Propositional Logic And First Order Logic.

1. Propositional Logic And First Order Logic. An Introduction

2. Logic Logic is the science of reasoning. Logic is a process by which we arrive at a conclusion from known statements with the use of laws of logic. Mathematical logic is called Boolean Logic.

3. Sentence and Proposition Sentence: It is sensible combination of words. Statement or Proposition: A statement is a sentence in the grammatical sense conveying a situation which is neither imperative, interrogative nor exclamatory. It is a declarative sentence which is either true or false but not both. The truth or falseness of a statement is called its truth value.

4. Sentence and Proposition The difference between an ordinary sentence and logical statement is that whereas it is not possible to say about truth or otherwise of an ordinary sentence. It is an essential requirement for a logical statement. In logical statement or Proposition the result is in true or false. In ordinary sentence the result is other then true or false.

5. Check for Proposition “3+3=8” Is this a statement or sentence? This is a statement. But is a false statement. Its false value will be denoted by the letter F or 0. “Why are you going to Bangalore ?” Is this a statement or sentence? This is a sentence. Because it is a interrogative sentence.

6. Try This “May God bless you with happiness !” “(x-1) 2 =x2 - 2x + 1” “x+5=10” The truth of the sentence is open till we are told what x stands for. Such a sentence is called an open sentence. An open sentence is, thus, not a statement.

7. Definition of Statement or Proposition A statement is any meaningful, unambiguous sentence which is either true or false but not both. A statement cannot be true and false at the same time. This fact is known as the “law of the excluded middle”

8. Basic Sentential Connectives A compound statement is a combination or two or more simple statements. In order to make a statement compound we have to use some connectives. Sentential Connectives: The phrases or words which connect two simple statements are called “sentential Connectives”, ”logical operators” or simply “connectives”.

9. Basic Sentential Connectives Some of the basic connectives are “and”, “or”, “not”, “if then”, “if and only if”. When simple statements are combined to make compound statements, then simple statements are called “Components”. Simple statements are generally denoted by small letters p, q, r, s, t,…..

10. Logical Equivalence Two propositions (i.e. compound statements) are said to be logically equivalent (or equal) if they have identical truth values. The symbol “= “ or “=” is used for logical equivalence.

11. Conjunction Any two statements can be combined by the connective “and” to form compound statement called the “conjunction” of the original statements. E.g. He is practical. He is sensitive. The conjunction is “He is practical and sensitive” The conjunction is denoted by p^q and read as ”p and q” p^q is true when both p and q are true.

12. Disjunction Any two statements can be combined by the connective “or” to form compound statement called the “disjunction” of the original statements. E.g. There is something wrong with the teacher There is something wrong with the student The disjunction is “there is something wrong with the teacher or with the student” The disjunction is denoted by pVq and read as ”p or q” pVq is false when both p and q are false.

13. Negation or Denial Negation refers to contradiction and not to a contrary statement. We should be very careful while writing the negation of the given statement. The best way is to put in the word ”not” at the proper place. Or To put the phrase. ”it is not the case that” in the beginning.

14. Negation or Denial E.g. If p stands for “He is a good student”. Negation of p, denoted by ~p or ¬p is either “He is not a good student” or “It is not the case that he is a good student” Note that: We cannot say “He is a bad student” is the negation of p.

15. Conditional Statement Any statement of the form “if p then q”, where p and q are statements, is called a conditional statement. Here p is sufficient for q but not essential. There can be q even without p. Let p: you work hard. q: you will pass. Now it is possible that a student may pass who has not worked had. Although p is not necessary for q, q is necessary for p. q is necessary for p. It will not happen that one who works hard will not pass.

16. Conditional Statement The conditional statement “if p then q” is denoted by p q (to be read as p conditional q) or (p implies q). The conditional statement p q is also read as “if p then q”, “p implies q”, “p only if q”, ”p is sufficient for q”, “q is necessary for p”, “q if p”. Rule: p q is true in all cases except when p is true and q is false.

17. Truth Table for conditional statement

18. Bi-conditional statement The statement “p if and only if q” is called a bi-conditional statement and is denoted by p q . The bi-conditional is also read as. q if and only if p. q implies q and q implies p. p is necessary and sufficient for q. q is necessary and sufficient for p. p iff q. q iff p.

19. Rule for Bi-Conditional statement Rule: p q Truth if both p and q have the same truth value i.e. either both are true or both are false. False if p and q have opposite truth values.

20. Truth Table for bi-conditional statement

21. Tautologies and Contradictions (or Fallacies) Tautologies: A tautology is a proposition which is true for all the truth value of its components. In a truth table of tautology there will be only T’s in last column. Consider the proposition p V ~p. Its truth table is: The proposition is always true whatever be the truth values of its components. It is a tautology.

22. Tautologies and Contradictions (or Fallacies) Contradictions: A contradiction (or fallacies) is proposition which is false for all truth values of its components. In a truth table of contradictions there will be only F’s in the last column. Consider the proposition p^~p Its truth table is: It is a contradiction.

23. Converse, Inverse and Contrapositive If p q is a direct statement, then. q p is called its converse. ~p ~q is called its inverse. ~q ~p is called its Contrapositive. Since p q = ~q ~p And q p = ~ p ~q contrapositive = direct statement And converse = inverse

24. Exercise For You Give the truth table for the statement (p q) (~p V q) (p ^ q) (p V q) (~p V q) ^ (~p ^ ~q)