 discussions on first order logic so far whatever we have discussed involved propositions and some logical connectives. Let us recall a declarative sentence which is either true or false is called a proposition now we denoted propositions by lower case letters p q r and so on we also introduced logical connectives which are used to combine several propositions to obtain compound propositions and sometimes we call those compound propositions as propositional functions assuming that the original propositions are in fact variables so we introduced compound propositions which are sometimes referred to as compound statements we also introduced propositional variables which we denoted by p q r and so on and propositional functions now the logical connectives that we introduced are conjunction disjunction negation conditional and at the end by conditional conjunction is also called and denoted by a wedge disjunction is called or denoted by a v negation is either denoted by over line or a tilde or a symbol like this conditional by a right arrow or a right arrow like this and a biconditional as a two sided arrow or a symbol like this we also introduced another idea of logical implication that is an implication which is always true and rules of inference now a logical inference involves are of two type valid and invalid so valid inference and faulty inference now whatever logical system that we have discussed based on this and the proofs methods of proof all together is called propositional logic or 0th order logic now what we see at this point is that by using propositions simple propositions and this framework of propositional logic we cannot express everything that we would like to express that is why we introduce something which is more general than a proposition which is called an open proposition or a predicate we will soon discuss what we mean by a predicate but what happens is that with these more general form of propositions we can use the logical connectives and inferences and methods of proof and build up a more powerful logical framework which is called the first order logic or predicate logic now first let us look at what we mean by a predicate now we often have some propositions like this 3 by 4 is a rational number we can have another proposition like half is a rational number yet another as root 2 is a rational number now what we note over here that in all these cases we are considering rational numbers and we are putting some numbers in the beginning and making a statement that that number is a rational number let us see this again 3 by 4 is a rational number half is a rational number root 2 is a rational number now what we see over here is that we are considering numbers and we can probably fix our discourse to a set you of real numbers we are picking up numbers from you and asking and stating that that number is a rational number sometimes this proposition is true and sometimes a proposition is fault depending on the number that we have chosen for example in this case the first proposition 3 by 4 is a rational number is true half is a rational number is true but root 2 is a rational number is false we can write this in a more compact way as stating that our universe of discourse is you the set of real numbers and we are building up a statement x is a rational number where x can take any value from you we can also write this as px now we see that the px that I have written here is not a proposition it is not a proposition because it is not meaningful to specify a truth value to this statement if we do not fix x so x is a rational number where x may take any value from you sometimes it will be true and sometimes it will be false however if I keep on putting values of x from you then I will get propositions which has got specific truth values so this px can be thought of as a function from the set you that is the universe of discourse that we have to fix before we start any discussion and to the set containing two symbols t and f and we can suppose that t designates true and f designates false so here we are looking at functions from you the universe of discourse to the set f, t and this function is p which can take the values x and the so the function is x is a rational number when I put that then the truth value of that will be f or t and this is how I am thinking p as a function now it is quite possible that I have got more than one variables varying over a universe of discourse for example let us consider the universe of discourse to be r cross r where r is the set of real numbers now we consider a proposition an open proposition or a predicate of the type s x, y x plus y equal to 5 so here the sentence is x plus y equal to 5 where x, y varies over the universe of discourse or simply the universe u so if we want to formalize s will be associated to a function from u cross u to the set f, t this is also a predicate now we are in a position to define predicate in a general framework so a predicate or an open proposition in n variables from a set u is a function f from u to the power n which is essentially the Cartesian product of u taken n times to a two symbol set tf as where t stands for true and f stands for false the set u is called the universe of discourse or simply universe of the predicate now let us look at some examples of predicates now we have all already seen that rx, x is a rational number is a predicate gy y greater than 5 is also a predicate 3 s x, y x plus y is equal to 5 this is also a predicate now here we have to remember that whenever we talk about predicate we have to be careful in what we are assuming as the universe in case of rx the universe can be rational numbers it can be real numbers it can be complex numbers and so on in case of gy these are also we see that it is tacitly assumed that it is a number y is a number and in case of x the second one it is also assumed or understood that x y are going to be numbers it may be real complex or any other numbers on the other hand we can have predicates like this ex, ex climbed mount everest here we see that it is of course possible to have the universe as numbers but it is not going to be very meaningful because if I say that x varies over real numbers then no real number has ever climbed mount everest so it makes no meaning in this case the universe might be the set of all human beings and among those set of human beings there are some who climb mount everest and of course many others who have never climbed mount everest like this we can have other predicates let us look at cy x is a lawyer and so on we can have many other predicates like this once we have understood the definition of predicate we will move on to a definition of another very useful notion which is called a quantifier we observe that declarative sentences use words indicate quantity such as some none one etc for example we have statement like all human beings have two legs or we can say at least one human being has two legs or there is no human being without a head here we are seeing that our universe of discourse is the set of all human beings and we are sometimes saying that all of them has two legs or at least one of them have two legs or none of them is without a head so these are somehow indicating some quantities of human beings having something now this idea gets formalized to two specific quantifiers which are called universal quantifier and existential quantifier quantifier all is called the universal quantifier it is denoted by the symbol inverted a if u is our universe of discourse then the phrase for all x belonging to you or simply for all x we will mean well we write this as for all x in you so this inverted a x belonging to you or inverted a x will designate the phrase for all x in you or if you is already specified and understood we may simply say for all x now suppose px is a predicate we can construct a proposition denoted by this which will read as for all x I am writing within bracket u which we usually omit while writing px is true now note that for all x in you px is true now we note that this although px is a predicate for all x px is a proposition this is true if for each and every x in the universe of discourse px is true otherwise this proposition is false moving on from the universal quantifier we come to existential quantifier suppose px is a predicate we can construct a proposition there exists x in you such that px is true this phrase there exists is written in a compact notation as there exists x in you or simply there exists x and the complete proposition is written as there exists x px this symbol which is which signifies there exists is called the existential quantifier now once we have these quantifiers and predicates and of course the universe of discourse we can form several sentences now let me list down some sentences before I end this lecture and in the next lecture we will look at these sentences more carefully and more in more details so we will have sentences like this for all x fx this is essentially all true then we have things like there exists x fx this means at least one true now I can have negation of this that is negation of there exists x fx this means that there exists x in you such that fx is true negation of that so this means if this is true then that means that none true for all x not of x negation of fx this means all false then there exists x such that negation of fx this means at least one false then negation of there exists x negation of fx this means none false negation of for all x fx this means not all true and lastly negation of for all x negation of fx this means not all false in the next lecture we will consider sentences of these types and in more details this is the end of the present lecture.