 Welcome back in the last lecture we discussed something about the basic building blocks of predicate logic where we introduced various things such as what you mean by a predicate what you mean by a term what you mean by a functional symbol etc and on. So all these things which we have discussed in the last class so in this class another major building block of predicate logic apart from predicates etc terms relational symbols etc and on the quantifier so in fact predicate logics are also called as quantification logic or it is also called as another name for this one is the first order logics. So in this class what we will be doing is we will be discussing something about what we mean by quantifier why we need these quantifiers and then we talk about some of the important laws of this quantifiers and then we will discuss some other important things which come under the category of syntax the syntax of predicate logic. So basically we are in the syntax of predicate logic basically we are discussing about some of the building blocks of predicate logic. So let us start with what we mean by a quantifier so there are two quantifiers that we use in the predicate logic the first one is called as universal quantifier it is represented as for all x the symbol for this one is written in this way for all x it is read as for all x so it claims that the formula that follows is true for all values of x for example if you say all men are mortal mortality is attributed to all the human beings so that mortality is attributed to all the human beings so whatever follows after the quantifier is that formula is going to claim that we are going to claim in that formula that that follows what follows after the quantifier is true for all values of x for example if you say all human beings are happy this is simply represented as this thing for all x h stands for the human being and x stands for any individual human being and then we have some kind of domain so that is universe of discourse we usually call it as Ud means universe of discourse which usually consists of human beings of people and then this stands for a quantify and this stands for usually predicate so being mortal being happy is considered to be the predicate and that is attributed to a one single variable x and if that happens for all the human beings then we represented in this way for all x hx so another quantifier that we will be using every very often frequently is the existential quantifier so it is represented as there exists an x it claims that the formula that follows after this quantifier is true for at least one value of x so that means suppose if you say that at least once van is considered to be usually fans are white in color suppose if you find if you figure out figure it out in such a way that figure it out in a way that the span that you looked at is considered to be black so you want to say represent that particular kind of thing usually represented in terms of there exists some x such that that particular x is considered to be a span which is considered to be black. So now assuming that the universe consists of real numbers so depending upon what you take into consideration the formulas represented by quantifier changes in our formula is going to be true sometimes the same formula is going to be false some other occasion for example if you consider the universe of discourse as real numbers then suppose if you want to represent this particular kind of sentence for all x x multiplied by 0 is equal into 0 any real number x which is multiplied by 0 obviously will give you 0 so x into 0 is equivalent to 0 that happens for all x whatever x that you are going to take into consideration which falls within the domain of real numbers the proper that x into 0 is equal to 0 holds for all x and so that is the reason why we wrote it in this way for all x x multiplied by 0 is equivalent to 0 so this is one way of representing this particular kind of thing again if you consider the same universe of discourse as real numbers and the other thing which you can represent it in this one is forever for all x there exists some way that means at least one way such that x multiplied by y is equivalent to one so here we have used two quantifiers the first one is considered to be the universal quantifier the second one is the existential quantifier I am just stating that you know these are the some of the things which you commonly come across I mean later we will be talking about the translation part where we will be talking about how to translate the sentences appropriately into the language of predicate logic little bit later. So right now there are two quantifiers that we need to study in detail because predicate logics are called as quantificational logics so usually we represent it in this particular kind of thing suppose if you want to represent universal quantifier it is considered to be like this for example if we have this particular kind of thing a x in place B x let us say all human beings are mortal where a is represented as human beings and B is represented as mortal so now let us consider that there are only two individual human beings it makes sense to talk about simple formulas in this way for example this can be written as a a implies B a where x is x is replaced by a and a B implies B so if there are only two this is universe of discourse is considered to be people all people are human beings in that we have taken into consideration only two human beings A and B let us say Aristotle and Socrates so if you want to represent this particular kind of thing because there are only two people in this universe I mean the domain so we can represent this formula simply as this one but when the number increases in all C D E F etc and all there is no way in which you can represent it in this particular kind of form because this string will go on and on and all so in order to represent this particular kind of thing this property is going to be true for all x then we require this particular kind of universal quantified so that means here another thing which you need to note is that if any one of this thing is false then the whole this particular kind of formula is going to be automatically false that is as good as saying this particular kind of thing suppose you want to say that all crows are black I mean you represented it like this you found a crow white black crow a if that is a crow a is a crow then a has to be black in the same way you found another crow you are naming it as B B is considered to be a crow and B has to be black like that it goes on and on and on suppose if you find an instance where for example a is false so the third instance you found a white crow then you cannot say that for all x if x is a then x has to be black black because that particular kind of thing is false even if one instance is false you cannot represent it in this particular kind of way usually mainly last statements in science are usually represented as universal generalizations but you need to note that all universal generalizations have its own has its own exceptions so so this is the way in which you represent the universal quantifiers whereas existential quantifiers same thing AX and BX so this is for example if you have again to we are taking into consideration two people from this universe of discourse that is n number of people are there out of that we have selected only two and I assume that there are only two human beings existing in this world for example then you represent this formula as represent this formula in this way a a and b a are a a a b and b b so here it is a conjunction here it is a disjunction that means at least at least one of the things should satisfy this particular kind of thing then this formula is going to be true so later when we talk about semantics of predicate logic we will be discussing in detail how to interpret this particular kind of formulas you know at this moment it is like this that for example if you want to represent for all X some PX then it is usually a conjunction of all these things I was to 1 to n etc P I so that means P 1 P 2 P 3 etc and all each P is considered to be a formula if you want to represent this thing there exists some X PX it is considered to be a disjunction on I 1 goes to n or infinity also then P I etc so each and every formula will be like this this is some X X 1 X 2 X 3 etc so even if at least one of these formulas is true then that is going to universal existential quantifier is going to hold that means it is going to be true so these are the two quantifiers that we come across and these two quantifiers are interrelated to each other in this particular kind of way so these are considered to be dual the duals so for example for all X PX can be represented in this particular kind of by definition it can be written in this sense so universal quantifier can be defined in terms of existential quantifier in this particular kind of way so it says that for all X PX means it does not exist some X such that it is not PX so that is what it says and then the same way there exists some X PX by definition it is same as not for all X not PX so this is what we have and then suppose if you negate this particular kind of thing universal quantifier I mean it is not the case that for all X PX means you need to push this negation inside and then negation of universal quantifier will become existential quantifier and you need to push this negation inside so that means it becomes not BX and then negation of X PX that means if you negate the existential quantifier for example this stands for X is happy or something so there exists some X such that X is happy or there exists some S van which is considered to be black in color if you negate that one this is going to be a universal quantifier for all X and this you push it inside and then it will become PX so this says that for all X for all birds etc now there exists at least one S van which is not considered to be white that means it has to be black in color so that is what it says so this is a relation between universal quantifier and the existential quantifier it always exist in duals so existential quantifier can be defined in terms of universal quantifier in this way in the same way suppose if you negate this universal quantifier you can talk about this thing in terms of existential quantifier negation of existential quantifier you have universal quantifier so let us talk about something in detail for example if you take the domain of discourse as natural numbers so in the you have to note that in the predicate logic everything makes sense only with respect to some kind of domain if the domain is not there it does not make any sense to talk about any formula because something happens something which is true in some kind of domain that you are taking into consideration may be false in some other kind of thing suppose if you take if you take into consideration real numbers and then you talk about some kind of formula which might hold and the same kind of formula might be false in terms of in case of some other numbers such as complex numbers etc so domain is considered to be the most important thing which we will talk about it when we discuss the semantics of predicate logic in greater detail in the next few classes so now if you consider domain of discourse is considered to be natural numbers like 1 2 3 4 etc and all 0 is not there if you add 0 to it will become whole numbers so another we have a predicate Phi which relates X and Y there is some kind of relationship between X and Y and that relation is defined as you define in this way X is less than Y so then we are taking into consideration natural numbers and then we have a function of XY which is considered to be a binary function which is defined in this way X plus Y plus is considered to be a binary function because it connects X and Y multiplication plus divided by etc are all binary functions in other meeting and ABCs are considered to be the constants which stands for some kind of numbers 0 1 2 etc so now we can talk about one particular kind of formula there exists some X Phi X Phi in the context of natural numbers so this is considered to be an unary predicate which essentially says that there exists some kind of actually should be there exists some Y Phi X Y which says that which says of Y that there is a natural number less than that particular kind of thing X where usually we consider it as since you are considering natural numbers definitely Y is not equivalent to 0 so it might hold as long as you do not take into consideration Y is equal to 0 so there is always a number which is less than suppose if you take X as 1 and Y as 2 and you have a situation where it satisfies this particular kind of formula there exists some X Phi X Phi holds in this particular kind of domain the domain of natural numbers provided Y is not equivalent to 0 and the second one let us consider another example where for all X there exists some Y and a predicate X Y which is stating that for any natural number X that means for all X means for any natural number whatever number that you are taking into consideration so that number has to be natural number X there is always there is there exists some Y it is not saying that for all way there exists some kind of way that means at least one particular kind of Y is also a natural number Y which is greater than X so you take two numbers in a domain and all pick up two domain two numbers from a domain so then you take any such kind of natural number for all X there is always a kind of natural number Y which is greater than X for example if you take two into consideration there is always another number which is greater than two that is three might be three is greater than two the same way for fix three and there always another number which is greater than that one so this holds for the natural numbers but if you take into consideration the real numbers that means all the irrational rational all these numbers and all then it may not hold the same kind of formula may work in some kind of domain it might be false in another domain so these are some of the equivalences between this quantifiers some of the things which are already explained on the board and all it connects this equivalence relations connects universal quantifier with the existential quantifiers so now so far we have discussed in a some kind of detail about the quantifiers if you do not have this quantifiers and then there is no way in which you can express this particular kind of thing you will be recursively writing all the formulas without any end example if you want to represent all gross are black then you will be writing there are suppose in the domain there are 10,000 gross and all which represent which is represented by ABC etc and all the A1 B1 B2 etc and all then if you start writing about if you start representing that particular kind of formula then suppose there are 10,000 birds that you are taking into consideration then your string your well-formed formula will have 10,000 sub formulas and all is very difficult for us to manage so for that reason you require this particular kind of quantifiers another important thing which you need to note is that in the case of universal quantifier there is a difference between this particular kind of thing for example if you represent this particularly AX implies BX for example in this one AX is considered to be empty that means this is false or empty or it is false and all for example you can talk about this particular kind of thing that all unicorns are wise so this particular kind of thing you can express it in terms of quantifier that might hold in you that might be true or might be false also so depending upon what values how you interpret this thing so suppose this particular kind of thing AX is false then irrespective of your consequent BX this whole formula is going to be true and hence this formula holds so although this formula can be true even without the existence of the unicorns in the actual world so you can still talk about a universal quantificational formula without talking about whether or not they exist in the world AX can be false AX if AX can is false then this whole formula is automatically going to be true and all but in the case of existential quantify suppose if you say that unicorns are for example if you say that all unicorns are wise because unicorns are doesn't exist in all so from this suppose if you infer that some unicorns are are intelligent wise so now this has no problem and all as such because this this statement can still be true provided BX AX is false if antecedent is false the conditional is going to be automatically true even if it doesn't exist also it doesn't make big difference but once you say that some unicorns are wise and all this presupposes the existence of unicorns in the actual world so this doesn't require we don't have any commitment that you know unicorns actually exist in the world you know but yet you can talk about this particular kind of formulas but once you talk about this particular kind of formula this presupposes some kind of existence so that means unicorns have to actually exist in the world you know so this is another interesting issue which is which we which we talk about it in the next few classes this car this again is a problem which is raised by Aristotle the problem is called as existential kind of fallacy suppose this is considered to be existential fallacy in the modern logic but Aristotle has taken into consideration that from a proposition this is considered to be a proposition that is all unicorns are wise from that if you can in you can still infer whether or not you can infer some unicorns are wise or not so this is if you infer in this way it is called as existential fallacy what is the problem here is that the problem here is that we are importing the existence in the conclusion which is not there in the premises for example if you say all unicorns are wise that doesn't presuppose any existence of unicorns but once you talk about some unicorns are Intel intelligent or wise it presupposes that unicorns are unicorns actually exist in the world so we will talk about this problem of existential import the end as a limitation of the first order logics as a quantification logic so now let us talk about what we mean by the scope of the quantifier so you just trying to talk about the basics of predicate logic still we are in the part of syntax itself so what do you mean by scope of a quantifier suppose if you say for all x ? is considered to be sub formula of ? that means that ? consist of some kind of sub formula ? then ? is called the scope of a particular occurrence of a quantifier that is for all x in a particular kind of sub formula ? the same applies to the occurrences of the quantifier there exists some x in so whatever is within the scope of this quantifier is usually considered as the scope of the quantifier whatever falls outside the scope is not considered to be bound so now we can talk about depending upon the scope of the quantifier to what extent this quantifier operates in a given formula we can talk about whether a given variable is free or a given variable is considered to be bound so there are only two things which exist here so an occurrence of an individual variable is considered to be bound if and only if it is within the scope of the quantification expression that contains the individual occurrences of the individual variables so whereas an occurrence of a variable is considered to be free if and only if it is not considered to be bound which is not in the scope of the quantifier is considered to be a not bound that means it is a free variable so it is like something which is in the room is considered to be bound for example a professor is teaching a class whatever whosoever is in the class are bounded by that particular kind of instructions are teaching etc and on but those whosoever is walking outside etc and on they are not they do not have to be they are not to follow the instruction of a teacher so they are not bound by the particular kind of instructor who is teaching in that particular kind of class so now let us consider some examples with which you can say which variable is considered to be bound and which variable is considered to be free suppose in the formula that that is shown in this slide for all x fx and cy implies there exists some y z y and hx are mz so in this particular kind of formula now there are various occurrences of this variables what are these variables x y z are considered to be the variables that means if you can replace with any constants etc and on it will you can replace these variables with some kind of constants these constants are considered to be some of the things which which are some kind of objects in the domain it can be people it can be anyone it can be gross or it can be anything so now in this particular kind of formula so x y and z these are the variables that exist in this particular kind of formula and it occurs in various places and all so now the first y and z are considered to be free for example if you take into consideration if you read it from left to right and all in this particular kind of formula so this whole formula is whole formula is within the scope of for all x so that means fx is considered to be the formula fx is obviously bounded by this quantifier x but whereas cy in the formula cy y is considered to be a variable which is not bounded and only x is bounded in that particular kind of formula the first sub formula so now wherever x occurs here that is obviously it will be bounded and all for example in the second formula there exists some y z y and hx are mz the first occurrences that means y y in the first occurrence means it occurs in fx and cy that is considered to be free whereas the same occurrence of y in the second sub formula that is there exists some y z y and hx are mz so z y in the formula z y it is bounded by the quantifier there exists some y so in the first occurrence y and z is considered to be free and of course rest of the individual variables are considered to be bound so whatever is within the scope of the quantifier is the one which we are trying to talk about let us take another example there exists some y g x y and f a in this particular kind of formula y is considered to be bound as both occurrences are considered to be bound so x is free because it is not in the scope of the x quantifier x quantifier is for all x we have only there exists some y n however the term a which is considered to be constant which is neither bound nor free such kind of terms are called as it is not considered to be a variable at all so depending upon the scope of this particular kind of quantifier we can talk about whether or not a given variable is x y z etc are considered to be free sometimes it may be considered to be bound and sometimes the occurrences of that variable in that particular formula can be bound and the same occurrence of that particular kind of formula y can be bound as well so now let us consider another example so that you can understand this scope of this quantifier to what extent this particular kind of quantifier operates in that particular kind of form the formula is read like this in this case it does not there does not exist x and there exists some y this formula is going to hold what is the formula for all z there exists some w such that a z w z and w are related in some way implies a y z and y z are also related in some way and a x y we are not talking about what how this x and y are related to each other it may be greater than it may be less than etc or x is a father of y or x is a brother of y or anything some kind of predicate a so now in this one suppose if you look into the innermost quantifier so that is there exists some w and the scope of that one is the next immediate kind of formula that exists after that that is a z w so that is the scope of that particular kind of formula beyond that it won't operate so there is another term which follows after that one that is a y z it won't operate on that particular kind of thing now if it consider the innermost quantify for all z and that is going to operate on the scope of that one is whatever is there in the brackets that is there exists some w a z w implies a y z so we are not trying to talk about which one which variable is considered to be bound which variable is considered to be free enough so we want to talk about that particular kind of thing with respect to this universal quantifier for all z no no for all z there exists some w a z w implies a y z in that w is z is considered to be no everything is considered to be bounded and all because there is no term which is considered to be free so now with respect to the next quantifier that is there exists some y the entire thing is considered to be within the scope of that particular kind of formula whatever follows after that particular kind of thing is within the scope of that quantify and then with respect to there exists some x the entire formula is going to be within the scope of that particular kind of formula so we have we have what we have done so far is that we understood what we mean by a scope to what extent the quantifier operates and based on that once it operates in all to which for which variable is considered to be free which variable is considered to be bound is the one which we have seen it has its own significance especially when you are trying to talk about some of the important inferences in predicate logic some of the important decision procedure methods that we will be using where we will we will need information about which formula is considered to be free etc which formula is considered to be bounded so we will make use of it in those decision procedure methods where we can check the validity of a given formula or you can talk about when two sentences are considered to be satisfiable etc. So now so far we talked about what we mean by quantifier and then what is the scope of a quantifier and then what you mean by a bound and free variable etc so now let us talk about what we mean by a proper sentence in predicate logic in propositional logic we said that a sentence is considered to be usually sentences means declarative sentences anything which can be spoken as is a true or false usually considered as declarative sentence so for example if you if you are referring to sentences like shut the door or open the door etc and all the sentence can neither be true nor false in so in the same way if you are talking about some questions etc what is your name etc and all this can be neither true nor false in so in the same way what we mean by a sentence a complete sentence in predicate logic is the one which we we should be interested to know a sentence is a formula in your language the language of predicate logic which lacks free variables. So your formula is so constituted in such a way that there is no free variable and that particular kind of formula is considered to be a proper sentence in predicate logic for example if you take into consideration for all x a y this is not considered to be a sentence because you have a free variable y suppose if you had said that for all x a x then it is considered to be could have been considered as a sentence but here the existence of free variable will make it is not a proper sentence in your predicate logic. So why we are talking about whether or not is a particular sentence etc and all just like in the case of propositional logic only sentences can be a statement can be true or false and all you can talk about truth or falsities false of that particular kind of sentence in the same way for example if you take into consideration for all x a x and there exists some x b x is considered to be a sentence because there is no free variable that exists in this one because a x is bounded by this quantifier for all x and b x is already bounded by this existential quantifier there exists some x and the entire thing is bounded which is within the scope of the universal quantifier so there is no free variable which exist in this second formula so that is why it is also called as a sentence. So let us consider the third one a x and there exists some x b x b x is bounded by this particular kind of existential quantifier there exists some x so x is not free the occurrence of x in that particular kind of formula is not free this x is bound in that particular kind of occurrence and x occurs x also occurs in the first term that is a x so in that occurrence of x in that occurrence a x x is considered to be free so whenever you have a free variable that is not considered to be a sentence as occurrence of that particular kind of formula is considered to be free in the same way for all x for all y a y y where y is considered to be free so it is not considered to be a proper sentence and of course b x is anyway in that particular kind of formula b x is obviously considered to be free I mean the variable x in the second term is considered to be free so now what we mean by complete and incomplete sentences in the predicate logic and what it signifies in particular just like in the case of propositional logic only statements can be only declarative sentence are the ones which we are going to take into consideration all the other sentences which we cannot draw a clear line between let us say mortal and non-motel etc and all we do not take those sentences into consideration so it said some kind of limitations which we talk about it at the end of this course so now what we mean by complete and incomplete sentences in predicate logic the expressions that is the formulas which are represented in terms of formulas are said to be complete if they contain no free variables that means everything is bounded by within the scope of the quantifiers and they are the variables that exist within the scope of the quantifier also considered to be bounded and you do not find any free variables then that particular kind of sentence is considered to be a complete sentence and they are incomplete if they do not contain if they do contain some kind of free variables you know so we have seen some examples earlier so now one of the important consequence of this particular kind of division that complete sentence and incomplete sentence is that complete sentences are considered to be fully meaningful and they can we can talk about whether or not their tautology is and they therefore have some kind of truth value that is true or false incomplete fragments by contrast are not meaningful you can only talk about satisfaction under some interpretations that formula is going to be true and some interpretation is going to be false and they therefore are incapable of having truth value you cannot clearly say that it is a tautology or you cannot say that it is a contradiction it just like some kind of contingent kind of statement to be true it may be false so this is one of the important significance of remarketing between complete and incomplete sentences in predicate logic a formula which consists of no free variable is considered to be a complete sentence in the sense of predicate logic do not take it into consideration in terms of English language but we are taking the context of taking this in the context of predicate logic so that means here the important message is that the formulas in any formula that you are going to take into consideration that constitutes a complete sentence provided if it has no variables if it has variables free variables then it is not considered to be a complete sentence it is usually called as incomplete sentence let us consider some examples with which you can understand this idea in a better way there exists some X X where X is considered to be happy for example if you write there exists some X H X that particular kind of sentence is considered to be a complete sentence because there is no free variable here in the same way there exists some X such that X is happy and X is bold that is what is both X the occurrence of X in this particular kind of formula is bounded so there are no free variables that is the reason why it is called as a complete sentence in the same way the third sentence for all way there exists some X E X Y is considered to be some kind of predicate and it can be we can talk about any such kind of predicate if in within in the context of a domain and this formula is read in this way there exists some Z not E Z X so all the variables that exist in this particular kind of formula are bounded by either universal quantifier or the existential quantify so that is why there are no free variables so there are no free variables in this particular kind of thing so that is why it is considered as a sentence whereas incomplete sentences are like this there exists some why such that Y runs and Y is old suppose if you represent it in this way there exists some X and RX and O Y the second occurrence of this variable Y only one once it occurs and Y in the second in the second term that is O Y is free so wherever you find a free variable and it is not called as a complete sentence in the context of predicate logic so H X and RX both occurrences of X are considered to be free because not bounded by any quantifier not so since it is the occurrences of X in this particular kind of formula going to be free so it is considered as an incomplete sentence in the same way P X implies for all X P X the first occurrence of X is free is not bounded by any quantifier etc and all so that is why it is called as an incomplete sentence so incomplete sentences you can only talk about satisfiability and all whereas complete sentences you can talk about tautology or you can even definitely say that it is a it is false or it is definitely you can say that it is true that is what we are interested in other we are interested in knowing that the particular kind of formula is true under all interpretations that is a tautology or it is for it is false in all interpretations and all that is a contradiction so far we have discussed about what we mean by quantifier and then scope of a quantifier and when we we also said that when a given formula is free when a given formula is bound etc and then we also talked about what we mean by a complete sentence an incomplete sentence in the context of predicate logic this we are building up our things so now just like in the case of propositional logic where you have discussed we discussed in greater detail that whatever way you combine will not constitute a well-formed formula and all and we need to know how some kind of rules for judging whether it whether or not a given formula is considered to be a well-formed formula so in the predicate logic which is usually considered as an extension of prepositional logic so most of the rules of prepositional logic will apply here also except that there is another additional rule that is the rule with respect to the quantifiers so now what what you mean by saying that a given well-formed form given formula is considered to be a well-formed formula in the first order logic or the predicate logic so now every atomic formula that is P Q R etc and all considered to be a well-formed formula you just try it like then it is considered well-formed formula if some x is considered to be well-formed formula not x is also considered to be well-formed formula and if circle is considered to be binary operator the binary operators are there are four in number like R and implies and if and only if if a and b are considered to be formulas then a circle be where a circle is represented as R and implies if and only if is also considered to be well-formed formula it does not tell us much except that is this is going to be useful when you are feeding some kind of information in the machine in particular the machine should know how which one is called as a syntactically correct kind of formula which one is syntactically incorrect formula this happens in the case of programming language as well while you are writing a program there is any syntax syntactical error it will clearly show that there is a error in you in our program in the same way these are the things which are important in the context of machines in particular so the fourth rule is that fourth rule is the one only thing which is new here in the case of predicate logic if a is considered to be a formula then x is a variable in that particular kind of formula a that means a x etc then for all x a x is also considered to be well-formed formula in the same way there exists some x a x is also considered to be a well-formed formula whereas a x there exists some x is not considered to be well-formed this is just tells us how these formulas how various strings are combined and forms some kind of well-formed formula nothing isn't tell tell us anything extra so now anything the fifth rule is like formality and all so that it says that all formulas generated by the finite number of applications of the above rules is automatically it will be treated as a kind of well-formed formula is talking about all the formulas that you have to judiciously use the above formulas and all it doesn't talk about anything new example if you say just px and all it is let us say it represented by some predicate x let's say human socrates is mortal for example p is considered to be predicate that is mortality is attributed to some kind of x that is x is considered to be some kind of socrates are Plato etc so that is considered to be a well-formed formula there exists some x q x c etc they are all considered to be well-formed formulas so just like in the case of prepositional logic suppose if the parenthesis is not given then we need to follow our own conventions so there is an order of presidents which is used widely in most of the textbooks so the order of presidents is slightly different in case of predicate logic when compared to the prepositional logic so the first reference is usually given to the universal quantifies so we have to put brackets whenever you come across this particular kinds of symbols for all why there exists some why bind most tightly and then followed by that rest of the things are same as the case of prepositional logic negation and are implies and double implies for example if you take into consideration this particular kind of example for all x px implies there exists some y there exists some z q y z and there isn't exist some x or x so there are no brackets nothing is given here so now in that context the first preference in this one you have to look for the universal quantifier so the universal quantifier occurs in the first string first formula sub formula is for all x px that's why we had put brackets there so now that is taken care of now we need to come to existential quantify that is the one which is needs to be preferred so now whatever follows after there exists some why you have to put bracket so that is what is happened here the second state and then again there is another existential operator which exists in the inner sub most formula and all so that is there exists some z etc so where that is where you have to put brackets and then another one which exists inside so the innermost you have to note that the inner occurrence of x is bound to the innermost existential quantifier not by the other external kind of quantifier so that is the reason why we have put there exists some x or x bracket there and the whole thing there exists some may z and the whole thing is in the brackets another what you get it from this particular kind of formula is is that the first preference is given to universal quantifier followed by that existential quantifier and you need to operate with all the existential quantifier once it gets over then you move to negation and put brackets there and then followed by that as usual in the case of preposition logic you follow and are etc in most of the good textbooks usually this parenthesis is already given but in some textbooks suppose if it is not given to you then we need to follow our own in this convention that you know first you need to take into consider universal quantifier existential quantifier and followed by this particular kind of rules is more or less similar to that of preposition logic except that we have two more operators there for all y and there exists some by universal and existential quantifier so we will talk about what do you mean by saying that a given formula in the predicate logic is considered to be a ground formula or it is when it is considered to be a closed kind of formula a formula F is considered to be ground if it does not contain any variables so like you know usually you refer it as constants etc and all A B C etc and all these things are they are not considered to be variables in all they are referring to fixed individual in the domain so they are they do not contain any variable that formula is called as the ground variable R C R B etc all these things is closed formulas are those formulas if it does not contain free variables so those things which doesn't have some kind of free variables means every all the formulas all the variables that exist in the given formula are considered to be bound then these are considered to be free variables sorry the closed formulas that means you don't have any free variables which exist in that particular kind of formula so let us talk about some examples of this ground and closed formulas and then we will close this particular kind of lecture so these things are important later so we will make use of these things a little bit later so for example if you say that some book that you are trying to read so that name of this book is something like Hamlet or something and if you want to represent that this thing so this is simply represented as B H N O the book Hamlet is boring for you so this is considered to be a ground formula it doesn't consist of any variable at all suppose if you are represented in this thing boring and some kind of X and you represented in this way there exists some X such that that particular kind of book is called as boring that particular kind of thing can be any other thing it can be Romain Mahabharth or Hamlet or any other book and all so X is considered to be variable here but here is a fixed kind of thing so it is in that sense this particular kind of thing is called as a ground formula it has no variables at all so now this is considered to be ground form so now suppose if you represent some other sentence like where there are variables here in this particular kind of formula really you will see it here X X and all here X is considered to be a variable so now here X is not free so now this particular kind of formula a formula which doesn't consist of free variables is considered to be a closed formula so this is considered to be a closed formula there are some other kind of formulas which are considered to be neither closed nor ground kind of formulas so for example if you say this particular kind of thing R X B Y X B is a predicate and then this Y X X are variables and then R is some kind of X is having some kind of property R so now in this one there are two occurrences of X here and here so this is bounded by this particular kind of quantifier X for all X so that's why X is not free here but what about Y here Y is considered to be free so it is in that sense whenever you have some kind of free variables which exist in a given formula so this is not considered to be a closed formula closed formula is not the case that it is a closed formula so now is it a is it considered as a ground formula that means a ground formula is a one it doesn't consist of any variables but you have variables here X Y etc so in that sense it is not even called as a ground formula so now usually what we would be interested in is this particular kind of formulas so mostly these formulas can be you can discuss about it as a tautologies etc and all you can talk about whether or not this formulas are true or false so in the next class what we will be doing is we will be discussing about some of the substitution instances of it and then we will also talk about so every formula will come up with some kind of some kind of diagram tree diagram a unique kind of tree diagram with which you can read the particular formula so what we have discussed in this class is simply is that we discussed about what we mean by a quantify and then we introduced two quantifies for all X there exists some X if you don't have this particular kind of quantifies things will be very difficult because you will be keep on writing it recursively n number of times and all without even coming to know what it says so we need this universal quantifiers and existential quantifiers and then we discussed about the relationship between universal and existential quantifier and then we discussed about when if a given formula is within the scope of the quantifier and based on that we can judge whether a given formula is given variable in that formula is free or bound etc and then based on whether or not you have free variables and variables etc and all then we discussed about what we mean by grounded and closed formulas and all then we said that closed formulas here are of some kind of interest to us because you can discuss many interesting things about interesting things about satisfiability tautology validity etc with respect to the closed formulas so in the next class we will continue with the syntax only we will finish with syntax and then we will move on to semantics and then we discuss about some of the important decision procedure methods which exist in the predicate logic they are first we start with the semantic tableaux method there is a one which we have discussed in the case of propositional logic and then we move on to one of the important proof procedure method that is a natural deduction method and then as usual in the case of proposition logic we use resolution refutation method so we will be talking about the same thing a little bit in the next few classes.