# 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. (1) In order to represent knowledge on epistemic, such as believe, know, desire,
intention, goal and commitment, various modal logics were proposed.
2. (2) In order to represent knowledge which is related to time, various temporal
logics were proposed.
3. (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
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. (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. (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. (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
7. .