Chapter 2. An Axiomatic Basis for a Logical System

G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, London and New York: Routledge, 1996, pp. 23-24.

 

(1) Systems of Modal Logic

① The axiomatic method: it allows us to define a class of wff without any reference to their meanings.

② An axiomatic basis for a logical system

• (a) a specification of the language in which the formulae of the system will be expressed - i.e. a list of primitive symbols, together with any definitions that may be thought convenient, together with a set of formation rules specifying which strings of symbols are to count as wff;
• (b) a selected set of wff, known as axioms (공리, 하나의 이론에서 증명 없이 바르다고 하는 명제)
• (c) a set of transformation rules, licensing various operations on the axioms, and also (normally) on wff obtained from the axioms by previous applications of the transformation rules.
• (d) The wff obtained from the axioms in this way, together with the axioms themselves, are known as the theorems (정리, 증명을 통해 참임이 밝혀지는 명제) of the system.

③ All the systems of propositional modal logic which we shall consider will have the same language, the one specified in the previous chapter on p. 16; so in stating their bases we shall merely list their axioms and transformation rules.

An axiomatic basis must be formulated in such a way that we can determine effectively
(i) of any arbitrary string of symbols whether or not it is a wff,
(ii) of any wff whether or not it is an axiom, and
(iii) of any purported application of a transformation rule whether or not it is a genuine application of that rule.

④ A semantical approach to logic vs. A syntactical approach to logic

A semantical approach	의미론적 접근
A syntactical approach (= An axiomatic approach)	구문론 or 통사론적 접근

※ 구문론 Syntax

• [수학] 의미를 무시하고 기호 사이의 형식적 관계만을 취급
• [언어] 문장을 기본 대상으로 하며 문장의 구조나 기능, 문장의 구성요소 따위를 연구하는 학문

All this, however, does not mean that in choosing the axioms for a system we ought to keep all thought of interpretation out of our minds. For although we could in theory take any wff whatsoever as axioms, in practice our reason for choosing certain wff as axioms will usually be either that they are valid by some criterion of validity that we have in mind, or at least that they are plausible or interesting in some way which leads us to want to explore their consequences; and these are matters which involve the interpretation we give to our symbols and formulae.

Analogously, when we are constructing a system with a certain criterion of validity in mind, we see to it that its transformation rules are such that when they are applied to valid wff the theorems they yield are always valid too. Such transformation rules are said to be validity-preserving (with respect to that account of validity). (p. 24)

⑤ Terminology

(1)

When a formula is a theroem of a given system:
A formula	belongs to	that system
	is contained in
	is in

(2)

If two axiomatic systems, S and S', have different bases but contain exactly the same theorems:
S and S' are	deductively equivalent
	equivalent
※ Two systems are deductively equivalent iff each contains each other

(3)

If S' contains all theorems of S and other theroems as well

S' propely contains S

S' is a proper extension of S

S' is the stronger and S is the weaker of the two systems

 

Appendix: Axiom and Theorem

일반적으로 공리(axiom)는 하나의 이론에서 증명없이 바르다고 하는 명제이며^[각주:1], 정리(theorem)는 공리를 바탕으로 하여 증명을 통해 참임이 밝혀지는 명제를 뜻한다. 그런데 Hughes&Cresswell은 이 책에서 공리 또한 정리로 간주할 것이라고 말하고 있다("The wff obtained from the axioms in this way, together with the axioms themselves, are known as the theorems of the system.") 이는 언뜻 보기에 공리의 정의와 모순되는 것처럼 보인다. 그러나 Hughes&Cresswell에 따르면, 우리는 공리를 그 자체로 a one-line proof로 간주할 수 있고 바로 이러한 의미에서 공리 또한 정리들에 포함시킬 수 있다.

We have said that the theorems of a system are those wff which can be derived from its axioms by applying its transformation rules. To prove a theorem is therefore to derive it in this way. More precisely, a proof of a theorem α in a system S consists of a finite sequence of wff, each of which is either (i) an axiom of S or (ii) a wff derived from one or more wff occurring earlier in the sequence, by one of the transformation rules or by applying a definition, α itself being the last wff in the sequence. (Note that by this account of what constitutes a proof of a theorem, every wff in a proof is itself a theorem; and also that one reason why we count the axioms themselves as theorems is that any axiom can be thought of as a one-line proof of itself.) (p. 26)

 

1. 바로 그렇기 때문에 영어 axiomatic은 '자명한self-evident'이라는 뜻을 지닌다. [본문으로]