style formal system, or a Gentzen natural deduction system, or a Gentzen sequent calculus system. A semantics of a formal logic system is an interpretation of the formulas of the logic into some mathematical structure, together with an interpretation of the logical consequence relation of the logic in terms of the interpretation. 3. RELEVANT REASONING AND ENTAILMENT Generally, in any valid argument of our ordinary logical thinking, the premises must be relevant to the conclusion. Informally, we say a reasoning is relevant if and only if in every argument or inference of that reasoning the premises is relevant to the conclusion; a reasoning is irrelevant if and only if it is not relevant. In this section, we will explain that it is the notion of entailment that plays the fundamental role in relevant reasoning. A formal logic system L is said to be e x p l o s i v e if and only if {A, ¬A} |− L B for any two different formulas A, B∈F(L); L is said to be paraconsistent if and only if it is not explosive [8]. 3.1 Entailment and irrelevant reasoning i n CML and its extensions Let (F(L), |−L, Th(L)) be a formal logic system and P⊆F(L) be a non-empty set of sentences (i.e., closed well formed formulas). A formal theory with premises P based on L, called a L theory with premises P and denoted by TL(P), is defined as follows: The CML is established based on a number of basic assumptions. Some of the assumptions concerning with our subject are as follows: The classical abstraction : The only properties of a proposition that matter to logic are its form and its truth-value. TL(P) =df Th(L) ∪ TLe(P) e TL (P) =df {A | P |−L A and A∉Th(L)} where Th(L) and TLe(P) are called the logical part and the empirical part of the formal theory, respectively, The Fregean assumption : The truth-value of a proposition is determined by its form and the truthvalues of its constituents. 451 The classical validity assumption : An argument is valid if and only if it is impossible for all its premises to be true while its conclusion is false. With the above definition of material implication and the inference rule of Modus Ponens for material implication (from A and A→B to infer B), any valid reasoning based on CML must be truth-preserving, i.e., the conclusion of a valid reasoning must be true if all premises are true. However, as a result of defining the material implication as a truth-function of its antecedent and consequent but ignoring whether or not there is a necessarily relevant and/or conditional relation between its antecedent and consequent, a reasoning based on CML may be irrelevant, i.e., the conclusion of some argument or inference in that reasoning may be irrelevant at all, in the sense of meaning, to the premises. For example, from “snow is white” and “snow is white → 1+1=2” we can infer “1+1=2” by using Modus Ponens for material implication. However, conclusion “1+1=2” is obviously irrelevant at all to premise “snow is white” even if it is true. Moreover, because of implicational paradoxes, the conclusion in the form of implication formula of an argument or inference based on CML even may be not true in the sense of entailment. For example, from any given proposition A and logical axiom A→(B→A), which is the most well known implicational paradox, we can infer B→A by using Modus Ponens for material implication. However, since there may be no necessarily relevant and/or conditional relation between A and B, in general, we cannot say “if B then A” in the sense of entailment. Taking above assumptions into account, in CML, the notion of entailment is represented by the extensional notion of material implication (denoted by → in this paper) which is defined as A→B =df ¬(A∧¬B) or A→B =df ¬A∨B. However, the material implication is just a truth-function of its antecedent and consequent but not requires that there must exist a necessarily relevant and/or conditional relation between its antecedent and consequent, i.e., the truth-value of the formula A→B depends only on the truth-values of A and B, though there could exist no necessarily relevant and/or conditional relation between A and B. As a result, for example, sentences “snow is white → 1+1=2,” “snow is black → 1+1=2,” and “snow is black → 1+1=3” are all true in the logic. However, if we read “→“ as “if ... then ...,” then “if snow is white then 1+1=2,” “if snow is black then 1+1=2,” and “if snow is black then 1+1=3” are all false in human logical thinking because there is no necessarily relevant and/or conditional relation between the “if” part and the “then” part of each sentence. Obviously, in meaning the notion of entailment used in human logical thinking is intrinsically different from the notion of material implication in CML. Using the material implication as the entailment is problematical in pragmatics. The well known “implicational paradox problem” in CML is that if one regards the material implication as the entailment and every logical theorem of CML as a valid inference form in human logical thinking, then a great number of logical axioms or theorems of the logic, such as A→(B→A), B→(¬A∨A), ¬A→(A→B), (¬A∧A)→B, (A→B)∨(¬A→B), (A→B)∨(A→¬B), (A→B)∨(B→A), ((A∧B)→C)→((A→C)∨(B→C)), and so on, present some paradoxical properties and therefore they have been referred to in the literature as “implicational paradoxes” [1, 2, 7, 9]. For example, in terms of CML, formulas A→(B→A) and B→(¬A∨A) mean “a true proposition is implied by anything” formulas ¬A→(A→B) and (¬A∧A)→B mean “a false proposition implies anything” and formula (A→B)∨(B→A) means “for any two propositions A and B, A implies B or B implies A” respectively. However, it is obvious that we cannot say “if B then A” for a true proposition A and an arbitrary proposition B, “if A then B” for a false proposition A and an arbitrary proposition B, and “if A then B or if B then A” for any two irrelevant propositions A and B, respectively, in our ordinary logical thinking. On the other hand, the following proof-theoretical and model-theoretical deduction theorems hold in CML: Γ∪{A} |−CML B iff Γ |−CML A→B Γ∪{A1, ..., An} |−CML B iff Γ |−CML A1→(...→(An→B)...) Γ∪{A1, ..., An} |−CML B iff Γ |−CML (A1∧...∧An)→B Γ∪{A} |=CML B iff Γ |=CML A→B Γ∪{A1, ..., An} |=CML B iff Γ |=CML A1→(...→(An→B)...) Γ∪{A1, ..., An} |=CML B iff Γ |=CML (A1∧...∧An)→B What these mean is that the notion of entailment in meta-logic of CML is “equivalent” to the notion of material implication in CML. Therefore, in the framework of CML, even if a reasoning is valid, there is no guarantee that its premises are necessarily relevant to its conclusion. All formal logic systems (including various modal logic systems, intuitionistic logic, and those logic systems developed in recent years for nonmonotonic 452 reasoning) where the notion of entailment is directly or indirectly represented by the notion of material implication have the similar implicational paradox problem as that in CML. A reasoning based on these logics also may be irrelevant. 3.2 Entailment relevant logic and relevant reasoning of getting to know whether A or whether B” (Geach 1958); “A1& … &An→B should not only be itself a tautology, but should also be a substitution instance of some more general implication A1'& … &An'→B', where neither B' nor ¬(A1'& … &An') are themselves tautologies” (Smiley 1959) [1]. However, it is hard until now to know exactly how to formally interpret such epistemological phrases as “coming to know” and “getting to know” in the context of logic. During the 1950s~1970s, Anderson and Belnap extended the work of Ackermann and proposed variable-sharing as a necessary but not sufficient formal condition for the relevance between the antecedent and consequent of a logical entailment [1, 2, 9]. in Historically, implicational paradoxes have been studied many years. The main aim of Lewis's work beginning in 1912 on the establishment of modern modal logic was to find a satisfactory theory of implication which is better than CML in that it can avoid those implicational paradoxes, though his plan was not successful in the sense that some implicational paradoxes in terms of strict implication remained in modal logic [1, 2, 9]. Relevant logics were constructed during the 1950s~1970s in order to find a mathematically satisfactory way of grasping the notion of entailment [1, 2, 7, 9]. The first one of such logics is Ackermann's logic system Π'. Ackermann introduced a new primitive connective, called “rigorous implication,” which is more natural and stronger than the material implication, and constructed the calculus Π' of rigorous implication which provably avoids those implicational paradoxes. Anderson and Belnap modified and reconstructed Ackermann's system into an equivalent logic system, called “system E of entailment”. Belnap proposed an implicational relation, called “relevant implication,” which is stronger than the material implication but weaker than the rigorous implication, and constructed a calculus called “system R of relevant implication”. E has something like the modality structure of a classical modal logic S4, and therefore, E differs primarily from R in that E is a system of strict and relevant implication but R is a system of only relevant implication. Another important relevant logic system is “system T of ticket entailment” or “system T of entailment shorn of modality” which is motivated by Anderson and Belnap. There are some neighboring logic systems of T, E, and R. All of these logic systems are usually called “entailment logics,” “relevance logics,” or “relevant logics” [1, 2, 7, 9]. In this paper, we will call these logics “relevant logics.” A major feature of the relevant logics is that they have a primitive intensional connective (i.e., it cannot be defined by other connectives) to represent the notion of entailment. Variable-sharing is a formal notion designed to reflect the idea that there be a meaning-connection between the antecedent and consequent of an entailment [1, 2, 9]. What underlies the relevant logics is the so-called “the relevance principle”, i.e., informally, if A⇒B, where ⇒ denotes the notion of entailment, is a logical theorem of a relevant logic, for any two propositional Sugihara 1955 provided the first general characterization of implicational paradoxes [1]. Relative to a given connective, →, intended as implicational, a formula A is said to be strongest if one can prove A→B for every formula B, and a formula A is said to be weakest if B→A is provable for all B. Thus, a logic system is paradoxical in the sense of Sugihara just in case it has either a weakest or a strongest formula, and a logic system is paradox-free in the sense of Sugihara if it is not paradoxical in that sense. Obviously, CML has ¬A∧A as strongest formula and ¬A∨A as weakest formula. Note that in both B→(¬A∨A) and (¬A∧A)→B the antecedent and consequent share no sentential variable, and therefore, there cannot be a meaning-connection between the antecedent and consequent. Ackermann 1956 pointed out: “Rigorous implication, which we write as A→B, should express the fact that a logical connection holds between A and B, that the content of B is part of that of A. … That has nothing to do with the truth of falsity of A or B. Thus one would reject the validity of the formula A→(B→A), since it permits the inference from A of B→A, and since the truth of A has nothing to do with whether a logical connection holds between B and A” [1, 2, 9]. During 1957~1959, Von Wright, Geach, and Smiley suggested some informal criteria for the notion of entailment, i.e., the so-called “Wright-GeachSmiley criterion” for entailment: “A entails B, if and only if, by means of logic, it is possible to come to know the truth of A→B without coming to know the falsehood of A or the truth of B” (Von Wright 1957); “A entails B if and only if there is an a priori way of getting to know that A→B which is not a way 453 formulas A and B, then A and B share at least one sentential variable [1, 2, 9]. As a result of requiring the relevance principle, the relevant logics include no implicational paradoxes as logical theorems. therefore we cannot say “if A and C, then B” in the sense of entailment. In order to establish a satisfactory logic calculus of entailment to underlie knowledge representation and reasoning, the present author proposed some new relevant logics, named Tc, Ec, and Rc. As a modification of T, E, and R, Tc, Ec, and Rc rejects all conjunction-implicational paradoxes and disjunctionimplicational paradoxes in T, E, and R, respectively, and therefore, they are free not only implicational paradoxes but also conjunction-implicational and disjunction-implicational paradoxes [4-6]. What underlies our relevant logics Tc, Ec, and Rc is the following strong relevance principle. However, although the relevant logics have rejected those implicational paradoxes, there still exist some logical axioms or theorems in the logics which are not natural in the sense of entailment. Such logical axioms or theorems, for instance, are (A∧B)⇒A, (A∧B)⇒B, (A⇒B)⇒((A∧C)⇒B), A⇒(A∨B), B⇒(A∨B), (A⇒B)⇒(A⇒(B∨C)) and so on, where ⇒ is the primitive intensional connective in the logics to represent the notion of entailment. The present author first named these logical axioms or theorems “conjunction-implicational paradoxes” and “disjunction-implicational paradoxes” [3]. Let A, B, and C be well formed formulas. A is a consequent part of A; if ¬B is a consequent part {antecedent part} of A, then B is an antecedent part {consequent part} of A; if B⇒C is a consequent part {antecedent part} of A, then B is an antecedent part {consequent part} of A, and C is a consequent part {antecedent part} of A; if either B∧C or B∨C is a consequent part {antecedent part} of A, then both B and C are consequent parts {antecedent parts} of A [1]. Why the conjunction-implicational paradoxes and disjunction-implicational paradoxes are not natural? Let us see some examples. A simple conjunction-implicational paradox is an implication and/or entailment formula such that its antecedent is a conjunctive formula and its consequent is a proper subformula of its antecedent as a conjunct. For example, simple conjunctionimplicational paradox (A∧B)⇒A is a logical axiom of almost all relevant logics and therefore is valid in the logics even if B may be irrelevant to A or B may be the negation of A. However, propositions in this form such as “if snow is white and 1+1=2, then snow is white,” “if snow is white and 1+1=3, then snow is white,” and “if snow is white and snow is not white, then snow is white” cannot be considered as to be valid in human logical thinking. If A is a theorem of Tc, Ec, and Rc, then every sentential variable in A occurs at least once as an antecedent part and at least once as a consequent part. We say that Tc, Ec, and Rc satisfy the strong relevance principle. Now, in the framework of paradox-free relevant logics Tc, Ec, or Rc, the conclusion of an argument or inference based on Tc, Ec, or Rc must strongly relevant to its premises. As a result, for a conclusion reasoned based on Tc, Ec, or Rc, we can accept it directly and do not need to investigate whether it is relevant to its premises or not. It is the notion of entailment that plays the fundamental role in relevant reasoning. A simple disjunction-implicational paradox is an implication and/or entailment formula such that its consequent is a disjunctive formula and its antecedent is a proper subformula of its consequent as a disjunct. Similar to the above discussion on simple conjunction-implicational paradoxes, we can know simple disjunction-implicational paradoxes are also not natural in the sense of entailment. 4. RELEVANT REASONING AS THE LOGICAL BASIS OF KNOWLEDGE ENGINEERING (A⇒B)⇒((A∧C)⇒B) is a more complex conjunction-implicational paradox. It is a logical theorem of almost all relevant logics and is valid in the logics. Therefore, from any given entailment A⇒B and the logical theorem, we can infer (A∧C)⇒B by using Modus Ponens for entailment. However, from the viewpoint of human logical thinking, this inference is not necessarily considered as to be valid in the sense of entailment because there may be no necessarily relevant and/or conditional relation between C and B and In this section, we will give some examples to explain why one should take relevant reasoning as the logical basis of knowledge engineering. 4.1 Entailment and ampliative, non-circular, and/or non-tautological reasoning First of all, from the viewpoint to regard logically valid reasoning as the process of drawing new and 454 correct conclusions from some premises which are known facts and/or assumed hypotheses, any meaningful reasoning should be ampliative, noncircular, and/or non-tautological, i.e., the truth-value of conclusion of the reasoning must not be used in deciding the truth-values of premises of the reasoning. 4.2 Entailment and paraconsistent reasoning In general, our knowledge about a domain may be inconsistent in some ways, i.e., it directly or indirectly includes some contradictions. Also, even if our knowledge about a domain seems to be consistent, in some cases, we may find a new fact or rule that is inconsistent with our known knowledge, i.e., we find a contradiction. In these cases, we neither doubt “logic” we used in our ordinary logical thinking nor reason out anything from the contradictions, but we must consider that there are some wrong things in our knowledge and will investigate the causes of the contradictions. Indeed, in scientific research, the detection and explanation of an inconsistency between a new fact and known knowledge often leads to formation of new concepts and/or discovery of new principles. How to reason with inconsistent knowledge is an important issue in scientific discovery and theory formation. As an example, let us see the most typical human logical inference form so-called “Modus Ponens.” The natural language representation of Modus Ponens is “if A holds then B holds, now A holds, therefore B holds.” When we reason using Modus Ponens, what we know? We know “if A holds then B holds” and “A holds.” Before the reasoning, we do not know whether or not “B holds.” If we knew, then we would not need to reason at all. Therefore, Modus Ponens should be noncircular. Thus, how can we know “B holds” by using non-circular Modus Ponens? Indeed, by using Modus Ponens, we can know “B holds,” which is unknown until the reasoning is done, based on the following reasons: (i) “A holds,” (ii) “There is no case such that A holds but B does not hold,” and (iii) we know (ii) without investigating either “whether A holds or not” or “whether B holds or not.” Note that the WrightGeach-Smiley criterion for entailment is corresponding to the above (ii) and (iii). From this example, we can see that the key point in non-circular reasoning is the primitive and intensional property required by the notion of entailment. For a logic with Modus Ponens as an inference rule, the paraconsistence requires that the logic does not have “(¬A∧A)→B”, which is the most typical implicational paradox, as a logical theorem where “A” and “B” are any two different formulas and “→” is the relation of implication used in Modus Ponens. If a logic is not paraconsistent, then infinite propositions (even negations of those logical theorems of the logic) may be reasoned out based on the logic from a set of premises that directly or indirectly include a contradiction. For example, CML is explosive and not paraconsistent because it uses Modus Ponens for material implication as an inference rule, and has “(¬A∧A)→B” as a logical theorem. On the other hand, the relevant logics are paraconsistent because they do not have the implicational paradox “(¬A∧A)⇒B” as a logical theorem. Obviously, as a logic system to underlie reasoning with inconsistent knowledge, a paraconsistent logic is by far better than a paradoxical logic, and therefore, we can say that formalizing the notion of entailment satisfactorily is a key point to the issue of reasoning with inconsistent knowledge. Because the material implication in CML is an extensional truth-function of its antecedent and consequent without the primitive and intensional property required by the notion of entailment, a reasoning based on the logic must be circular but not ampliative. For example, Modus Ponens for material implication is usually represented in CML as “from A and A→B to infer B.” According to the extensional truth-functional semantics of the material implication, if we know “A is true” but do not know the truth-value of B, then we cannot decide the truth-value of “A→B.” In order to know the truth-value of B using Modus Ponens for material implication, we have to know the truth-value of B before the reasoning is done! Obviously, Modus Ponens for material implication is circular if it is used as a inference form, and therefore, it is not a natural representation of Modus Ponens. On the other hand, because relevant logics have a primitive intensional connective to represent the notion of entailment which satisfies the Wright-Geach-Smiley criterion, Modus Ponens for entailment in these logics is non-circular. Based on the above discussion, we can say that the notion of entailment is indispensable for ampliative and non-circular reasoning. 4.3 Entailment and abductive reasoning Now, we argue what role the notion of entailment plays in abduction and/or abductive reasoning which is also an important issue in knowledge engineering. The term “abduction” was coined by C. S. Peirce. Peirce represented an abductive inference form as : 455 The surprising fact, C, is observed. But if A were true, C would be a matter of course. Hence, there is reason to suspect that A is true. shows that both the proof-theoretical relation |−CML and the model-theoretical relation |=CML are equivalent to the material implication. Therefore, we can say that the monotonicity of CML is a direct result of the truthfunctional definition of the material implication. In terms of logic, the abductive inference form can be represented as: From “C” and “if A then C” to infer “A.” The above discussion suggests that a nonmonotonic deducibility relation may be obtained by formalizing the notion of entailment satisfactorily, and indeed this is right. The relevant logic R is naturally nonmonotonic [10]. Its deducibility relation is already nonmonotonic and no new logical primitives, such as a minimization notion, a modal operator, or default inference rules introduced by current research on nonmonotonic reasoning, need be introduced to get the desired effect. To get a nonmonotonic deducibility relation by formalizing the notion of entailment satisfactorily is an important research direction for establishing a satisfactory logic system underlying nonmonotonic reasoning. Obviously, since “C” is an observed fact and “A” is the result of the inference, the logical validity of such an inference is totally determined by the validity of entailment “if A then C.” Those implicational, conjunction-implicational, and disjunctionimplicational paradoxes, if they are used as entailments in abduction, are not only useless but also harmful, because an abduction used the paradoxes may give us some results that are not relevant to the observed fact at all. Therefore, we can say that the key point in abduction is how to get and use genuine logical entailments that are certainly relevant to the observed fact. 4.5 Entailment and inference rule generation and verification 4.4 Entailment and nonmonotonic reasoning Now, let us discuss the relationship between the notion of entailment and nonmonotonic reasoning which has been actively studied in recent years by the researchers working on fundamentals of AI. Finally, let us see some practical issues in research and development of knowledge-based systems. There is a basic fact that the notion of entailment is indispensable for any practical knowledge-based system. In fact, the entailment has been used, in various explicit or implicit forms, in every kind of approaches to knowledge representation and reasoning such as logicbased, associational, frame-based and procedural representations and reasoning. This is inevitable because there is no knowledge-based system which works without reasoning and there is no inference form which does not invoke the entailment. However, using which kind of entailments as logical entailments and/or inference forms is a key point here. Obviously, unrestrictedly using those implicational, conjunctionimplicational, and disjunction-implicational paradoxes (e.g., A→(B→A), B→(¬A∨A), ¬A→(A→B), (¬A∧A)→B, (A→B)∨ (¬A→B), (A→B)∨(A→¬B), (A→B)∨(B→A), ((A∧B)→C)→((A→C)∨(B→C)), (A∧B)⇒A, (A∧B)⇒ ((A∧C)⇒B), A⇒(A∨B), (A⇒B)⇒(A⇒(B∨C)), and so on) as logical entailments and/or inference forms must lead to the combinational explosion. Therefore, if we want to have a mechanism to filter those paradoxical entailments automatically, then the only way is to construct a completely paradoxfree calculus of entailment. The paradox-free relevant logics Tc, Ec, and Rc have provided a basis for the further work on this issue. It is well-known that CML is naturally monotonic. The monotonicity is caused by two fundamental facts which hold for any set P of formulas and any two formulas A and B. The two facts can be equivalently represented either in a proof-theoretical form or in a model-theoretical form as follows: if P |−CML B then P|−CML A→B if P |=CML B then P |=CML A→B (1) (1') P∪{A} |−CML B iff P |−CML A→B P∪{A} |=CML B iff P |=CML A→B (2) (2') As a direct result of these two facts, CML has the following proof-theoretical and model-theoretical monotonicity: if P |−CML B then P∪{A} |−CML B if P |=CML B then P∪{A} |=CML B (3) (3') Indeed, the above first fact means that if B is right then so is A→B. This is a direct result of defining the material implication as a truth-function of its antecedent and consequent, i.e., A→B =df ¬(A∧¬B) or A→B =df ¬A∨B, and ignoring whether or not there is a necessarily relevant and/or conditional relation between its antecedent and consequent. The above second fact, which is the so-called “deduction theorem” of CML, 456 A common inadequacy of the current knowledgebased systems is that they cannot automatically generate new and valid inference rules from those existing rules that are programmed in the systems by their developers. As a result, the systems fail to have the ability to apply the knowledge, which are programmed by their developers for some situations and purposes, to other situations and purposes, which have not been considered previously by the developers, even if there is little difference between the considered situations and/or purposes and unconsidered situations and/or purposes. To make knowledge-based systems more powerful and flexible, it is indispensable to establish a domain-independent fundamental theory that underlies the automatic generation of new and valid inference rules. Obviously, a paradoxical logic cannot be used as the fundamental theory, because from those facts and entailments given as premises, an inference based on the logic may result invalid entailments as conclusions. On the other hand, the conclusions of an inference based on a paradox-free logic must be valid, if all premises of the inference are valid. Therefore, the notion of entailment and its calculus are serious and crucial to the issue of inference rule generation in knowledge-based systems [4, 6]. Note that a calculus of entailment not only can be used as a tool for the automatic generation of new and valid inference rules, but also can be used as a tool for incremental knowledge acquisition. regard the notion of entailment as the most fundamental notion in knowledge representation and reasoning, should establish a satisfactory logic calculus of entailment to underlie the knowledge representation and reasoning, and should take relevant reasoning based on paradox-free relevant logics as the logical basis of knowledge engineering. REFERENCES [1] A. R. Anderson and N. D. Belnap Jr., “Entailment: The Logic of Relevance and Necessity,” Vol. I, Princeton University Press, 1975. [2] A. R. Anderson, N. D. Belnap Jr., and J. M. Dunn, “Entailment: The Logic of Relevance and Necessity,” Vol. II, Princeton University Press, 1992. [3] J. Cheng, “Logical Tool of Knowledge Engineering: Using Entailment Logic rather than Mathematical Logic,” Proc. ACM 19th Annual Computer Science Conference, pp. 228-238, 1991. [4] J. Cheng, “Entailment Calculus as a Logical Tool for Reasoning Rule Generation and Verification,” in J. Liebowitz (ed.), “Moving Towards Expert Systems Globally in the 21st Century,” pp. 386-392, Cognizant Communication Co. (Hardbound), Macmillan New Media (CD-ROM), 1994. [5] J. Cheng, “The Fundamental Role of Entailment in Knowledge Representation and Reasoning,” Journal of Computing and Information, Vol. 2, No. 1, pp. 853873, Special Issue: Proceedings of the 8th International Conference of Computing and Information, 1996. The lack of formal and rigorous verification technique is a major reason of that many experimental knowledge-based systems cannot be reliably used in practices for solving business and industrial problems in the real world. The issue of inference rule verification also requires a domain-independent fundamental theory to underlie justifications. Obviously, what we need here is also a satisfactory logic calculus of entailment [4, 6]. [6] J. Cheng, “EnCal: An Automated Forward Deduction System for General-Purpose Entailment Calculus,” in N. Terashima and E. Altman (Eds.), “Advanced IT Tools, IFIP World Conference on Advanced IT Tools, IFIP 96 14th World Computer Congress,” pp. 507-514, Chapman & Hall, 1996. Based on the above discussion, we can see that the role played by the notion of entailment in practical knowledge-based systems is also essential. 5. [7] J. M. Dunn, “Relevance Logic and Entailment,” in D. Gabbay and F. Guenthner (eds.), “Handbook of Philosophical Logic,” Vol. III, pp. 117-224, D. Reidel, 1986. CONCLUDING REMARKS [8] G. Priest, R. Routly, and J. Norman, “Paraconsistent Logic: Essays on the Inconsistent,” Philosophia, 1989. We have pointed out why a reasoning based on CML may be irrelevant, shown that a reasoning based on paradox-free relevant logics is relevant and that it is the notion of entailment that plays the fundamental role in relevant reasoning, and given some examples to explain why one should take relevant reasoning as logical basis of knowledge engineering. From the above discussions, we can conclude that we should [9] S. Read, “Relevant Logic: A Philosophical Examination of Inference,” Basil Blackwell, 1988. [10] P. B. Thistlewaite, M. A. McRobbie, and R. K. Meyer, “Automated Theorem Proving in Non-Classical Logics,” John Wiley & Sons, 1988. 457