propositional calculus

Slide 1: 

Unit 1 Statements and Notations Propositions Connectives, well formed formulas Truth tables Tautology, Contradiction Logical Equivalences Normal forms DNF, PDNF Normal forms CNF, PCNF

Statements and Notations : 

Statements and Notations A statement is a form of an assertion. Types of statements Declarative Exclamatory Interrogative Imperative A primary statement is a declarative sentence which can not be further broken down or analyzed into simpler sentences. Every primary statement must posses one of the two possible truth values TRUE or FALSE also denoted as 1 or 0. Logic involving such statements is called two-valued or binary logic.

Primitive or Primary or Atomic Statements : 

Primitive or Primary or Atomic Statements Notation : A, B, C, …… Z Examples: 5 is a prime number. sum of all angles of a triangle is 360. Today is Monday. 1 + 100 = 101 Hyderabad is Capital of Andhra Pradesh. 25 is greater than 20.

Slide 4: 

Test - 1 Check whether the following are statements or not - Close the Box. Sum of 2 and 5 is 8. What is your name ? Wow !! What a beautiful Car. Tallest boy in my class has height 5’6’’. Mathematics is an interesting subject. Do you speak Hindi ? Please give some water.

Slide 5: 

Propositions In philosophy and logic, a proposition is a string of sounds or symbols that have a unified meaning. In mathematical logic a proposition is usually a statement and is therefore necessarily either true or false.

Slide 6: 

Aristotelian logic identifies a proposition as a sentence which affirms or denies the predicate of a subject. An Aristotelian proposition may take the form "All men are mortal" or "Socrates is a man." In the first example, which a mathematicial logician would call a quantified predicate (note the difference in usage), the subject is "men" and the predicate "all are mortal". In the second example, which a mathematicial logician would call a statement, the subject is "Socrates" and the predicate is "is a man". The second example is an in Propositional logic, the first example is a statement in predicate logic. The compound proposition, "All men are mortal and Socrates is a man," combines two atomic propositions, and is considered true if and only if both parts are true.

Connectives : 

Connectives In logic, two sentences (either in a formal language or a natural language) may be joined by means of a logical connective to form a compound sentence. The truth-value of the compound is uniquely determined by the truth-values of the simpler sentences. The logical connective therefore represents a function.

Five Major Connectives : 

Five Major Connectives Primary statements can be combined by using five major connectives not, and, or, if….then, and if & only if. These five connectives of English language have special names in logic as not negation ~ or ┐ and Conjunction ^ or Disjunction V if – then Conditional → if & only if Bi-conditional ↔

Statement Formulas : 

Statement Formulas Statements which do not contain any connectives are called simple or atomic statements. Compound statements derived from atomic statements (components) and connectives are called statement formulas. Truth value of a statement formula can be derived if truth value of its component atomic statements is known.

Well Formed Formula (WFF) : 

Well Formed Formula (WFF) A WFF can be generated by using following rules : A statement variable standing alone is a WFF. If A is a WFF then ┐A is a WFF. If A & B are WFF then A ^ B, A V B, A → B and A ↔ B are WFF. A string of symbols containing the statement variables, connectives, parenthesis is a WFF if it can be obtained by finitely many applications of rules 1, 2, 3.

Truth Tables : 

Truth Tables

Slide 12: 

Conjunction Operation P ^ Q P ^ Q

Disjunction Operation : 

P V Q P V Q Disjunction Operation

Negation Operation : 

Negation Operation

Conditional Operation : 

P → QP P ↑ ¬Q¬P Conditional Operation

Bi-Conditional Operation : 

P P ¬Q¬P Q¬P Bi-Conditional Operation

Slide 17: 

Tautology

Contradiction : 

Contradiction

Slide 19: 

For example, the statements it is raining and I am indoors can be reformed using various different connectives to form sentences that relate the two in ways which augment their meaning: It is raining and I am indoors. If it is raining then I am indoors. It is raining if I am indoors. It is raining if and only if I am indoors. It is not raining. If we write 'P' for It is raining and 'Q' for I am indoors and we use the usual symbols for logical connectives, then the above examples could be represented in symbols, respectively: P ^ Q P → Q Q → P P ↔ Q ¬P