 Welcome back so far we are discussing about the basic basics of predicate logic and we started with what we mean by a quantifier and we introduce two different quantifiers that is one is for all x and the second one is there exists some x. So in a way we are trying to extend a preposition logic with these two quantifiers then in the last few classes we discussed about various properties of quantifiers and then we introduced a concept called as scope of a quantifier when a particular kind of variable is considered to be free when a particular variable is considered to be bound and these are the things that we have discussed in the last few classes. So today we will be talking about some of the other important properties of quantifiers and then basically we will be talking about the syntax of predicate logic. So what we will be doing today is we will be talking about various things related to quantifiers like what do you mean by saying that substituting a term with a variable sorry when do you say that a variable is substituted by another term a constants etc so what are the ways to substitute etc and then we will talk about instantiation etc and then we will go into the details of various kinds of translations so translations in predicate logic. So given an English language sentence how do we translate it into the language of predicate logic is the one which we are going to see in this class and then at the end of this lecture we will be talking about a particular thing which is called as I mean each and every formula will have its own corresponding tree diagram so each and every formula comes up with its own unique tree diagram which we will be drawing in a while from now. So to start with we will begin with the concept of substitution or instantiation so these are the instantiation is the one which you often come across in the next few classes especially when an universal quantifier is instantiated then we call it as universal instantiation and then the same way existential quantifier is instantiated that means one particular instance of this existential quantifier we call it as instantiation so what do you mean by saying that you mean by saying that substitution so let us consider a simple formula Phi if Phi is a formula and V is considered to be a variable so you have to note that we have constants we have variables we have predicates terms etc and on so if Phi is a formula and V is a variable then we usually write Phi of V to denote the fact that V occurs free in Phi suppose if you write like this Phi and V so then this V occurs as free so only when this variable occurs as free then we can substitute with another term another constant a bc etc and all so Phi of V denotes a fact that V occurs free in that particular kind of formula Phi suppose if you take T as your term then Phi of T are to put it more explicitly it will be Phi V given T so it is a result of substituting T for all the free occurrences of V in Phi so this is considered to be a free variable and this variable whenever you have a variable like x y z etc and all just like saying that some men all men etc and all so that if we present with some kind of constant such as Socrates are one more thing or anything so then it will become T so usually we represent it as this thing a variable V is represented by another term T so usually we write it like this V given T this means that the variable V is substituted by T so this is to be precise we will write it in this particular kind of V so this means a formula which consists of a variable V is substituted by a term T when you can substitute a term T especially when this variable is considered to be free so when do we say that variable is considered to be free in a within the scope of a quantifier a variable is considered to be free if it is not within the scope of this particular kind of quantifier so then that that variable is considered to be free. So now this is a substitution instance of this thing now we will be talking about some kind of strategy for substituting these terms for the given variables so we call that Phi T as an instance of Phi Phi of V so if Phi of T contains no free variables then we call it as ground instance of Phi so in the last in the last class we discussed about this particular kind of thing a ground formula is a formula which does not contain any variables so if Phi of V V is not considered to be a variable that means like x y z etc in our language of predicate logic then we call it as a ground term ground kind of formula or the term exist in that kind of formula is called as a ground term so in the same way closed formula is a formula which does not contain any free variables so that means you can make substitution only when it is not a ground formula or a closed kind of formula so it has to have a free variable and that free variable will is going to be substituted by a term T and that is represented as an instance of that particular kind of formula Phi of V so now if Phi of T contains no free variables then we call it as ground instance of Phi it would be like another constant Phi of C etc so that is called as a ground instance of Phi so now if the term T contains an occurrence of some variable x which is not necessarily free in T then we say that T is substitutable for that particular kind of variable V in that formula Phi of V if all occurrences of x in T remains free in that particular kind of formula Phi usually we represent it as Phi V given T is what we have discussed earlier in the last slide so now there is some kind of procedure which we follow for making this kind of substitution instance of a given formula which consists of a free variable so the first step that we will be following is this that first we will be dropping the initial kind of quantifier and then after dropping that quantifier then we will be talking about the instance of that particular kind of quantifier so now we replace all the free kind of variables with some kind of desired constants for example let us consider this particular kind of formula to start with we start with simple kind of formulas let us say you have a formula there exists some x Px so now one instance of first what we will do is we will drop this kind of quantified then we will talk about this particular kind of term which follows after this quantified now so now what we are doing is you are replacing this x with another term T so now in the second step what happens with this find a formula is this P T so it is like there exists some x such that some Px x is intelligent some it k students are intelligent for example so if you one instance of that one is this that some ROM Ramesh etc are in are considered to be intelligent so what exactly we are trying to do here is that first what we are doing is we are eliminating this quantifier and then we are substituting the variable that occurs here that is x with some kind of constant usually all constants are also considered to be terms here so that is why x is replaced by some kind of constant T this constant represent some kind of individual objects in the domain so for example if you want to do this particular kind of thing Px ? Qx and then one substitution instance of this one is like this x is substituted by T so now first what we will do is do like this will eliminate this quantifier and then this is going to be the thing and then in the second step what we will do is we substitute it with some kind of constant so this is what it becomes so now this is considered to be an instance of this one it is like for saying that all crows are black and one instance of that one is like this you might have seen you have seen one particular kind of crow which is considered to be black so that is considered to be one instance of that in the same way if you say all metals expands upon heating you observe one particular kind of metal and that also that started expanding and that is considered to be an instance of all metals expands upon heating so in this way we can substitute it with this particular kind of in this way first you drop the quantifiers and then replace the variables with some kind of terms then it will become substitution instance of a given formula so only quantified sentences can have substitution instances either it should be starting the formula should be starting with either existential quantified or the universal quantified so that you can substitute for example if you take this into consideration this one will not have any substitution substitution instance for example not for all x Px you cannot simply say that this is big this will become not PA so this is not permitted here first what you need to do is you need to convert it into some kind of standard form so that is so this will become there exists some x not of Px this can be substituted this will have some kind of substitution instance but not this particular kind of formula so now this will become not of PA where this a has to be new so we will talk about these roles a little bit later so there is a lot of difference between not of for all x Px there is a lot of difference between this thing for all x Px and there exists some x Px so so these kinds of formulas will not have any substitution instance so we need to simplify this formula then only you can substitute x for x with some kind of constant you cannot straight away substitute and say that it is not PA or something like so that is kind of wrong substitution so that is what we are trying to say not of for all x fx is not considered to be a quantified sentence you have to simplify that formula and then it will become some kind of quantifiable sentence because it starts with negation of the quantified so so this formula will become there exists some x not of fx and then you can start substituting for this ground variable x with some kind of term now let us consider some other examples of substitution there is some kind of strategy which we follow here for example if there exists two quantifiers for all x and for all y and you have fxy then what is this procedure that we followed earlier first you need to drop these quantifiers and then you have to substitute it with some kind of term which is considered to be a constant so now if you take this example into consideration for all x for all y fxy and then you drop the second quantifier and then substitute it y with some kind of constant a and that is considered to be a wrong kind of substitution so why because you need to drop this should be some kind of convention that will be following and that convention should be like this so the formula is like this p x y so something like p x y and all so now what I am trying to say here is is that suppose if you write like this you drop this kind of quantifier and then you substitute wherever you have y with some kind of letter B or something like that so then this is considered to be wrong and all so what is the correct kind of substitution is this one first you need to drop the quantifier that exists in the starting point and not the innermost quantifier so this is the innermost one so you need to drop this one first we need to move from left to right so that we will be following some kind of convention so initially what we will be doing is we will be substituting this one for all y you drop this particular kind of quantifier and then wherever you wherever you find x you substituted with C and then you keep it as it is now in the second step you can substitute the variable that exist here y with some kind of console so now this will become PC another letter D so now this will become an instance of this one so for all x for all y p x y x and y are related in some way so that one instance of that one is PC D so there should be some kind of convention that usually we will be following so that is first you drop the initial whatever occurs in the beginning and then you go you move towards right hand side and then you drop this quantifiers and then make these kinds of substitutions but this is considered to be a wrong substitution so in the same way if you consider the third second example there exists some y for all z for all x u y and L x z implies L x y etc and all and then you have given one substitution instance wherever you find a variable y you are substituting it with a constant C so now in this case what will happen is that so you need to drop the quantifier that occurs in that you will find it in the beginning of this formula that is there exists some way so when you drop that particular kind of formula wherever you find a formula with this subscript y is substituted with this let this constant C so now this formula will become first you need to eliminate there exists some way then this formula will become for all z for all x and u y y becomes C now that is why it becomes u C and L x z remains as it is and then in L x y you substituted y with a letter C so that is why it becomes L x C so this is the way we substituted with some kind of variables are substituted with constants so there is some kind of order which we follow so what we got from this one is is that the procedure is simple so first you need to eliminate the quantifiers and then you substitute the variables with some kind of constants and that will become a substitution instance of some kind of generalized kind of state for example if you say all men are mortal and one substitution instead of instance of that one is some there exists some x socrates is mortal for example is considered to be an instance of that where socrates is considered to be the constant which is substituted in substituted for the variable x so this is what we mean by substitution and then we will let us discuss something about different kinds of laws of quantifier distribution so these laws we will use it some of the decision procedures that we will be using it later where we will be talking about validity consistency etc and on so there we will make use of this particular kind of loss so the first law says that if you negate the quantifier universal quantifier followed by a formula px and that is the same as there exists some x not px so this formula needs to be written in this way there is some mistake in that slide so for all x px so this is same as what you do here is that you push this negation inside and the negation of the quantity quantifier will become the other one there exists some x and then you push this quantifier push this negation inside and this will become this one so now we can write this formula in this way so the moment if I write like this that means both sides it happens so this will become like this some x not of px so the negation of the universal quantifier will become an existential quantifier with the negation of a particular kind of formula so the other thing which you are going to notice in this formula is this thing as a distribution over the conjunction for example if you say for all x px and qx if and only if that is same as for all x px and for all x qx so it is just nicely distributed over the conjunction universal quantifiers are distributed over the conjunction whereas existential quantifier the next one third formula is distributed over the disjunction that is there exists some x px or qx is same as there exists some x px if you taken it alone in isolation that is same as this one there exists some x px or there is some x qx and the other way round also it happens there exists some x px are sorry for all x px or for all x qx is same as for all x px or qx so in the same way existential quantifier PA there exists some x px and qx is same as there exists some x px and there exists some x qx so it is distributed when when you are trying to use the same kind of quantifiers in all it distributed over conjunction as well as disjunction so now there will be some kind of problem if you take into consideration two different kind of quantifiers in all if the quantifiers are same if you are using the same universal quantifiers it does not make any difference in which order you use for example if you say for all x for all y p x y is more or less same as for all y for all x p x y so let us consider a domain which consist of natural numbers and x and y are considered to be to any two numbers then if you take any number into consideration one and then the relation let us consider relation as greater than a greater than or less than for example for time being you take it as less than so now if you take any number into consideration let us say two which is always going to be less than the other number if you are taking natural numbers as your domain so two is always less than three so for all x if it happens for all y if p x is less than y then if it is same as for all y for all x p x y then this particular kind of property holds so here the idea here is that the order is not going to cause us a kind of problem here so for all x for all y p x y is same as for all y for all x p x y in the same way if the quantifiers that you are using are more or less same that means they are the same type either existential quantifier or the universal quantifier and that is not going to make a big difference there exists some x there exists some y p x y is same as there exists some y there exists some x p x y it is like let us say x and y are related in this way x is a brother of y for example so you are saying that there exists some x there exists some y x p x y means x is brother of y that is same as there exists some y there exists some x again x is brother of y it does not make any big difference when you interchange the quantifiers provided when you are when you are using the same kind of quantifiers so if you use different kind of quantifiers then as you see in the third kind of inference it happens only in one way the other way round it will not happen so that is there exists some x for all y p x y implies for all y there exists some x p x y but the other way round it will not happen that is for all y there exists some x p x y does not imply there exists some x for all y p x y again you take into consider the same example x is a brother of y p x y stands for let us assume that x is a brother of y so there exists some x for all y where x is considered to be brother of y it is like in the context of church for example that fellow is considered to be brother of that is why they they call him as brother a father or something like that so that is same as for all y there exists some x that is same p x y but the other way round for all y there exists some x p x y does not imply there exists some x and for all y so the explanation is like this the first one says the relative scope of two universal quantifiers is going to be irrelevant that happens in the second case also as long as you use the same quantifiers it is not going to make a big difference so the relative scope of the universal quantifiers does not make any big difference in all so there is all irrelevant the second one says that relative scope of existential quantifiers in the same way is also considered to be irrelevant what is relevant here is that whenever you use two different kind of quantifiers then the meaning changes so that is why it happens only in one way in the case of third example there exists some x for all y p x y implies for all y there exists some x p x y but it is not the case that vice versa is not true so that is for all y there exists some x p x y does not imply there exists for all y p x y the scope is going to be relevant only when you use two different kinds of quantifiers otherwise there is going to be the same thing as long as you do not change the p x y kind of the formula p x y does not change so in whatever way use in whatever order that you are going to use is going to be the same thing so this is what with respect to loss of quantifiers with respect to the scope so now let us consider talk about some kind of translations that you commonly come across in the language of predicate logic so before that we will talk about the two quantifiers for all x and there exists some x so for all x is represented as this thing for example if you say that for all x x is mortal then you represent it as for all x just a letter p x so that means what does it mean to say that for all x x is mortal that means for every x whatever x that you are going to take into consideration x that x has to be mortal it cannot be the case that is one particular kind of x you have chosen and that x is not considered to be mortal so that means whatever you pick it up and that has to be having that has to be have this particular kind of property that is mortal so that means for each x x is considered to be mortal or the other way round of saying this thing is that for any x that you are taking into consideration x has to be more so exists x is like this it happens only for some x at least one x is considered to be more than we represent it as there exists some x so that means there exists an x such that x is considered to be mortal that is as good as saying the same thing as there is at least one x the sum is usually represented as at least some at least one particular kind of thing has a property something then you say that we call it as there exists some x px so let us try to talk about some kind of translations so we have to familiarize ourselves with the translation why because given an English language sentence we should be in a position to unambiguously transfer the language English language sentence into the language of predicate logic and then once you translated into the language of predicate logic then all the other things will follow whether it is the formula is considered to be a valid formula that means it is true in all interpretations in a given domain or whether that formula is called considered to be a contingent sentence contingent or consistent all this kind of things one can talk about only provided we have some good translation so let us consider some examples of this translation. So consider this particular kind of sentence it says like this there are mental things that aren't physical and there are physical things that are not mental whatever is pertaining to physical domain will not be in the mental kind of domain the same way whatever is there are physical things that are not considered to be mental so now this is this consists of two sentences and all the first one is there are mental things that are not physical things so this is a conjunction if you represented in terms of prepositional logic this is simply becomes P and Q so that is not going to give us the full information about what is there in this particular kind of sentence so we need to represent in terms of quantifiers then it makes some sense to talk about the inner structure of this particular kind of sentence so now each part that is there are two parts here separated by end each part has his own quantifier so there are two quantifiers also there are two negation operators that have their own location so one is talking about not they are not physical that means the negation is already there in that and the other one is saying that they are not mental that means another kind of operator is there so now this particular kind of statement can be translated in this so you so there are mental things that means not all the things are considered considered to be not physical and all but there at least one particular kind of mental thing that is not that is considered to be not physical so that in that sense you represented this this sentence as there exists some X where X has X is considered to be a mental thing and then that particular kind of X is not considered to be a physical that is represented as not be X and this takes care of the first part of the sentence and the second one there are physical things that are not mental again this is represented as there it exists at least one particular kind of X that X has to be a physical thing that is P X and at the same time X has to be not mental thing that is X not M X so this whole formula is represented in this particular kind of thing so why we are not writing it like there exists for all X MX not P X because the sentence is talking about only one particular kind of instance and that instance is like this there are some kind of mental things that are not physical and there are some physical things that are not meant the word some is not involved in this particular kind of thing so basically we will be so since it is not talking about the talking about all the things so we usually mean it as the there is the usage of there is some kind of usage of the phrase some in this particular kind of sentence so let us consider some more examples and then we will talk about how we are going to translate these things into something so let us consider another example is like this everyone admires at least one person that one particular kind of person admires everyone is very ambiguously stated here so now we need to go a little bit slow in this while translating this particular kind of formula by breaking this sentence in an appropriate way so now what this sentence says is that everyone admires at least one person let us say is considered to be the father of the nation or something like that so we admire that particular kind of person like Nelson Mandela Mahatma Gandhi etc. who admires everyone that particular kind of person admires everyone so now this can be broken into different parts and all so that the translation you will get some kind of justification for this particular kind of translation so first thing is it why there exists some kind of why at least one person is there that particular kind of person is why so who admires everyone so that means for all y a y y for all z a y z is the thing which we need to write it here it is not a y y but a y z so a y z means y admires z so that z has to be for all z and all so whatever z you take into consideration that y y has to admire that particular kind of z so now in the second step let us consider there exists some another x and all x admires y if x admires y then y has to admire everyone so that means the sentence is translated as a x y and this particular kind of sentence for all z a y y so what we mean by saying that here in a x y a stands for the predicate admires and then x and y they are in one particular kind of order so x y simply means that x admires y it is not the case that y admires x and all there is some kind of order which we follow in the predicate law if it is written as a y x then we say that y admires x but here everyone admires at least one person who admires everyone so who admires everyone is written as for all z a y z and then this is the sentence is a conjunction of this thing x admires y and y admires everyone so that is why a x y and for all z a y z so now there is at least one person y whom x admires and in the same way y admires everyone so that means there exists at least one y is represented as there exists with the existential quantifier there exists some y and then whatever sentence that we got it till now that is a x y and for all z a y z so this will become there exists some y a x y and for all z a y z so now for each x there is at least one y whom y whom x admires and y admires everyone so now we need to add another universal kind of quantifier that is for all x there exists some y and whatever sentence that is a x y and for all z a y z so this is the way to translate this particular kind of ambiguous sentence into an appropriate form in this particular kind of way so let us consider some more examples so that we will understand this idea in translation in a better way so just we will consider some examples so that we will get used to this particular kind of translation no woman no woman loves every man you not be necessary that no woman loves every man and all he might hate some human beings also so it need not be necessary that women always love all the time love every man and all need not be the case and all so how to translate this particular kind of thing in various step so now the as a first step you write it like this no woman so that particular woman you consider it as x it is such that such that just put it in bracket so that separating this sentence and we are handling the sentences by piece by piece so now we have something called every man and all so every such that every man so now woman is represented as x no man you represent it as y is such that so this is a second sentence and then whatever is left here is is that so that particular woman x loves y and all so now you write it like this x loves y so what we have done here is is that there exists some kind of woman there is some kind of y that is considered to be man and then the relation between x and y is like this x loves y it is not the case that y loves x and all so this is one particular kind of order which we fall so this sentence is translated in this particular kind of thing we have we are trying to consider it in piece by piece so no woman x is such that there exists some kind of y and that y is meant for all the man and all and that x loves all kind of man for all why so now you keep it as it is only no woman x is such that you keep it as it is now we translate this thing into appropriately into the language of predicate it says that every man y is such that that means for all y suppose if y is considered to be a man then x is y if y is considered to be a man then x loves y L stands for love f stands for man so this happens for all y and all so that will take care of this particular kind of sentence needs to sentences so now this translation is not it over so now we need to represent this no woman x such that all these things should happen so now in the third step no woman x such that means there doesn't exist some kind of for x x stands for woman and vice-transfer man so this will be like that doesn't exist x such that you have to take another property into consideration g so now gx implies so gx and there doesn't exist x gx and for all y this sentence whole sentence is for all a f y lies L x 1 so what it essentially says is that we just broken this sentence into this thing every man y is such that x loves y this is represented as this thing for all y for if y is considered to be a man then x loves man so this happens for all y and then in the first sentence no woman x is such that is represented as this thing that doesn't exist some x such that x is g and at the same time for all y if y is a man and x loves y so now this can further be translated in this sense so this says that they doesn't exist some x all this formula so now we have some kind of for translations if you come across a formula like this there doesn't exist some x px is same as this negation goes inside and the negation of the existential quantifier will become so now this this will become like this so I will write it here that doesn't exist this thing means for all x you have to push this negation inside and then so this is conjunction negation this will become not gx and the negation of conjunction will become disjunction and then it is negation of for all y f y implies L x y so now you can further simplify it and then you can write it like this so this is like not x or y so not x or y same as x implies y so now we can write it like this gx implies for all y this will be same and not f y implies L x y so this is the translation of this particular kind of thing so no woman loves every man in some three or four steps is translated in this particular kind of way so so this essentially says that for all x if x is having some kind of property g and then doesn't mean that x is a woman then they don't exist is not for all way if y is a man then that particular kind of x has to love this particular kind of man so that means no woman needs to love every man so this is a way to translate it is just one more example we take into consideration and then we will move on to some other kind of translations so now let us consider one more example no man no man who loves Rajesh are no man who loves Rani loves Rajesh are Rajesh are couple let us say let us try to translate this one so now you need to represent this constants with there are three people who exist here Rani Rajesh and couple we need to represent it with some kind of symbols so now this is same as no man who who loves Rani is such that so this is going to be the first sentence we are breaking it into three parts so that it will become convenient to translate this particular kind of sentence into the language of predicate logic no man who loves Rani is such that so the idea here is that no man who loves Rani loves Rajesh are couple so this is the one which we are trying to translate it so x x loves x loves Rajesh are x loves couple so now this x loves Rajesh is represented in this thing L stands for a predicate L x x loves Rajesh is represented as R R in the second sentence x loves couple is this thing k stands for couple and x loves k is represented in this sense so now we are taking care of the sentence which is on the right hand side so now we need to take care of this particular kind of sentence no man who loves Rani is such so now so this is represented as this thing that does not exist some x where f of x and this particular kind of so where this is represented as n is represented as R and this is k so the three constants that we have the individuals which exist in this particular kind of sentence Rani Rajesh and couple so now this says that there is there does not exist some x so that x is considered to be that particular kind of man and then x loves n and this happens for and this particular kind of sentence so that is L x R L x k this says that there does not exist some that particular kind of person x so that you know x x is a man and x loves Rani and at the same time he will do this particular x loves Rajesh and x loves k so now this is same as this thing not for all x there exists some x will become for all x and you push this negation inside and this will become x L x n and negation of conjunction will become disjunction and it will become L x k so this is going to be the translation of this sentence no man who loves Rani loves Rajesh are couple so like this one can translate given English language sentence into the language of predicate logic by breaking that particular kind of sentence into one or two different parts and all first you manage the right right whatever exists in the right most of this particular kind of sentence and then you extend it to the whatever is there in the left hand side of this particular kind of sentence so while discussing the traditional logic we discussed about four different kinds of sentences and then it will have its own translations in the modern logic like this so there are four particular kind of sentences which we call it as categorical statements categorical propositions so they are a I E and O so suppose if you suppose that universe of discourse to be everything let us say you are talking about people all kinds of people will come into that particular kind of domain and let S X B X is having some kind of property S and P X stands for X is having some kind of property P can be mortality it can be beautiful handsome except all these things so now I E I O propositions are represented in this particular kind of way I proposition it is like some men are mortal so it is represented in this sense there is at least one X such that X is having property S and X is also having property P so this is simply represented as there exists some X S X and P X so this is as good as saying that some birds cannot some birds flies O statement is exactly O sentence can be represented in the sense there is some X such that X is S but it is false that X is P so it is simply this represented as there exists some X S X and not P X so these are considered to be particular kind of categorical propositions and then there are universal propositions such as A and E a proposition is stated in this sense for any X for every X if X is having property S and X is also having property P you say that all birds flies if X is a considered to be a bird and X has to fly it cannot be the case that X is considered to be a bird and it does not fly now so it is represented as for all X if X is having property S implies X is having property P so now E proposition is also considered to be universal categorical proposition it is it states that for any X if X is having property S then it is false that X is having property P if that is a case then you write it in this way for all X S X implies not P text so now let us talk about a relation between these four kind of categorical propositions and then we will try to stop this lecture so this is the square of opposition which you which we discuss it while doing traditional logic so it is like this on the one hand we have categorical universal propositions a preposition and E preposition and then you have an I preposition and you have O preposition usually diagonals are considered to be contradictory to each other so these are all contradictory to each other so let us represent these things a preposition is represented in this sense for all X P X implies Q X so before all these things you need to talk about some kind of domain etc and on where P and Q are considered to be properties and X are some kind of individual some kind of variables which can be further represented by some kind of individual objects so now this is for all X P X implies Q X an I preposition is represented in these things existential quantifier P X and Q X and then both preposition there exists some X X is having property P and then this is a thing X is not having property Q and then E preposition is like this for all X P X X is having property P implies it doesn't X is not having property Q so these are the things which we need to note a and O are contradictory to each other if you take the conjunction of this thing that is going to become you are going to have the value F in the same way there exists some X P X and Q X and for all X P X not Q X so these are two these are contradictory to each other so for example if you say that all birds flies and all suppose if you come across one particular kind of bird X and X doesn't fly then that contradicts this particular kind of preposition that all birds flies so in the same way if you say that some birds flies and all and then you come across a proposition that for all X if X is a bird and X doesn't fly so this is exactly contradictory to this particular kind of thing so that means a and O are contradictory to each other and I and E are contradictory to each other so this is what we consider to be square of opposition in the predicate logic but there are some other kinds of inferences which might be of interest to us whether from a proposition that is for all X P X implies Q X can we deduce that there exists some X P X and Q X or for example if you say that for all X P X you deduce a proposition that you infer there exists some X P X it looks simple for us but it leads to some kind of problems which are which we discussed it partly while doing traditional logics that is called as the problem of existential import like this there are some other kind of translations one one can do we have to we need to practice a lot for doing this particular kind of translation let's consider some some one or two examples and then we will close this lecture and all so let us consider this example all animals that can fly are either not humans are not fish so here we need to break this sentence into break the sentence like this either not humans are not not fish is represented in this sense not H X are not L X so that is what is the thing and all and then all animals that can fly is represented in these things animals that can fly is represented in this sense if X is an animal and X has to fly so that's why this whole formula is quantified over all animals that's why for all X A X and F X implies not H X are not I X because we are talking about not humans and not fish so in the same way for example you want to represent no persons on the moon can talk or sing so usually that should be in this particular kind of format for all X P X and not Q X so the last sentence should be negation of that particular so no persons on the moon is persons on the moon is represented as P X and M X and then they can talk or sing is represented in this particular kind of say not of T X or S X so like this one can translate various kinds of sentences that occur in the language English language into the language of predicate logic so some of the things will look ambiguous for us but if you break that particular kind of sentence into two or three parts and all and things will become easy to handle some examples which we have taken into consideration but it requires lot of practice so this translation is considered to be a very important because once you translate a given language a given sentence into the language of the predicate logic and then things will become simpler and then you can talk about validity etc consistency etc later so in this class what we have discussed is that we are first we began with when do we when can we say that a particular kind of formula is an instance of an universal kind of formula which consists of universal kind of point so we have talked about substitution instances of a given formula so we can substitute only when we have some kind of variables so then we moved on to some kind of loss of quantifies then we talked about an interesting observation we gave an instance we come up with an interesting observation that whenever you have two different quantifies then it makes the scope is going to become relevant scope of a quantifier is going to become relevant whenever you do not have this particular kind of you do not have the same kind of we have the same kind of quantifies then does not make any big difference it is going to be irrelevant to the it would not play any role in that particular kind of formula that means it is as good as saying that for all x for all y is same as for all y for all x some kind of p x y is a case so in the so then we talked about some kind of translations and then we need to practice a lot with this particular kind of translations and then in the next class what we are going to do is we will be talking about the semantics of a given semantics of predicate logic and then we will move on to various kinds of decision procedure methods.