THE LOGIC OF MATHEMATICS

THE LOGIC OF MATHEMATICS

By : Muhammad Muhibbuddin

Logic is a method of thinking or an art of thinking. Definitively, logic is a field of study explaining or dealing about evaluation of arguments (Wesley C. Salmon: 1977, 673). Logic is also a rule guiding human being to think correctly and well. Logic is like grammar in language. The grammar is arranged to give a rule for someone in practicing of a language. With a grammar, someone has a guidance and a rule in using language correctly. Without grammar, someone will use language blindly. 

That is the same with logic. Without the logic, human being will think at random or blindly, becuase he does not have a guidance to think truly and precisely. He will not know how to think correctly, especially in analyzing something. If someone does not know logic, he will be difficult to evaluate whether an argument or a statement logically is true or false. For this purpose, logic is invented to provide a guidance and a rule in activity of thinking. We need a logic to arrange or compose a valid argument or statement. 

Short History of Logic

Historically, logic was born in Greek era. The first person who arranged logic systematically is Aristotle. Thus, Aristotle is well-known as father of logic. Although  Aristotle is well-known as a pioneer of logic, he did not use term “logic” when he early composed his logic. At that time, Aristotle used term ‘analytic’ and ‘dialectic’ (The Liang Gie et.al:1980, 16).

Rational knowledge (episteme), according to Aristotle consists of three kinds : practical knowledge, productive knowledge, and theoritical knowledge. Analitic and Dialectic (logic), in Aristotle’s opinion was not part of the such episteme. Because, according to him, analytic and dialectic are tools or instruments funtioned mainly to understand the such kinds of rational knowledge. Aristotle’s logic in modern era is called with a traditional logic. One of examples of traditional logic is:

p is q (major premise)

r is p  (minor premise)

r is q  (conclusion)

So, based on form of logic above, we know that conclusion is derived from premises. The validity of conclusion depends on validity of the premises. If the premis is wrong, the conclusion is wrong, and conversely, if the premise is true, the conclusion is true. Besides, in the rule of the traditional logic, the subject in major premise (p) cannot be a subject in conclusion. The subject of conlusion is derived from subject of minor premise (r).   

But, in medieval age, the logic became a branch of philosophical knowledge and even including one of kinds of philosophy. And today, in modern age, logic is not only to be philosophical study, but also to be mathematic study. Thus, it is not surprising if we can see some experts of logic in department of philosophy and mathematic. Until in 19th century the famous logic used in the world was traditional logic invented by Aristotle and developed by medieval age philosophers or thinkers. And in this era, the developing of logic study is more complicated. Such as nowdays we know about modern logic as a transformation of traditional logic. For example :

x is derived from y (the first premise)

y is derived from z  (the second premise)

x is derived from z  (conclusion)

The form of logic above, based on traditional logic, is not valid because the subject  in the first premise (x), according to traditional logic cannot be subject in conclusion. But, in modern logic, it can be valid.

In modern logic, we can also do a immediate inference based on relation among premises. Because a relation of things comprises some logical principles. For example :

7  > 5

5 > 3

7 > 3

Deduction and Induction

Generally, logic is classified into two things, namely deductive and inductive. Deductive logic, according to Donald C. Williams (1997:612) is used to make a conclusion with demonstrative arguments, namely arguments whose premises formally cause a conclusion. In deductive logic, the conclusion is a result or effect of the presmises. Deductive has two main elements, namely antecedent and consequent. If antecedent (premises) is true, the consequent (conclusion) must be true. It is impossible or contradiction if antecedent is true, but the consequent is worng. In other word, deductive inference is a logic or reasoning deriving a conclusion as a consequence of the premises. For example :

All creatures in the world need water (major premise)

All Human beings need water    (minor premise)

All human beings are creatures       (conclusion)

The next, inductive logic or inductive reasoning is a logic based on particular entities to make a universal conclusion. This reasoning is conducted by doing immediately an observation for  real or empirical entities. Thus, this reasoning is true if it is in accordance with reality. Because of  the conclusion of this logic is universal and created based particular facts, this reasoning is also called by generalization. Therefore, the truth of inductive reasoning is probability, not necessity. For example :

John breathes air ( major premise)

John is human being   (minor premise)

All human beings breathe air (conclusion)

Operator of Logic

In oppositional logic, there are some operators or connectors of logic. These connectors of logic are functioned to make easily statement and to simplify several statements. In other word, this connector is used to arrange compound sentence from simple sentence. For example :

-It is a book

-It is a pencil

Those simple sentences becomes a compound sentence with connector “and” :

-It is a book and a pencil.

In logic at least there are fours connectors of logic :

Conjunction  is the other name of connector ‘and’, symbolized by ‘˄’  

Disjunction is the other name of connector ‘or’, symbolized by ‘˅’

Implication is the other name of connector  ‘if... then... ’ symbolized by ‘→’

Bi-Implication is the other name of connector ‘if and only if’’ symbolized ‘↔’

a.Conjunction (˄)

Conjunction  is a compound statement using connector “and”.  If we read structure  p ˄q, it means : p and q.

If we want to make a compound sentence from theses sentences :

• - Indonesia is one of countries at Asia (p)

and (˄)

• - Inggris is one of countries at Europe (q)

We can simplify those sentences to be : p ˄ q

b.Disjunction (˅)

Disjunction is compound statement using connector ‘or”.  p˅ q means ‘p or q’.

If we want to make compound sentence from these sentences :

Ronald Reagen  is an actor  (p)

Or (˅)

Ronald Reagen is a politician (q)

We can simplify those to be : p ˅ q

c.Implication (→)

Implication is compound sentence using connector: ‘if... then... ’. p →q means if p then q.

For example :

‘if she is a student of EEC (p), then she studies English at Sanata Dharma University  (q) or  (p→q)

d.Bi-Implication (↔)

Bi-implication is compound sentence using connector: ‘if and only if’.  p ↔q means if p then q and if q then p. For example :

‘He is an atheist person (p) if and only if he disbelieves in God (q)’ or (p↔q). It means that  if he is an atheist person, then he disbelieves in God and if he disbelieves in God, he is an atheist person. 

References :

Gie, The Liang, et.al., Pengantar Logika Modern, Yogyakarta: Penerbit Karya Kencana,1980

Salmon, Wesley C., “Logic” in Encyclopedia Americana, New York: Americana

Corporation, 1977

Soesianto, F. & Dwijono, Djoni, Logika Proposisional; Seri Logika Matematika, Yogyakarta:

Penerbit Andi, 2003

Williams, Donald C., “ Induction” in Encyclopedia Americana, New York: Americana

Corporation, 1977

_________________, ‘Deduction” in Encyclopedia Americana, New York: Americana

Corporation, 1977