﻿ [SL01] Introduction

§1. Formal systems of logic

Sentential logic (SL) is a formal system of logic. It is a very simple system of logic. When people study formal logic this is usually the first thing that they would study. Other more complicated systems include for example predicate logic (PL), and modal logic.

So what is a system of logic? Basically, it is a set of rules that tell us how to make use of special symbols to construct sentences and do proofs. To define a particular system of logic, we need to specify :

1. The formal language of the system
2. The semantic rules for the formal language
3. The rules of proof for the language

A formal language in a system of logic is a language with precisely specified rules that tell us how to construct grammatical sentences. Such rules are called syntactic rules. They are equivalent to the rules of grammar you find in English or Cantonese.

The semantic rules are rules for interpreting the sentences in the language. They tell us what the sentences mean and the conditions under which the sentences are true or false.

The rules of proof are rules that specify how logical proofs are to be constucted. They tell us what conclusions can be derived given certain initial assumptions.

§2. Why study formal systems of logic?

There are many reasons for creating and studying such formal systems of logic:

• Systems of logic can be used to formalize arguments in natural languages. A natural language is a language that is used for normal everyday communication in a human society. So languages such as Japanese, Irish, and French are all natural languages. By formalization we refer to the process of translating arguments or sentences in natural languages into the notations of formal logic. The reason for carrying out formalization is that very often they can help us understand the logical structure of arguments better, by identifying patterns of valid arguments. Also, the rules of proof in a formal system of logic are precisely specified. By formalizing an argument we can use the rules of proof to check whether the argument can indeed be proved to be valid.
• Because the rules of formal systems of logic are defined very clearly, we can program them into a computer and get a computer to construct and evaluate proofs quickly and automatically. This is particularly important in areas such as Artificial Intelligence, where many researchers teach computers to use formal logic in reasoning.
• Linguists are scientists who study natural languages. Many linguists also study formal languages and use them to compare and contrast with natural languages.
• Many philosophers are also interested in formal systems of logic. One reason is that natural languages are sometimes not precise enough to express certain ideas clearly. So sometimes they turn to formal systems of logic instead.
• Formal systems of logic are also interesting in their own right. Logicians and mathematicians are interested in finding out what they can or cannot prove, and also their many other logical properties. Formal systems of logic also play an important role in understanding the foundations of set theory and mathematics.