SYMBOLIC LOGIC NECESSARY FOR RATIONAL THINKING AND UNDERSTANDING

BASIC LOGIC1 a) A Proposition is a True(T) or False(F) Statement. Example: It is raining now. Note: If Heavens/Skies are open ( rains are falling ) , then TRUE(T). Else FALSE(F). b) Let A be a proposition. Not A (~A ) [ wave A ] is NEGATION of A. If A is T then ~A is F. c) CONJUNCTION (^) and DISJUNCTION (U). Taking A and B to be two propositions , A Conjunction B is A and B [ A^B ] , A Intersection B. A Disjunction B is A or B [ AUB ] , A union B. TRUTH VALUES d) CONJUNCTION: Truth Value of A^B is T if A is T and B is T. Else A^B is F. e) DISJUNCTION: Truth Value of AUB is F if both values of A and B are each F. That is A is F and B is F. Else AUB is T.. In other words T value for at least one of the propositions in the Statement AUB makes AUB True(T). IMPLICATION f) Consider A Implies B [ A»B ]. Taking » for Implication sign. g) TRUTH VALUES: A»B is F if A is T and B is F. Else A»B is T for all other combinations for values of A and B in the Implication Statement. It may not be difficult to see A»B is T when A is T and B is T. And when A is F and B is F. NOTE: A»B is T if A is F and B is T. h) Example: A False (F) Statement may lead to a True(T) Conclusion. Let A be 1=9. This is a False(F) Statement. From A , 9=1. Let B represent 9=1. A+B is 1+9=9+1 which is 10=10 , a True(T) Statement. From a False(F) Theorem 1=9. We have argued correctly and arrived at a True(T) Conclusion 10=10. BASIC LOGIC II METHOD OF PROOF Consider a Theorem: A(t)=B , t is a condition say t= time. To prove it we are to establish A(t)=B for all t. To disprove the Theorem One Counter Example of Truth of A(t)=B is Sufficient. For example if A(a) is NOT equal to B for t=a , then A(t)=B is False(F) for t=a. Hence Theorem is disproved. One Counter Example here is A(a) is NOT equal to B , where t=a. CONVERSE and CONTRAPOSITIVE of THEOREM: If A Implies B [A»B] , then the Converse is B Implies A [B»A]. The Contrapositive is Not B Implies Not A [~B»(~A)] ( wave B Implies wave A ). Truth Value of a Converse of Theorem is NOT always the same as that of the Theorem. Truth Value of the Contrapositive is same as Truth Value of the Theorem. In otherwords Truth Value of Converse is NOT deducible from the Truth Value of Theorem. Whilst Truth Value of Contrapositive is deducible from the Truth Value of Theorem. It is the same Truth Value as the Truth Value of the Theorem. For a Theorem A»B , where A is F and B is T , the Truth Value of A»B is T. The Truth Value of B»A is F , when A is F and B is T. But B»A is Converse of A»B. With different Truth Values. A»B is T and B»A is F when A is F and B is T. Contrapositive of A»B is ~B»(~A) where A is F and B is T. But ~A is T and ~B is F. And the Truth Value of ~B»(~A) is T. Which is same as the Truth Value of A»B. That is Contrapositive and the Theorem have the same Truth Value T. Mathematical or Symbolic Logic finds wide range of Applications in Computing , Artificial Intelligence and General Software Development. Without it most logic devices and underlying Coding are are impossible. Beauty and Elegance in Symbolic or Mathematical Logic is found in its profound simplicity and very easy use once its basics and principles are mastered.