 discussing fundamentals of logic now in logic we will be dealing with statements rather than numbers now a particular type of statements are called propositions let us define propositions more formally any declarative sentence to which it is meaningful to assign one and only one truth value true or false is said to be a proposition now we have to be careful that each and every sentence is not a proposition the key feature of a proposition is that it has to be either true or false and if a proposition is not true then it is false and if it is not false then it is true let us look at one example of a proposition we have written down three two propositions so first proposition is denoted by P and the second proposition is denoted by Q now the proposition P states that there are seven days in a week now this is of course a sentence and this is true so we know that this cannot be true and false at the same time and we know that it is meaningful to say that it is true thus P is a proposition the next sentence is Q which states that a week has more number of days than a month now this sentence is obviously false but it is a proposition because it is a declarative sentence which has a truth value which in this case is false let us look at another sentence we denote by R which is an interrogation what is your name now this sentence cannot have a truth value because that is meaningless and hence this sentence is not a proposition therefore we see that all propositions are sentences but all sentences are not propositions another thing that we notice here is that we can write propositions we can denote propositions by symbols like P Q and R we generalize this fact and introduce propositional variables a variable which can take propositions as values are called propositional variables we will denote the propositional variables by small letters P Q R and so on now if we have some basic propositions we can connect this propositions by using the so-called logical connectives and build up compound propositions the basic connectives are 5 in number we will note down these 5 basic connectives the first connective is called conjunction it is also called and and denoted by a wedge the second connective is called disjunction it is also called or and denoted by V the third connective is not or negation and it is denoted by either an over line or a tilde or a symbol like this prefixed before a propositional variable for conditional this is also known as if dash then dash it is denoted by an arrow like this or sometimes an arrow like this the fifth one is by conditional this is also known as if and only if or just if with two F's it is denoted by a both sided arrow or a both sided arrow like this we will now one by one check the effect of these connectives to the propositions one conjunction suppose P and Q are two propositions the conjunction of PQ is the statement P we denote this compound statement or this compound proposition as P wedge Q now since this is a proposition we must know definitely when it is true and when it is false P wedge Q is true only when P and Q are true otherwise P wedge Q is false now we can translate these things to a table which is called a truth table and which is very useful in understanding these connectives and more complicated compound propositions we write the propositional variables PQ and also write the statement P and Q the possible values of PQ are FF that is false false T FFT and TT P conjunction Q or P and Q will have the truth values F F F and T here T means true and F means false this table is called a truth table and this specifies the truth values of the compound proposition P and Q next we move on to disjunction again we take two propositions P and Q and P or Q is called disjunction of PQ or simply P disjunction Q which is denoted by P V Q now this statement is true only when at least one of P or Q is true so we write now we go to the truth table of P or Q the third connective is called negation this is a unary operation that is it involves only one propositional variable suppose P is a proposition negation of P or simply not of P denoted by either over line or tilde P or this is a proposition which is true P is false and false when P is true the corresponding truth table will look like this we have P false and true and negation of P true when P is false and false when P is true next we have conditional now this is called in common delanguage as if then so if something then something else the proposition P implies Q denoted by P arrow Q is if Q is true whenever P is the truth table of P implies Q is like this now when P is true and Q is true that means that P implies Q is true now when P is false then I cannot prove that P implies Q is false because if we have to prove that P implies Q is false we have to show one instance where P is true but Q is false since we cannot prove that P implies Q is false it will have the truth value true so in case P is false in both these cases P implies Q is true whereas if P is true and Q is false that means that P implies Q is not true because P implies Q forces Q to be true when P is true therefore we will write false over here and of course when P is true and Q is true P implies Q is true the last conjunction last connective is called by conditional now if P and Q are propositions P if and only if Q denoted by is called by conditional and P if and only if Q is essentially conjunction of P implying Q and Q implies P now if we consider the truth table of by conditional we will have P Q taking all the possible values and P implying Q is true true F and true Q implying P is true F true and true and therefore the by conditional which is conjunction of these two propositions is true F F and T now next we will take propositional variables and use these logical connectives to build up compound statements or compound propositions now let us see examples of that suppose we have propositional variables P Q R consider a compound proposition F P Q R P and Q or not of R now these type of expressions will be called propositional functions we can find out the truth table of these propositional functions for example for the one that I have written just now we can build up the truth table in this way we write the propositional variables then we start writing the terms P and Q and are not now all the possible we list down all the possible truth values of P Q R which are F F F F F F F T F T F T T F T F T F T F T F T T now first column at the right hand side is P and Q which is true if and only if both P and Q are true therefore here it will have F the next row also F then F F F F F T T and R not is T F T F T F T F T F now in the last column we will calculate the function F P Q R which is equal to P and Q or not of R this is T F T F T F T and T once we have discussed the propositional functions which are also called compound statements or compound propositions then we introduce the idea of equivalence of propositional functions propositional functions logically equivalent if they have the same truth table let us give an example we start with two propositional variables P and Q and consider two propositional functions P implies Q and not of P or Q let us draw the truth table of each of them so P implies Q we have already seen that it is T T F T and not of P or Q is let us look at the first row P is F over here therefore not of P is T so the R will be T similarly the next row the R will be T this one is false because not of P is going to be false and Q is false and lastly it is going to be true T therefore we see that P implies Q and not of P or Q both have the same truth table therefore these two statements are equivalent we write equivalent statements as P implying Q equivalent to that is we write three parallel straight lines equivalent to not of P Q further we observe another point if we consider the biconditional between these two equivalent statements that is P implying Q biconditional not of P or Q we will see that this will be always true now we introduce one more notion that is of a tautology if there is a propositional function which is true irrespective of the values of the propositional variables involved in it then it is called a tautology from what we have discussed it is not difficult to see that if two propositional functions are equivalent then if we form another function by connecting these two propositional functions by a biconditional then that resulting function is going to be a tautology now if we have a propositional function which is never true then it is called a contradiction or an absurdity and the usual propositions which are sometimes true and sometimes false are called contingency by this we come to the end of today's lecture thank you