 Welcome back in the last lecture we presented preliminary concepts of predicate logic where we introduce syntax of predicate logic so in this class I will be in continuation with the last class we will be talking about some of the basic building blocks of predicate logic the basic building block of blocks of predicate logic this is that predicates first thing to start with and you have objects and you have terms etc and all. So we will define all these things one by one in greater detail with some examples so this class is dedicated to syntax of predicate logic so what we will be doing is when do you say that a given well-formed formula is given formula is considered to be a formula well-formed formula or when do we say that a particular formula is considered to be a kind of formula and predicate logic etc and all so to start with these are some of the building blocks of predicate logic to start with we need to have some kind of objects there the things which exist in the world it can be cats dogs or anything we can give these names such as some will person John etc and all these things are referring to some kind of objects and we have also relations and properties relations between these objects such as the relationship between right Ravi and Sita is wife siblings etc and all. So we need to have relations and we need to have functions such as a type of relation that maps an input to some kind of value they are called considered to be functions we will start with usually we represent predicates with capital letters actually one what we need to know what note before this is that in the context of predict prepositional logic we represented all the sentences as some kind of prepositional variable so that is so for example if I represent X that means stands for maybe like it is raining etc so the compound sentences are formed by combining these atomic sentences which are represented by some kind of prepositional variables so our logical connectives and our implies negation etc will take care of I mean combining these simple sentences into a compound kind of sentences so in the predicate logic the story is little bit different so we need to go into the deep structure of sentences so where what is essential for us is the predicate and the relations the objects etc all these things are very important for us so all these things are important especially when you are trying to interpret a given sentence so sentences in predicate logic are not so simple like the ones which you have already seen in the case of prepositional logic suppose if you are asked to say if you are asked to find out the truth value of it is raining and it is not raining it is so simple whatever value that it is raining and not raining takes it is p and not p obviously the truth value of the particular kind of sentence is false because it is a contradiction but it is not so simple in the case of predicate logic predicate logic we need to go into the details deep structure of these particular kinds of sentences where we need to interpret especially predicates relation relations objects etc so the semantics of predicate logic we will talk about it in the next class but we will focus our attention on our basic building blocks to start with predicates the predicate letter will usually be associated with the list of at least one variable that is suppose if you have a x then it is considered to be a unary predicate and if you have let us say if you represent x and y x and y are considered to be variables and then a is considered to be a predicate so that is considered to be a binary predicate a can be some kind of relation etc suppose if you want to say in between x y z etc are in between x and y you need to have a tertiary kind of predicates b x y z so usually a predicate is used to represent a property of its variable or a relationship between its variables when you have a unary predicate it is talking about the property of a particular human being for example if you say socrates is mortal then mortality is a property which is attributed to the variable that is socrates here in this case is specific so you represent it as s and x for example if I want to represent all men are mortal then mortality is attributed to all human beings that is H x etc so a predicate is used to represent a kind of property of its variable or a kind of relationship between its variables in the case of father x y in a particular kind of order x is considered to be father of y suppose if you represent like this in the first case it is like this so now in this one x is a variable this is considered to be a variable and some kind of property is attributed to this one H is considered to be a predicate and then this is considered to be a quantifier so this is a structure of this particular kind of thing so a predicate can also be represented in this particular kind of way so now in the second case it is talking about the relationship between x and y so it can be less than or greater than y if you are talking about natural numbers suppose if you are talking about human beings you can say that let us say if I write like this there is a particular kind of order which is followed here so this means x is a wife of y if you write like this x is father of y so this is this is different from wife y and x if you change the order here order of your variables terms that is what you call it as terms representing some kind of variables if you replace this one like this these two are not the same so this is a kind of binary kind of predicate and then if you want to represent more than two then you represent it in this way x y z for example someone is sitting in between x y and z and in between this for example requires let us say more than three variables so like this you can go on and on you can talk about an energy kind of predicates and so as usual we have the connectives and or implies double implies negation etc. These are the same as in the case of propositional logic and they mean exactly the same that is n stands for or n stands for this upside down v stands for n and v stands for or and implies arrow stands for implies and if only if stands for this thing a bi-directional kind of connective and negation is as it is so we have unary binary and tertiary kind of predicates and different textbooks so it is represented in a different way so sometimes so you can write like this a x y and z are in some textbooks it is written in this way so they are more or less the same or in the same way for example L x y some kind of property L let us say L x loves y and all so in some textbooks it is written as love is considered to be the predicate which comes first we give a lot of emphasis to the predicates and these are the objects so they represent some kind of individual variables and when you represent replace this x and y with some individual variables then for example it takes value a and b a and b are referring to some specific kind of individuals like John Ravi Sita etc so these are the basic building blocks to start with after the predicates predicate consists of terms and all terms in particular and what do you mean by a term in the predicate logic so the definition is like this so we will be following some kind of definitions some of the definitions sound technical but I will try to explain or try to make it as simple as possible so now to start with what do we mean by a term so every variable is considered to be a term so any variable that you take into consideration so that is going to be a term so suppose if you represent this thing so this is considered to be a term or an individual constant like this can also be treated as a term okay so every constant symbol is also considered to be a term like a b c's etc so that is the thing so these are the preliminary things one need to note either if you write individual variables like x x stands for anything like Sita Gita Ramam etc and all an individual constant a that is also considered to be a term usually a predicate consists of terms so if f is a kind of energy function symbol that means n is equal to 0 to 1 to n and then t1 to tn are the terms then f t1 to tn is also considered to be a term so it is like a energy function means for example it is suppose if you are representing only two variables it is a binary function and all it is a relationship between x and y so these are considered to be terms and all so now we have t1 t2 tn are terms so these represents either individual variables x x yz etc and all this x yz can be replaced later or it can be individual constants like a b c etc and all specifically referring to individual objects like dust chalk piece etc so now these are considered to be terms and all and if f is energy function that means if it is a binary function if it is binary function then you have two variables x and y like this three means x y z etc fn means x2 you can take in this way x1 to xn if it is a energy function then these are also these are already considered to be terms then f of t1 tn is also considered to be a term so that is what it says so a term has to be defined at least in these three ways anything which is not defined in this kind of thing is not considered to be a term so out of this terms there are certain specific kind of terms which are considered to be ground terms so what are these ground terms these are the terms with no variables these are considered to be ground kind of terms so in the sense that you have already exhaustively replaced this individual variables with these variables with some kind of individual objects like x y y z are represent replaced by some constants like ABC they are referring to some specific individual objects and in that case these are considered to be variable free terms or they can also be considered as ground terms so in this case for example f of t1 t2 etc tn so now you replace these terms with some kind of individual variable individual constants like ABC etc then this is considered to be these terms are considered to be the terms which exist in that particular kind of energy function f are considered to be ground terms so sometimes you will have only one variable sometimes you can have more than one variable with one free variable etc now so now to make it more explicit a term is said to be closed or a ground term if it contains no free variables for example if t1 and t2 are considered to be term and x is considered to be variable we can use different kind of notation so a term t1 is such that t1 is like this a t2 is replaced x is replaced by t2 so that means t1 t2 oblique x it denotes a term resulting from the replacement of every occurrence of x in t1 by t2 so there are some examples which we take into consideration and we will try to make it clear so now let us say this is a notation that we are following p2 and x so you can so these are the terms in all t1 t2 and x is considered to be a variable so now so in this term t1 so t1 is resulted especially when wherever you have x that is replaced by another term t2 so then that is what is resulting by this particular kind of formula so this results in by replacement of every occurrence of x in this term t1 with t2 some examples which you can take into consideration for example if you have this particular kind of thing 3 x 2 y 2 x 3 x plus y and 2 x that means it stands for this thing that wherever you have x you are replaced it with 2 this is some of the notation that we can use so now this this is considered to be a formula in that you have a term x in stands for individual variable so now this x is replaced by 2 so now this will become where plus y let us for a time being you have a formula like this 3 x 2 y so now you replaced x with 2 so now this will become 2 square plus y so now this will become 12 plus y so this means wherever you have x you replaced it with 2 that is what it says so what is a 2 is in a kind of individual constant which you are replacing this y with x with 2 so then the formula will become 12 plus y in the same way in this formula 3 x square plus y you replace x with 2 y plus 1 so then this formula will become like this so wherever x is there you replace it with 2 y plus 1 whole to the power of 2 plus y so whatever formula which results in is considered to be the resultant term so a term is said to be closed if it contains no free variables so this is this tells us how to replace the individual variables with some kind of constants so now we spoke about what we mean by terms and when individual some variables can be replaced with some kind of constants in a term all these things which we spoke about it in a some kind of detail manner so now let us talk about what we mean by a formula in predicate logic so predicate logic is also considered to be termed logics just like you know in the case of Aristotelian logic in the case of classical Aristotelian logic it also talks about terms but it fails to explain those arguments in which it consists of complex terms are simple singular terms and complex terms and the terms which includes some kind of relations in all so predicate logic the Aristotelian logics in a way fall short of explaining these complex terms and singular terms in a satisfactory manner so in a sense predicate logic is also considered to be kind of term logics predicate logic may be even called as first order logic so we will talk about what we mean by first order logic little bit later so let us consider what we mean by saying that a given string of symbols are considered to be a formula so these are also considered to be assertions in the predicate logic the definition goes like this so we need to have some particular kind of language that is the language of predicate logic which consists of all the individual variables constants terms predicates etc and the functions etc the set of open formulas over L is given inductively by the following rules so let us consider for is a relation symbol of arity n that means unary predicate binary predicate etc and T1 to Tn are considered to be terms then R of T1 to Tn is also considered to be a formula so it is like P T1 T2 Tn is considered to be a formula is also considered to be atomic formula in the predicate logic so you have terms and you have a relation that is what you know for example if you can represent our T1 to Tn as simply w x yn where w stands for wife and x stands for Ravi and y stands for Sita for example Ravi is a Sita is wife of Ravi so w x y in that particular kind of order so we have T1 T2 Tn here so anything if R is a relation symbol which relates some kind of objects in your language and then our T1 T2 Tn is also considered to be a formula so as usual in the case of propositional logic if alpha and beta are two formulas alpha in plus beta alpha and beta alpha beta all these things are also considered to be terms and the third thing is that if alpha is considered to be formula and x is considered to be a variable then the only thing which is different in the case of predicate logic are these things so there exists some x alpha is also considered to be a formula these are the two things two quantifies that will be using in the predicate logic it is in that sense it is an extension of propositional logic anything which you extend it with two more extended extend the preposition logic with two more quantifies like for all x there exists some x it will become predicate logic so then there exists some x alpha or for all x alpha is also considered to be a formula these are the additional things which you find it in in the predicate logic and alpha is usually considered to be within the scope of your quantified example if you have so this essentially says that when you say that a given kind of sting of letters that I am going to write on the board is considered to be a formula so the first thing is that T1 T2 individual terms are also obviously considered to be formulas and all and anything which you write like this Tn if T1 T2 Tn are formulas anything which you write it like this F T1 T2 is also considered to be a term which obviously is considered to be formula then this usually we replace it with or sometimes you can even write it as P so now the second thing is that as usual in the case of prepositional logic if all five and beta are formulas and all these things are also considered to be formulas and negation of all five etc. This is the second thing and the third thing is that for all x so the other thing is that there are two quantifiers which you can use there exists some x alpha so this is considered to be quantified it talks about for all x alpha is alpha holds for all the property all the objects x so now alpha holds for some x so here this x stands for variable individual variable and alpha is within the scope of this particular kind of quantifier so that is what it says so here alpha is said to be within the scope of the quantifier the quantifier is in the case the first one it is for all alpha and the second case it is there exists some kind of alpha there is some x alpha so anything which is defined in the following three ways is considered to be a formula but anything which is defined not in this particular kind of sense is considered to be not a formula so this is just like in the case of prepositional logic we talked about some kind of formation to any string of formula is cannot be considered as a well-formed formula so like this we have in addition to these terms etc which are new here and we have two quantifiers for all x and there exists some x now we need to talk about what we mean what exactly these quantifiers are all about so some examples of formulas are like this so x is greater than y because it is greater than is property which is relating x and y so it can be written as r x y simply it shows that greater than is a sign which is represented by relation or a property or you can say for all there exists some x for all y x is less than y or you can simply write x as a particular kind of property that is the prime number so in that sense 17 is considered to be a prime number so prime 17 you can write sibling sibling can be sister brother anything x and y x is the brother of y y is a sister of x etc so if you want to represent anything between P1 P2 P3 that is B P1 P2 P3 so one can write it in the linear order or sometimes you can even write in this particular kind of thing suppose in some textbooks it is written like this P x y z and if I want to avoid this particular kind of notation I can write in this way also P x y z it means the same thing some textbooks they maintain this particular kind of notation in some other textbooks you will find the other notation so to continue with what we have discussed so far and the terms terms are either considered to be variables constants and functions which are functions which apply to terms that is f t1 t2 tn is considered to be a term just a summary of what we have discussed so far and variables are also considered variables are usually represented by small letters x y z usually these are replaced by terms t1 t2 a b c etc and all constants representing some kind of arbitrary objects from the universe like some people some man or some bright students of itk like that so it is referring to arbitrary objects within some kind of universe so you have to note that in predicate logic whatever we are trying to talk about makes sense only when it refers to some kind of domain or universe of discourse it does not make any sense if you do not have some kind of universe of discourse so for example something which is a which holds for natural numbers may not hold for real numbers etc are integers etc so we need to specifically talk about the domain that you are trying to talk about in the beginning of analyzing this sentences in predicate logic so constants are underlined letters such as a b c is etc representing particular object from the universe you want to refer to this particular kind of desktop it is this thing this particular kind of chalk this etc they are all referred by constants this is what we have discussed so far and functions are usually represented as again small letters except the predicates you will see all the things which you will find it in your vocabulary will have small letters f x y means f is a function which is applied to variables x and y so now sometimes there might be a lot of confusion between function and predicate the usage of function and a predicate so here is the distinction between a function and a predicate so the usually the value of a function is an object while the value of a predicate is considered to be a truth value so let us consider some examples so that this distinction will become clear so this is one of the important distinction that is the value of a function is considered to be an object whereas the value of a predicate is considered to be a truth value that means for example if you say Ravi is husband of Sita for example so this is usually represented as this thing so this is the predicate so being husband is considered to be a predicate and then let us say you write it in this particular kind of thing this says that x is a husband of y where x is considered to be Ravi we replace it replace it with individual terms then it will become this thing Sita Ravi is a wife husband of Sita so this is in particular kind of order suppose if x stands for Ravi and y stands for Sita and indeed these are considered with the states of efforts then obviously takes a value either E or F so that is what the predicate does in all the value of a predicate is always considered to be some kind of truth value either it has to be 0 or it has to be 1 for example if you say one Monsingh is let us say one of the prime minister of India so suppose if you say that particular kind of thing prime minister is considered to be the predicate and x is considered to be the prime minister of India x is referring to one Monsingh and y is referring to India then this is the property that is being the prime minister is considered to be the property suppose if you say some other name and then replace it with this one it takes the value F that means the predicate the value of a predicate is will take the value F so that the value of a predicate always takes some kind of truth value that is true or false suppose if you want to represent the father of x it makes sense to use a particular kind of function fx father of the nation etc it refers to some kind of object such as mathma Gandhi etc a prime minister of India is referring to one Monsingh some kind of object they can refer to f of x that is referring to some kind of object is mapping to some kind of object on the other hand if you want to say that x is a father then now it should take some kind of truth value so now you use px that is let us see if you want to represent this thing father of Javelal Nehru F of Javelal Nehru is referring to some kind of object I mean another object with which exist within your domain that is considered to be Mothilal Nehru we know the facts so that is why now predicate Mothilal for example that is p stands for is a father of Mothilal that can take only value either true or false if it refers to states of affairs that indeed Mothilal Nehru is considered to be father of Javelal Nehru and of course the statement is true which occurs in the predicate logic is true otherwise it is going to be false so this is the final distinction between the function and predicate we often make mistakes in using these things is in these things in our language that is function versus predicate is like this. So the fine the minute distinction that you need to note here is that the value of a function is always considered to be an object it maps to some kind of object whereas in the case of predicate the value of a predicate is always going to be some kind of truth value either it has to be true or false that is f of Javelal Nehru is referring to some kind of object that is Mothilal Nehru whereas p being a father of Mothilal whether or not he is a father of indeed if he is a father then the statement is going to be true otherwise it is going to be false the value of that one is either true or false that is the minute distinction between the function and predicate. So so far we discussed about some of the basic building blocks of predicate logic to start with we started with the terms and then predicates and now one of the important things that you need to discuss is the quantifies I mean are the quantifies so there are two quantifies that you will see in the predicate logic one is you want to quantify over the entire universe of discourse you use universal quantifier for all x or if you are trying to refer to some kind of individuals within your universe of discourse you refer to existential quantifier. So for all x is usually read as for that is the name for this for all x which is in the red so it claims that a formula that follows is true for all kinds of values x example if you want to say all human beings are mortal mortality is attributed to not a single human being but it is referring to the whole group of individuals that exist in the given domain the domain here is this the domain of universe of discourse which consists of all human beings of course if you refers to animals and that thing is going to be maybe that might also be true but if you are referring to some other kinds of domain maybe it might be false so there exist some x is usually read as there exist an x it claims that a formula that follows anything after this quantifier is going to be true for at least one value of x so for example if you say that all some IIT k students are intelligent then it refers to only if some IIT k students are intelligent means at least one is intelligent that itself shows that some are considered to be intelligent so that satisfies this particular kind of sentence if you have more it is good enough but definitely want to make a distinction between all IIT k students are intelligent and some students are considered to be intelligent so that is a difference between existential quantifier and the universal quantifier so there is some kind of important we need to maintain this particular kind of distinction and there are some efforts to say that whether for all x a x is true implies that there exist some x a x are not so these are the kind of inferences that a troublesome kind of inferences and all so usually our temptation is this that for example if you say for all x a x and from this can be infer there exist some x fx or the other way round if you have something like there exist some x fx from this can be generalized it and say that it is for all x fx whether these can be considered to be rules of inference and all these are things which will talk about later but there are lot of issues which are surrounding these particular kinds of inferences even Aristotle was also talking about this particular kind of inferences suppose if you infer something like for all unicorns all you all unicorns are intelligent and this stands for some unicorns are intelligent from that you obtain this particular kind of since intelligence is attributed to all human beings maybe some are also considered to be intelligent but once you infer this particular kind of thing there exist some x here the problem here is is that you are importing existence into the conclusion which is not there in the premises in the premises the there is no commitment for the existence of the unicorns here but you can still use in an abstract terms all unicorns are intelligent etc in the premises but still you not have to have unicorns to exist in the world but here once you say that there exist some x fx and there is some kind of we imported existence which is not there in the premises to the conclusion and that kind of inference is considered to be a kind of fallacy which Aristotle talks about it as existential and in the modern terms and the Boolean terms it is considered to be an existential fallacy so how they resolve this existential fallacy and all we will talk about later but so this is the minute distinction between for all x and there exist some x and let us consider some examples so that we will understand this concept in a better way so for all x x into 0 is 0 that means any term multiplied by 0 is going to be 0 that happens for any number that you are going to take into consideration whether it is a real number or complex number or any number that you take into consideration so now for this whether or not this statement is true etc and all we require a domain either you need to talk about natural numbers or real numbers or integers rational numbers etc and all within the domain you take x as some kind of number if you are talking about natural number you take 1 2 3 etc and all then for all x any x you take into consideration if this property x into 0 is 0 holds for all the letters that you have exhaustively listed out in that case it happens it holds for all the numbers all the members of x x time 0 equals to 0 so the in that case it is to going to be true we will talk about truth and falsity a little bit later but right now we are trying to talk we are focusing our attention on how we are using the quantifies so we are we did not enter into the details of what we mean by this quantifies the semantics will take care of what exactly we use this quantifies how we use this quantifies etc we will talk about it when we discuss about the quantifies second sentence for all x there exists some y x multiplied by y is equivalent to 1 so now your domain is real numbers here and there is a two real numbers x and y and if you multiply these two real numbers x and y that is it so happened that the multiplication leads to 1 and this holds for for all x at least for all x means you take 1 2 3 etc and all for all these elements and all there exists at least some y that y is also another real number for all these things there is some y and all for that if you multiply some of the things which are there here with some of the element here then you will generate a number 1 whether it is true or not we will talk about a little bit later but this this is what it conveys the information that it conveys is this thing for all x there exists some y if any number x is multiplied by y whatever number randomly you take into consideration and then they you choose another y from y which happens to be another real number you multiply it and it is the multiplication leads to 1 the some examples of quantifies so now one example could be like this we need to talk about the domain of discourse usually in the case of natural in the case of numbers you have to define it properly it can be real numbers it can be rational numbers complex numbers etc so now let us consider natural numbers as your universe of discourse so let ? x y denotes a kind of relation all between x and y that is x is less than y so now we have a function x which relates x and y where x is constable domain and y is a range which is dependent in terms of order pair x y that is considered to be a binary function which talks about this thing x plus y and the ABCs are considered to be constants naming some kind of numbers suppose if you are referring individual number and all that is 0 stands for a maybe 1 stands for B 2 stands for C etc and all like that you go on and all if you exels exhaust this constants and all you can use a1 a2 a3 etc so now in that context you given the universe of discourse natural numbers and then we have a relation x less than y and you have a function that is x x plus y etc so here x is less than y less than easy is considered to be a predicate that is a property which is on this two objects so these two objects are related by a predicate being less than so now there exists some x ? x y it is considered to be an unary predicate which says that which says of y that there is at least one natural number less than it where y is not equivalent to 0 so you take any numbers for example 1 2 3 2 n and all you take any number other than 0 let us say you take 1 that is obviously less than any other element that is 2 etc are in the same way 2 is less than 3 3 is less than 4 etc so now in the second context for all x there exists y ? x y starting that for any natural number x there is a natural number y which is greater than x it is in the context of the first one x is less than y the same thing is represented in terms of this thing suppose if you want to represent that for any natural number that you take into consideration there is always a natural number y which is greater than x so that is represented by this thing for all x at least there is some y such that ? ? he has stands for less than y and all ? he stands for a property being less than something x is less than y for all x there exists some y x is less than y is represented by ? x y so this is the way we represent this thing in this one for all x there exists some y are considered to be quantifier and ? stands for a predicate and x y stands for individual variables when it refers to individual constants like the numbers like 1 2 3 4 etc and all then it takes some kind of value so far we discussed about terms predicates etc and the quantifiers then given any formula in particular we can talk about to what extent this quantifier operates so that is considered to be the scope of the quantifier so depending upon the scope of the quantifier we can say that the variables exist in your formula are considered to be either free or bounded this is what we mean by bond bondage and freedom for example if you have for all x ? he is a sub formula of another formula ? ? is a big formula like a x or y or z etc and all in that you have taken into consideration the part of it and all that is some considered to be a sub formula of ? then ? is called as the scope of a particular occurrence of a quantifier for all x ? so that is considered to be the scope of it the same applies to the occurrences of when you are talking about there exists some x so you have a formula in that formula only the quantifier operates over some kind of sub formulas not over entire formula and all but it operates over some kind of sub formula so its scope is still that extent on so beyond that there is no the other variables that exist there are going to be free variables so we will talk about the scope of the quantifier with some examples and we will talk about what we what is the significance of this one so some examples which we can talk about so what we mean by first of all what we mean by free and bound variables and occurrence of an individual variable individual variables means x y z etc and all they are considered to be bound within the scope of a quantifier if and only if it is within the quantification within the scope of a quantification expression that contains the individual occurrences of that kind of individual variables an occurrence of variable is considered to be free if and only if it is not considered to be bound now let us consider one example so that we will understand what we mean by scope of a given quantifier so the example that is there here is like this for all x fx and Cy some terms which are represented in this way so there is one bracket here implies so I am just listing out one formula and then we are trying to see what is the scope of the quantifiers the quantifiers here are for all x and there exists some y z y there exists some y z y and hx some strange randomly also you can take into consideration m z so this is closed by this thing by this so these brackets are very important so this quantifier is operating or till this extent so this is till here and then we are closing this one with this and this is closed by this that is why there are three brackets here so now let us assume that this is the formula and all now we are trying to talk about the scope of the quantifier so now what are the variables that exist here x y and z these are the individual variables that you will see in this particular kind of formula so now these are considered to be quantifiers so now in this formula so this is over the whole thing so the occurrence now we need to talk about sub formulas so what are the sub formulas this is one and this can be one more and this whole thing can be said to be another kind of sub that means fx and cy can be one sub formula and there exists some y and all these things this can be one sub formula of the main formula or it can be only simply this particular kind of thing so now with respect to the universal quantifier for all x y and z are considered to be free because this is this bracket is till this extent so now with respect to this particular kind of quantifier so this is considered to be bounded because this is within the scope of this particular kind of quantifier and even this is also in the second second sub formula also this is bounded by this particular kind of quantifier so now what are considered to be free with respect to this universal quantifier these are the variables that are considered to be free with respect to this thing even then the second sub formula also so these z is also considered to be free with respect to this particular kind of quantifier the first y and z are considered to be free whereas the rest of the individual variables are considered to be bounded with respect to both the quantifiers that is for all x are there exists some way so now with respect to x the first occurrence this one this and z are considered to be free so now fx is bounded by for all x here y is free here and then with respect to this particular kind of quantifier and y is bounded by this thing and in with respect to this individual existential quantifier x and z are considered to be free so now we can list out this thing so it is like this so what are considered to be free here are these things in the first occurrence in this particular kind of formula x is considered to be free sorry next is considered to be bound in the second sub formula this second sub formula so this is considered to be bound and all but in this case y in the first occurrence y is free first occurrence is first sub formula y is considered to be free in this case y is considered to be bounded but in the second occurrence of this quantifier for all there exists some way x and z is also considered to be free so in this sense one can find out when we can say that a given quantifier binds the individual variable so in this example the first y and z are considered to be free first wise it means the moment you start reading the formula from the left y and z are considered to be free and in the second case in the second sub formula y is bounded whereas x and z are considered to be free and rest of the individuals are considered to be bound so in the same way one can talk about this particular kind of thing we can talk about some more example like the quantifier for example in this case are the three quantifiers there it does not exist that some x there exists some y for all z is a complex kind of thing you know there exists some w a z w implies say y z and a x y this is let us say this is considered to be a complex kind of formula with this kind of thing I will end this lecture so now here what are the quantifiers there exists some x there exists some y and there exists some w are considered to be the quantifiers to start with a z w is the one which you are seeing in this formula is bounded by the quantifier there exists some w so now the scope of that one is a z w so now with respect to for all z entire thing is considered to be bounded that is there exists some w a z w implies a y z that is considered to be bounded so with respect to the quantifier there exists some y and whatever follows after that one in the brackets that is for all z there exists some w a z w implies a y z and a x y that is considered to be within the scope of there exists some y now with respect to there exists some x and whatever follows after that one is considered to be within the scope of that particular kind of thing so this is the way in which the quantifier operates over a given formula so later we will see the significance of why we are talking about why we need to know that a given formula is bound with respect to a formula or when why we need to know that a given formula is free so in this class I discussed about some of the basic building blocks of predicate logic and we started with predicates which consists of some terms etc and we defined what we mean by a term and then I introduce two quantifiers and I did not discuss about the relationship between these two quantifiers this one quantifier can be defined with respect to another one can reduce to the other one so the other thing which I discussed is when do you say that given variable is considered to be bound it is considered to be free so we need this information especially in knowing the validity of a given formula or a given formula is considered to be a formula is considered to be a well-formed formula especially or a sentence in the predicate logic especially when it is bounded etc so we need to know some information about whether or not a given variable is bounded or not to have some kind of sentences in the predicate logic we will discuss about we will continue with the syntax in the next class and we will also talk about the semantics of the predicate logic what we mean by saying that for all x a x y is going to be true etc so these are things which we are going to discuss in the next class.