A Strong Relevant Logic Model of Epistemic Processes in Scientific Discovery

Transcription

1 A Strong Relevant Logic Model of Epistemic Processes in Scientific Discovery (Extended Abstract) Jingde Cheng Department of Computer Science and Communication Engineering Kyushu University, Hakozaki, Fukuoka, , Japan Abstract. This paper presents some significant fundamental observations and/or assumptions on scientific discovery processes and their automation, shows why classical mathematical logic, its various classical conservative extensions, and traditional (weak) relevant logics cannot satisfactorily underlie epistemic processes in scientific discovery, and presents a strong relevant logic model of epistemic processes in scientific discovery. 1 Introduction Any scientific discovery must include an epistemic process to gain knowledge of or to ascertain the existence of some empirical and/or logical conditionals previously unknown or unrecognized. As an applied and/or technical science, Computer Science should provide scientists with some epistemic representation, description, reasoning, and computing tools for supporting the scientists to suppose, verify, and then ultimately discover new conditionals in their research fields. However, no programming paradigm in the current computer science focuses its attention on this issue. In order to provide scientists with a computational method to program their epistemic processes in scientific discovery, we are establishing a novel programming paradigm, named Epistemic Programming, which regards conditionals as the subject of computing, takes primary epistemic operations as basic operations of computing, and regards epistemic processes as the subject of programming. Modeling epistemic processes in scientific discovery satisfactorily is an indispensable step to automating scientific discovery processes. This paper presents some significant fundamental observations and/or assumptions, which underlie our research direction, on scientific discovery processes and their automation, shows why classical mathematical logic, its various classical conservative extensions, and traditional (weak) relevant logics cannot satisfactorily underlie epistemic processes in scientific discovery, presents a strong relevant logic model of epistemic processes in scientific discovery as the logical foundation to underlie epistemic programming.

2 490 2 Fundamental Observations and/or Assumptions First of all, we present here some significant fundamental observations and/or assumptions, which underlie our research direction, on scientific discovery processes and their automation as follows: (1) Specific knowledge is the power of a scientist: Any scientist who made a scientific discovery must have worked in some particular scientific field and more specifically on some problem in a particular domain within the field. There is no universal scientist who can make scientific discoveries in every field. (2) Any scientific discovery has an ordered epistemic process: Any scientific discovery must have, among other things, a process that consists of a number of ordered epistemic activities that may be contributed by many scientists in a long duration. Any scientific discovery is nether an event occurs in a moment nor an accumulation of disorderly and disorganized inquisitions. (3) New conditionals are epistemic goals of any scientific discovery: Any scientific discovery process must include an epistemic process to gain knowledge of or to ascertain the existence of some empirical and/or logical conditionals previously unknown or unrecognized. Finding some new data or some new fact is just an initial step in a scientific discovery but not the scientific discovery itself. (4) Scientific reasoning is indispensable to any scientific discovery: Any discovery must be unknown or unrecognized before the completion of discovery process. Reasoning is the sole way to draw new conclusions from some premises that are known facts and/or assumed hypothesis. There is no scientific discovery that does not invoke scientific reasoning. (5) Scientific reasoning must be justified based on some sound logical criterion: The most intrinsic difference between discovery and proof is that discovery has no explicitly defined target as its goal. Since any epistemic process in any scientific discovery has no explicitly defined target, the sole criterion the epistemic process must act according to is to reason correct conclusions from the premises. It is logic that can underlie valid scientific reasoning. (6) Scientific reasoning must be relevant: For any correct argument in scientific reasoning as well as our everyday reasoning, the premises of the argument must be in some way relevant to the conclusion of that argument, and vice versa. A reasoning including some irrelevant arguments cannot be said to be valid in general. (7) Scientific reasoning must be ampliative: A scientific reasoning is intrinsically different from a scientific proving in that the purpose of reasoning is to find out some facts and conditionals previously unknown or unrecognized, while the purpose of proving is to find out a justification for some fact previously known or assumed. A reasoning in any scientific discovery must be ampliative such that it enlarges or increases the reasoning agent s knowledge in some way. (8) Scientific reasoning must be paracomplete: Any scientific theory may be incomplete in many ways, i.e., for some sentence A neither it nor its negation can be true in the theory. Therefore, a reasoning in any scientific discovery must be paracomplete such that it does not reason out a sentence even if it cannot reason out the negation of that sentence.

3 491 (9) Scientific reasoning must be paraconsistent: Any scientific theory may be inconsistent in many ways, i.e., it may directly or indirectly include some contradiction such that for some sentence A both it and its negation can be true together in the theory. Therefore, a reasoning in any scientific discovery must be paraconsistent such that from a contradiction it does not reason out an arbitrary sentence. (10) Epistemic activities in any scientific discovery process are distinguishable: Epistemic activities in any scientific discovery process can be distinguished from other activities, e.g., experimental activities, as explicitly described thoughts. (11) Normal scientific discovery processes are possible: Any scientific discovery process can be described and modeled in a normal way, and therefore, it can be simulated by computer programs automatically. (12) Specific knowledge is the power of a program: Even if scientific discovery processes can be simulated by computer programs automatically in general, a particular computational process which can certainly perform a particular scientific discovery must take sufficient knowledge specific to the subject under investigation into account. There is no generally organized order of scientific discovery processes that can be applied to every problem in every field. (13) Any automated scientific discovery process must be valid: Any automated process of scientific discovery must be able to assure us of the truth, in the sense of not only fact but also conditional, of the final result produced by the process if it starts from an epistemic state where all facts, hypotheses, and conditionals are regarded to be true and/or valid. (14) Any automated scientific discovery process need an autonomous forward reasoning mechanism: Any backward and/or refutation deduction system cannot serve as an autonomous reasoning mechanism to form and/or discover some completely new things. What we need in automating scientific discovery is an autonomous forward reasoning system. 3 The Fundamental Logic to Underlie Epistemic Processes Based on the fundamental observations and/or assumptions presented in Section 2, the fundamental logic that can underlie epistemic processes has to satisfy some essential requirements. First, as a criterion for validity of reasoning, the logic underlying scientific reasoning in epistemic processes must take the relevance between the premises and conclusion of an argument into account. Second, the logic must be able to underlie paracomplete and paraconsistent reasoning; in particular, the principle of Explosion that everything follows from a contradiction cannot be accepted by the logic as a valid principle. Third, for any set of facts and conditionals, which are considered as true and/or valid, given as premises of a reasoning based on the logic, any conditional reasoned out as a conclusion of the reasoning must be true and/or valid in the sense of conditional. Almost all the logic-based works on modeling epistemic processes are based on classical mathematical logic (CML for short) or its some classical conservative extensions [6], keeping as much as fundamental characteristics of CML. However, CML

4 492 cannot satisfy all the above three essential requirements for the fundamental logic. First, because of the classical account of validity that an argument is valid if and only if it is impossible for all its premises to be true while its conclusion is false, a reasoning based on CML may be irrelevant, i.e., the conclusion reasoned out from the premises of that reasoning may be irrelevant at all, in the sense of meaning, to the premises. Second, CML is of no use for reasoning with inconsistency, since the principle of Explosion is a fundamental characteristic of CML. Third, as a result of representing the notion of conditional, which is intrinsically intensional, by the extensional notion of material implication, CML has a great number of implicational paradoxes as its logical axioms or theorems which cannot be regarded as entailments from the viewpoint of scientific reasoning as well as our everyday reasoning. Traditional (weak) relevant logics [1, 2] have rejected those implicational paradoxes in CML, but still have some conjunction-implicational paradoxes and disjunctionimplicational paradoxes [4] as their logical axioms or theorems, which cannot be regarded as entailments from the viewpoint of scientific reasoning as well as our everyday reasoning. In order to establish a satisfactory logic calculus of conditional to underlie relevant reasoning, the present author has proposed some strong relevant logics and shown their applications [4, 5]. Since the strong relevant logics are free not only implicational paradoxes but also conjunction-implicational and disjunction-implicational paradoxes, we can use them to model epistemic processes in scientific discovery without those problems in modeling epistemic processes by CML, various classical conservative extensions of CML, and traditional (weak) relevant logics. 4 A Strong Relevant Logic Model of Epistemic Processes For a given L-theory with premises P, denoted by T L (P), and any formula A of L, A is said to be explicitly accepted by T L (P) if and only if A P and A P; A is said to be explicitly rejected by T L (P) if and only if A P and A P; A is said to be explicitly inconsistent with T L (P) if and only if both A P and A P; A is said to be explicitly independent of T L (P) and is called a explicitly possible new premise for T L (P) if and only if both A P and A P. For any given formal theory T L (P) and any formula A P, A is said to be implicitly accepted by T L (P), if and only if A T L (P) and A T L (P); A is said to be implicitly rejected by T L (P) if and only if A T L (P) and A T L (P); A is said to be implicitly inconsistent with T L (P) if and only if both A T L (P) and A T L (P); A is said to be implicitly independent of T L (P) and is called a implicitly possible new premise for T L (P) if and only if both A T L (P) and A T L (P). Let K F(EcQ), where F(EcQ) is the set of formulas of predicate relevant logic EcQ, be a set of sentences to represent known knowledge and/or current beliefs of an agent. For any A T EcQ (K) K where T EcQ (K) K, an epistemic deduction of A from K, denoted by K d+a, by the agent is defined as K d+a = df K {A}; for any A T EcQ (K), an explicitly epistemic expansion of K by A, denoted by K e+a, by the agent is defined as K e+a = df K {A}; for any A K, an explicitly epistemic contraction of K by A, de-

5 493 noted by K A, by the agent is defined as K A = df K {A}; for any A T EcQ (K), an implicitly epistemic expansion of K by A, denoted by T EcQ (K) e+a, is defined as T EcQ (K) e+a = df T EcQ (K N) where N F(EcQ) such that A K N but A T EcQ (K N); for any A T EcQ (K), an implicitly epistemic contraction of K by A, denoted by T EcQ (K) A, is defined as T EcQ (K) A = df T EcQ (K N) where N F(EcQ) such that A T EcQ (K N); a simple induction by the agent is an epistemic expansion such that for x(a) K and x(a) T EcQ (K), do K e+ x(a) ; a simple abduction by the agent is an epistemic expansion such that for B K, (A B) K, and A T EcQ (K), do K e+a. The basic properties of these epistemic operations can be found in [5]. An epistemic process of an agent is a sequence K 0, o 1, K 1, o 2, K 2,..., K n 1, o n, K n where K i F(EcQ) (n i 0), called an epistemic state of the epistemic process, is a set of sentences to represent known knowledge and/or current beliefs of the agent, and o i+1 (n>i 0), is any of primary epistemic operations, and K i+1 is the result of applying o i+1 to K i. An epistemic process K 0, o 1, K 1,..., o n, K n is said to be consistent if and only if T EcQ (K i ) is consistent for any i (n i 0); an epistemic process K 0, o 1, K 1,..., o n, K n is said to be inconsistent if and only if T EcQ (K i ) is consistent but T EcQ (K j ) is inconsistent for all j>i; an epistemic process K 0, o 1, K 1,..., o n, K n is said to be paraconsistent if and only if T EcQ (K i ) is inconsistent but T EcQ (K j ) is consistent for some j>i; an epistemic process K 0, o 1, K 1,..., o n, K n is said to be monotonic if K i K j for any i<j; an epistemic process K 0, o 1, K 1,..., o n, K n is said to be nonmonotonic if K j K i for some i<j. Any epistemic process K 0, o 1, K 1,..., o n, K n including an epistemic contraction must be nonmonotonic. The idea to model epistemic processes in scientific discovery using relevant logic rather than classical mathematical logic was first proposed in 1994 by the present author [3]. Other work by the author on this direction and a comparison with related work can be found in [5]. References 1. Anderson, A. R., Belnap Jr., N. D.: Entailment: The Logic of Relevance and Necessity. Vol. I. Princeton University Press (1975) 2. Anderson, A. R., Belnap Jr., N. D., Dunn, J. M.: Entailment: The Logic of Relevance and Necessity. Vol. II. Princeton University Press (1992) 3. Cheng, J.: A Relevant Logic Approach to Modeling Epistemic Processes in Scientific Discovery. Proc. 3rd Pacific Rim International Conference on Artificial Intelligence. Vol. 1. (1994) Cheng, J.: The Fundamental Role of Entailment in Knowledge Representation and Reasoning. Journal of Computing and Information. Vol. 2. No. 1 (1996) Cheng, J.: A Strong Relevant Logic Model of Epistemic Processes in Scientific Discovery. Working Notes of ECAI-98 Workshop on Machine Discovery (1998) Gärdenfors, P., Rott, H.: Belief Revision. In: Gabbay, D. M., Hogger, C. J., Robinson, J. A. (eds.): Handbook of Logic in Artificial Intelligence and Logic Programming. Vol.4. Epistemic and Temporal Reasoning. Oxford University Press (1995)