# Logic Foundation of Artificial Intelligence

### Introduction

Logic as a formal science was founded by Aristotle. Leibniz reaffirmed

Aristotle’s logical developing direction of mathematics form and founded the

mathematical logic. From the thirties of the last century, various mathematical

methods were extensively introduced and used in the mathematical logic; with

the result that mathematical logic becomes one branch of mathematics and is as

important as algebra and geometry. Mathematical logic has spread out many

branches such as model theory, set theory, recursion theory, and proof theory.

Logic is a primary tool in the study of computer science as well as in the

study of artificial intelligence. It is widely used in many domains, such as the

semasiology, the logic programming language, theory of software specification

and validation, theory of data base, theory of knowledge base, intelligent system,

and the study of robot. Objective of the computer science is essentially

coincident with the goal of logic. On the one hand, the objective of the computer

science is to simulate with the computer the function and behaviour of the human

brain, and bring the computer to be an extension of the brain. Here the simulation

of the function and behaviour of the human brain is infact to simulate the

thinking process of persons. On the other hand, logic is a subject focused on the

discipline and law of human’s thinking. Therefore, the methods and results

obtained in logic are naturally selected and put to use during the research of

computer science. Furthermore, the intelligent behavior of human beings is

largely expressed by language and character; therefore, simulation of human

natural language is the point of departure for the simulation of human thinking

process.

Language is the starting point for the study of human’s thinking in the logic,

as well as for the simulation of human’s thinking in the computer science. Topics

related to language are important issues that run through the domain of computer

science. Many subjects of the computer science, such as programming languages

and their formal semantics, knowledge representation and reasoning, and the

natural language processing, are all related to language. Generally speaking,

representation and reasoning are two basic topics in the computer science and the

artificial intelligence. Majority of the intelligent behavior relies on a direct

representation of knowledge, for which the formal logic provides an important

approach.

Knowledge, especially the so-called common knowledge, is the foundation of

intelligent behavior. Intelligent behavior such as analyzing, conjecturing,

forecasting and deciding are all based on the utilization of knowledge.

Accordingly, in order to simulate with computer the intelligent behavior, one

should firstly make knowledge represented in the computer, and then enable the

computer to utilize and reason about the knowledge. Representation and

reasoning are two basic topics on knowledge in the study of artificial intelligence.

They are entirely coincident with the two topics focused by the study of natural

language, i.e., the accurate structure and reasoning of natural languages.

Therefore, the methods and results obtained in logic are also useful for the study

of knowledge in the artificial intelligence. The ability of representation and the

performance of reasoning are a pair of contradictions for any logic system

applied to intelligent systems. A trade-off between such a pair is often necessary.

The logic applied in majority of logic-based intelligent systems is first order

logic or its extensions. The representation ability of first order logic is so strong

that many experts believe that all the knowledge representation problems arising

in the research of artificial intelligence can be carried out within the framework

of first order logic. First order logic is suitable for representing knowledge with

uncertainty. For example, the expression ∃x P(x) states that there exists an object

for which the property P holds, while it is not pointed out that which one is such

an object. For another example, the expression P ∨ Q states that at least one of P

and Q holds, but it is not determined whether P (or Q) really holds. Furthermore,

first order logic is equipted with a complete axiom system, which can be treated

as a standard of reference in the designing of strategies and algorithms on

reasoning. Although first order logic is capable for representing majority of

knowledge, it is not convenient and concise for many applications. Driven by

various requirements, lots of logic systems have been proposed and studied; in

the following we enumerate some typical examples.

- (1) In order to represent knowledge on epistemic, such as believe, know, desire,

intention, goal and commitment, various modal logics were proposed. - (2) In order to represent knowledge which is related to time, various temporal

logics were proposed. - (3) In order to represent knowledge with uncertainty, the so-called fuzzy logic

was proposed. As a system built upon the natural language directly, fuzzy

logic adopts many elements from the natural language. According to Zadeh,

the founder of fuzzy logic, fuzzy logic can be regarded as a computing

system on words; in another words, fuzzy logic can be defined by the

formula “fuzzy logic = computing with words”. - (4) Knowledge of humans is closely interrelated to human activities.

Accordingly, knowledge on behavior or action is important for intelligent

systems. Compared with various static elements of logic, action is

distinguished by the fact that the execution of actions will affect properties of

intelligent systems. Representation and reasoning about actions are classical

topics in the study of artificial intelligence; many problems, such as the

frame problem and the qualification problem, were put forward and well

studied. Many logic systems, such as the dynamic logic and the dynamic

description logic, were also proposed. - (5) Computer-aided decision-making has become one of the important

applications of computer. Persons always hold their predilections as while as

they are making a decision. In order to represent the rule and simulate the

behavior of people’s decision-making process, it is inevitable to deal with the

predilection. As a result, based on the management science, a family of

so-called partial logics was proposed and studied. - (6) Time is one of the most important terms present in intelligent system. Some

adverbs, such as occasionally, frequently and ofter, are used in the natural

language to represent time. Knowledge about the time which is described by

these adverbs can not be represented with classical temporal logic. Therefore,

an approach similar to the integral of mathematics was introduced into logic.

With the resulted logic, time that described by various adverbs can be

formally represented and operated

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