Statement or Proposition : A statement or proposition , is a declarative sentence that is either true or false , but not both. those declarative sentences will be admitted in the object language which have one and only one of two possible values called “Truth value”. The two truth values are true and false and are denoted by the symbols T and F respectively , Occasionally they are also denoted by the symbol 1 and o. Note that we do not admit any other types of sentences , such as Exclamatory ,Interrogative in object language. Declarative sentences in the object language are two types , The first type includes those sentences which are considered to be primitive in the object language.This will be denoted by distinct symbols selected from the capital letters A , B , …. , P ,Q ….

Second type are obtained from the primitive ones by using certain symbols ,called connectives , and certain punctuation marks, such as parentheses , to join primitive sentences. In any case , all declarative sentences to which it is possible to assign one and only of the two possible truth values are called Statements . These statement which do not contain any of the connectives are called Atomic (Primary Primitive) statement.1)Canada is a country. 2) Moscow is the capital of Spain.3)This statement is false.4) 1 + 101 = 1105) Close the door.6) Toronto is an old city.7)Man will reach Mars by 1980.

Statement (1) and (2) have truth values true and false . Sentence (3) is not a statement according to our definition , Because we cannot assign to it a definite truth value. If we assign the value true then statement (3) is false. if we assign the value false then statement (3) is true. Statement (4) is also statement but depends upon the context ; we are consider this sentence in decimal system then it is false , and if we consider in binary system then it is true sentence , so sentence (4) is statement.Statement (5) is not statement because it is interrogative statement. statement (6) is consider true in some part of the world and false in certain other parts. In statement (7) could be determined only in the year 1980 or earlier if a man reaches Mars before that date.

Connectives : in the case of simple statements ,their truth values are fairly obvious. it is possible to construct rather complicated statements from simpler statements by using certain connecting words or expressions known as “sentential connectives”. The statement that we consider initially are simple statements, called atomic or primary statements. New statement can be formed from atomic statements through the use of sentential connectives. The resulting statement are called molecular or compound statements.Thus the atomic statements are those which do not have any connectives. Capital letters are used to denote statements.

The capital letters with or without subscripts , will also be used to denote arbitrary statements.In the sense , a statement “P” either denotes a particularstatement or serves as a placeholder for any statement . This dual use of the same symbol to denote either a definite statement , called a constant , or an arbitrary statement called a variable.The truth value of “P” is the truth value of actual statement which it represents.It should be emphasizes that when “P” is used as a statement variable ,it has no truth value and such does not represent a statement in symbolic logic.

Negation : The negation of statement is generally formed by introducing the word “Not” at a proper place in statement or prefixing the statement with the phrase “It is not case that” and read as “not P”. Let P be a statement. The negation of P ,written P or P is the statement obtained by negating statement P. If the truth value of P is true then truth value of P is false , and If the truth value of P is false then truth value of P is true.This definition of Negation is summarized by truth table given below.

P P

T F

F T

Example :1)P : London is a city

P : It is not case that London is a city P : London is not a city.2) P : I went to my class yesterday P : I did not go to my class yesterday. P : I was absent from my class yesterday. P : It is not the case that I went to my class yesterday.Note : negation is called connectives although it only modifies a statement or a variable.

Conjunction : Let P and Q be statements .The Conjunction of P and Q ,written P Q , is the statement formed by joining statements P and Q using the word “and” .The statement P Q is true if both P and Q are true ; otherwise P Q is false. The symbol is called “and” . Let P and Q be statements . The truth table of P Q is given by : P Q P Q

T T T

T F F

F T F

F F F

Example :1)P : 2 is an even integer , Q = 7 divides 14. R : 2 is an even integer and 7 divides 14.2)P : It is raining today , Q = There are 20 tables in this room. R : It is raining today and there are 20 tables in this room.3) Jack and jill went up the hill.Form this statement we get two statement jack went up the hill and jill went up the fill.If we now write P : Jack went up the hill , Q : Jill went up the hill.Then the given statement can be written in symbolic from P Q.

Disjunction : Let P and Q be statements . The Disjunction of P and Q , written P Q , is statement formed by putting statements P and Q together using the word “Or”. The truth value of the statement P Q is T if at least one of statements P and Q is true. The symbol is called “Or” , For the statement P Q is given by :

Example : 1)P : 22 + 33 is an even integer Q : 22 + 33 is an odd integer Then P Q : 22 + 33 is an even integer or 22 + 33 is an odd integer OR P Q : 22 + 33 is an even integer or an odd integer.

P Q P Q

T T T

T F T

F T T

F F F

Conditional or Implication : Let P and Q be two statements. Then “If P , then Q” is statement called an Implication or conditional statement ,written P Q. The statement P Q has a truth value F when truth value of P is true and Q is false. Otherwise truth value of conditional or implication is T. The statement P is called the antecedent or hypothesis and Q is called consequent or conclusion in P Q.According to the definition , it is not necessary that there be any kind of relation between P and Q in order to form P Q.The statement P Q is also to be read as P implies Q Or P is sufficient for Q Or Q if P Or Q whenever P.

In the implication P Q ,p is called the hypothesis and q is called the conclusion. The truth table of P Q is given by :

Example : 1)If today is Sunday ,then I will go for walk.” Let P : Today is Sunday q : I will go for walk

p Q P Q

T T T

T F F

F T T

F F T

Biimplication and biconditional : Let P and Q be two statements , Then “P if and only if Q” , written P Q is called the Biimplication or biconditional of the statement P and Q. The statement P Q may also read as “ P is necessary and sufficient for Q” or “Q is necessary and sufficient for P” , or “Q if and only of P” or “Q when and only when P”.We define that the Biimplication P Q is considered to be true when both P and Q have the same truth values and false otherwise. It is also denoted by P Q . The truth table of Biimplication is given below.

P Q P Q

T T T

T F F

F T F

F F T

Let P and Q be statements.1)The statement Q P is called the converse of the implication P Q.Example : P: Today is Sunday Q : I will go for walk.Converse of P Q : If I will go for a walk ,then today is Sunday. 2) The statement P Q is called the inverse of the implication P Q.Inverse of P Q : If today is not Sunday then I will not go for a walk.3) The statement Q P is called the Contrapositive of the implication P Q .Contrapositive of P Q : If I will not go for a walk , then today is not Sunday

Statement Formulas (Formulas) Those statements which do not contain any connectives are called atomic or primary or simple statement.Those statements which contain one or more primary statements and some connectives are called Molecular or composite or compound statements.Let P and Q be any two statements . Some of the compound statements formed by using P and Q are P P Q (P Q) (P) P (Q) the compound statements given above are statement formulas derived from the statement variables P and Q.P and Q may be called the components or statement variables of the statement Formulas.The symbol , , , , are called logical connectives.

In the construction of formulas, The parentheses will be used in the same sense in which they are used in elementary arithmetic or algebra or sometimes in a computer programming language.This usage means that the expressions in the innermost parentheses are simplifies first.If there are n distinct components in a statement formula , we need to consider 2n possible combinations of truth values in order to obtain the truth table.In statement formula without parentheses that contains logical connectives , the logical connectives are evaluated in the following order ; i.e. the precedence of logical connectives is :

, , , ,