 Welcome back so far we discussed syntax of predicate logic here we discussed about what we mean by terms quantifiers etc and all what is the scope of a quantifier and when and we also discussed about particular things that a given well-formed formula comes up with a unique formation tree and each term also comes up with a formation tree etc. And now what we will be doing is we will be talking about the semantics of the predicate logic the semantics means giving the meaning of a given formula so meaning of a given formula means giving truth conditions of that particular kind of given formula following Frega we will be taking that into consideration so semantics of predicate logic is little bit different from that of prepositional logic in prepositional logic the meaning of a given complex formula molecular formula is solely determined by the meaning of its constituents that means whatever values that the individual constituents takes and in whatever way the connectives are and implies etc behaves based on that you can talk about the meaning of particular kind of given formula that means you are giving the truth conditions of the given formula for example if you want to know the meaning of P implies Q that means the truth condition of that one is whenever P is true Q is false P implies Q is going to be false so things are not say as simple as in the case of prepositional logic because in the predicate logic we will be we will be using variables predicates functional symbols etc so now once you give the meaning of a given formula we need to take into consideration all these symbols that we are trying to use we need to assign some kind of values to these these symbols that you come across in the predicate logic so a language of L of a predicate logic by its predicate symbols functional symbols variables and constants so functional symbols variables constants and predicates there are the four building blocks of predicate logic and we need to have quantifies there exists a mix for all x etc. So a single language will have many possible interpretations each suited to a different context or domain of discourse example if you have a particular kind of formula the same formula can be true of natural numbers and there is the same formula that we are trying to interpret in terms of a different domain let us say real numbers other than natural numbers whole numbers etc which includes 0 also so if you talk about whole numbers the same kind of formula may turn out to be false or if you are talking the same kind of formula the meaning of a formula with respect to integers and the meaning might change so it is dependent on the domain that you are using so suppose if you have P of x y it will have different meanings with respect to let us say natural numbers maybe it might be true it may false in natural numbers the same thing might be true in integers etc or maybe in the rational numbers it might be something else so what we need is we need to fix some kind of domain so in order to talk about the meaning of a given formula in the predicate logic two essential things are important first we need to fix the domain we need to fix that the domain consist of let us see if you are talking about numbers need to talk about either natural numbers are whole numbers are integers a non negative numbers positive numbers etc and the real numbers of course it includes all these things or if the same thing might be false with respect to irrational numbers etc so it might lead to multiple number of possibilities the same formula can be true in different interpretations so we may view a function F as representing a kind of multiplication or plus or something a relation such as x and y whatever is the maximum of x and y etc that will be considered as a function functional symbol and essentially what we require is the domain and there is some kind of thing which you require that is interpretation function which assigns some kind of values to constants variables predicates and functional symbols so the first and foremost thing which is essential for the semantics of formula in a given predicate logic is the domain of discourse and the intended meaning of meaning for predicates and the functional symbols so that is taken care by interpretation function I so the first essential thing which need is the domain and the other thing which you require is the interpretation function so let us talk about what we mean by this functional symbols there are four symbols that we are essentially talking about first is functional symbols FGH it is represented by FGH etc and we have constants which represent some kind of individual objects in the domain like Socrates Ravi Raju etc and all they are referring to some kind of individual entities and we have variables such as X Y Z etc it stands for anything as Socrates or Aristotle or anything and then we have predicates it which talks about some kind of relationship between some kind of objects like something is read something is white beautiful etc or X is brother of Y or Y is father of Z etc all these things are predicates which essentially have some kind of property so let us talk about what we mean by functional symbol so functional symbol F takes n arguments and since then F is called as nary function if it takes only one kind of thing it is called a unary function f of X is equal to Y f of X square is equal to Z etc. Suppose if you are talking about binary function like X plus Y for example it is a binary kind of function so if there are n kinds of arguments and all it is called as nary function so now individual or constants may be considered as functional symbol that does not take any argument so these things are considered to be individual constants if predicate P takes n arguments then it is called as n place predicate for example unary predicates are for example X is mortal that is M X suppose if you want to say that X is brother of Y so B X Y so X and Y are in some kind of order or if you want to talk about the tertiary predicates in all three place predicates then you can give you some examples for tertiary predicates etc so if you use n kind of arguments and all it is called as n place predicate so the thing minimal things which we need to note they are the four kinds of symbols that we are using individual symbols are constants which we have discussed just know there usually names of objects such as desktop chalk piece etc we India Kanpur all these things comes under referring to specific kind of entities in your domain so they are called as constants individuals so usually variables are replaced by these individual constants so now there are other kinds of symbols in the domain so they are variable symbols why we are discussing all these things because for interpretation for giving the meaning of a given formula what you require is a domain and then we need to talk about assigning some kind of values to these four kinds of symbols so variables are represented by X Y Z etc X can stand for anything so we are not just specifically mentioning what X is all about so there are variables so now the third thing is functional symbols represented by FGH usually plus minus multiplication all these things are called as functions and the predicate symbols usually they are represented as capital letters now greater than etc and all beautiful all these things are predicates mortal all these things so there are the four symbols that you come across you need to when you talk about meaning of a given formula you need to take care of all these symbols and we need to talk about some other things which are important for this one for defining the meaning of for all X FX and on so we require some of the basic concepts such as ground term so in the last class we have seen that in the formation of formation tree for a term we have seen that in that particular kind of formation tree for that thing we do not have any free variables then that particular kind of term is called as a ground term so a ground term is considered to be a term or an atom which is said to be ground if it contains no variables so a formula is ground if it has no quantifies and also no variables for example if you say something like F of G of CD and HC etc and all if you draw the formation tree for this one it is going to be like this G CD and HC and then it further reduces to C and D and C so now here all this terms are going to be constants so you do not have any free variable here so that is the reason why this term is called as a ground term a ground term is a term which can which does not consist of free variables and a formula is said to be ground if it has no quantifies as you see here it does not have any quantifier and it has even no free variables in free variables are XYZ etc so that is a thing which is considered to be a ground formula or a term is said to be ground in that particular sense it has no free variables it has no variables that is a thing which you need to talk about not free variables a term a formula is said to be closed when it has no free variables so this is the difference between closed term and the ground term ground term has no free variables no variables where as closed formula does not have any free variables like for example in this case this is a formula for example if you say for all X for all Y P X Y implies P Y X something like this one so all these variables are bounded by these two quantifies so that means there is no free variable in this particular kind of formula so it is in that sense it is called as a closed formula and all the closed formulas predicate logic they are considered to be sentences in the predicate logic and there are some formulas such as this one for all X for all Z for example if you write like this P X Z for all X and for all Y P X Z implies P something like Y Z so now if you observe this particular kind of formula X is bounded by this particular kind of quantifier whereas the occurrence of Z in both the terms is considered to be free so now it is in this sense it is called as a formula in predicate logic but it is not considered to be a closed formula because it has free variables whenever you a predicate logical formula has a free as free variables then it is considered to be it is not considered to be a closed formula it is considered to be just a formula and this is also not considered to be a sentence in predicate logic only closed formulas are going to be considered as sentences in predicate logic so this is one of the important distinction that we need to make out so the other thing is what we mean by saying that something is considered to be a ground instance for example there are two formulas a and a prime and a prime is considered to be ground instance of a quantifier free formula a if it can be obtained from a by substituting ground terms for some kind of free variables for example if you have something like this one the same formula which you can take into consideration now imagine that you have some kind of free variables like this some formula which is there like this and H X and for example if you take this into consideration now one of the instances of this one is this if you replace this particular kind of variable X with some kind of ground term just T or anything which you can use then one instance of this one is like this so this is a formula a let us assume that this is a formula a and one instance of this one when you remove this existential quantifier and then substitute it substitute X with some kind of ground variable like STUV whatever it is then it is considered to be the ground instance of this particular kind of formula so now this will become let us say you are uniformly replacing X with T now H T so now this is considered to be a ground instance of this particular kind of formula so since it is properly it is not called as a formula because X here there is no variable which is it has no free variables in all so we can introduce another thing called H Z or something like so now H of for Z you can replace it with something like you or something so this is one of the instances of this particular kind of formula so in it is in that sense a prime is an instance of ground instance of a formula a so what happened here is simply this is that the free variables X is is replaced by some kind of ground term so when free variables in a formula is replaced by some kind of ground term then it is called as a ground instance so now these are the examples of ground terms FAA it does not have any free variable so that is why it is a ground term FAA ZB F of F of A B ZA all these things are ground terms examples of ground formulas are like this not of PAA implies P F of A B B and implies PAA and the formula whatever is there down that is not of P F of A B B or P of A F of A X is a ground instance of this particular kind of formula so that is so in not P F of X B Y or P X F of X X in that particular kind of formula X is replaced by A and Y is replaced by B so it is in that sense it is a ground instance of that particular kind of formula so this is considered to be ground instance of a given formula so the variables X are replaced by ground terms then it will become ground instance of a given formula so now these are some of the definitions that we need to use before talking about the meaning of a meaning that means through condition of a given well-formed formula in the predicate logic first we need to have a domain so domain is usually considered to be a universe or we can also talk about domain the name or universe of discourse sometimes in some textbooks it is written as universe of discourse etc for the predicate variable predicate law variable is some set of values that may be assigned to a given kind of variables it can be natural numbers a domain can be natural numbers a domain can be set of people a set of animals etc or a set of rivers etc they all consider to be one particular kind of domain so X stands for a variable which stands for rivers that Ganga Krishna and all these things come under that particular kind of category so this is what we mean by domain it is considered to be an universe of discourse and the second thing which we need to notice something called truth set example if P X is considered to be a predicate where X is an individual entity which has that particular kind of property P like X is mortal etc is a predicate and X has this particular kind of domain you you can be anything it can be natural numbers it can be set of people etc and all then the truth set of P X that means we are talking about when this formula P X is going to be true for example if you say that all humans are mortal it is all will die at some day or other you represent it as H X or something like that if X is a human being then X has to be mortal H X in place MX so that particular kind of formula when that that is going to be true when you need to have a domain you the set of people in that context the set the truth set of P X is considered to be set of all elements of T of you such that the P and T has to be true that means a truth set is considered to be any term which belongs to the universe of discourse you such that the P when it is replaced by a ground term T and that has to be true for example if you say all men are mortal suppose X is considered to be all humans and MX is considered to be H is considered to be humans and M is considered to be mortal now this is going to be true when you have an instance where let us say something called HS stands for socrates for example it so happened that socrates is human being and then socrates is also mortal in that case this this is going to be true so this is this constitutes the truth set of that particular kind of predicate P X so when it is replaced by a ground term that PT has to be true it has to be true in all the cases then we represent it as this thing H X implies MX for all such kind of substitutions of X if this becomes T then we write it in this way for all X if X is a human being then X has to be mortal in the same way all crores are black so if X is a crow then X has to be black if it happens for all the crores that you have seen so far then it will be it should be written in this particular kind of sense so this is what we mean by truth set and for example if you say if you take the universe of discourse as natural numbers from 1 to 10 1 2 3 4 to 10 now P X is considered to be some kind of property which X has that is X is considered to be even number so then if you take this particular kind of thing P X is considered to be X is even and then we have a set such as universe of discourse is 1 to 10 so now now the truth set that means when this P X is going to be true only when you take this particular kind of numbers so when to when that particular kind of P X that satisfies this particular kind of thing X is even then only then this is considered to be truth set 2 4 6 8 10 all satisfies this particular kind of property that X is going to be even for example if it is X is odd then that particular kind of set is going to be this is considered to be the truth set of this one suppose if you take the predicate as X is odd then all these things will come into X 7 and 9 that's it suppose if you take this particular kind of set the same set 2 4 6 8 10 and all and then your predicate is this that X is odd then this is not considered to be the truth set with respect to find a predicate P X is going to be false in that case because it is any number that you take into consideration is not even it is not odd all are even so that's why that is not considered to be the truth set with respect to P X X is odd so truth set in a sense that when given a universe of discourse you and a predicate property which is attributed to the some kind of individual X then under what conditions P X property satisfies you know so then based on that you can talk about the truth set in some context it is true some other context it is going to be false if we take all odd numbers then suppose if you have universe of discourse as all natural numbers till 10 and P X X is even then this particular kind of set 2 3 1 3 5 7 9 etc that is going to be false so this is what we mean by truth set when the predicate is going to be true is the one which you have taken into consideration now let's talk about the quantifies now so basically essentially what we are trying to do is the building blocks of the predicate logic or variables constants functions and predicates we need to address all these four things before talking about the meaning of a given formula so now quantifies we may convert predicates into prepositions by assigning values to all the variables that means suppose if you have some predicates such as P X such that X is even you can convert it into some kind of prepositions suppose if replace it replace X with a ground term 6 then that P X P 6 and 6 is even that will turn out to be a preposition so all the predicates are reduced to prepositions when you replace X with some kind of ground terms like 6 7 etc so now universal quantified it is represented as for all X universal quantification on P X is considered to be a statement which is written in this sense it is considered to be a predicate P X and P X holds for all values of X in that particular kind of universe suppose if you take the universe of discourse as crows all the crows and then P X is considered to be something like X is black then that for all X P X has to hold for all the crows that you have taken in the universe of discourse even if for one particular crow which is turned out to be white and all then P X will not hold so for that property P X holds for all values of X then we call it as for all X P X and it is represented as universal quantified so which is written logical notation as for all X P X or sometimes in some other textbooks it is written as this thing for all X where X belongs to some kind of universe of discourse D so that P X that means P X holds for all X so different ways of reading this universal quantifier that is same thing it stands for the same thing for all X P X sometimes you can say that for every X P X for every X P X is considered to be true that means P X satisfies or the other way of saying is for all X some P X holds and all so there are some terms such as any the term any phrase any sometimes it act like universal quantifier sometimes it acts like a existential quantifier so depending upon the thing one may use it as universal or one might use it as existential quantified let us consider some examples and consider a domain to be natural numbers natural numbers are 1 2 all positive kind of numbers 1 2 infinity and if you add 0 to it it will become whole number and if you add all minus 1 to minus infinity that will become integers and then if you add all the rational numbers to that particular kind of thing all the fractional numbers including minus etc and all then it will become Q rational numbers and if you have real numbers whole numbers sorry rational numbers natural numbers and integers etc and all and that will constitute real numbers and then if something is called as complex number which is different from the real numbers so that is a different kind of domain other than real numbers real numbers has all these things whole numbers natural numbers integers rational numbers etc okay consider a domain to be natural number now consider a predicate P X Y X and Y are related in this sense in this way where P X Y is represented in this sense if X and Y are added to each other it will you will get a value 10 now assign value X to be 1 and Y to be 9 and you have taken 1 and 9 from this domain set of natural number the 1 and 9 are considered to be natural numbers only if you add 1 and 9 it satisfies this particular kind of property P X plus Y is equal to 10 so that means P 1 9 satisfies this particular kind of thing that is considered to be true so now if you take another proposition another thing into consideration another values 2 and 5 2 and 5 adds to 7 only it will not is not equivalent to 10 that means P 25 does not satisfy this particular kind of formula that is that means if you take the values 2 and 5 P X Y X plus Y is equal to 10 is not going to be satisfied in that sense it is false so if you take 5 plus 5 and of course that is going to be true so now if you change the domain to be negative integers integers also then for example if you take X as minus 1 and Y as 9 in some cases it might be the case that suppose if you take the natural numbers in some cases this holds and all but this is not going to hold for all the all X and all so whatever X that you are going to take into consideration and whatever Y that you are going to take into consideration P X Y that is X plus Y does not add up to 10 so it is in that sense you write it write this particular kind of formula as this thing since it holds for only some kind of properties that exist some X and there exist some Y P X and Y if it holds for all the properties and all which is not the case in this one then you can write for all X and for all Y P X Y where P X Y is defined as X plus Y is 10 it holds only if it holds for at least one particular kind of values of X and Y then you write it in this sense there exists X there exists some Y P X Y otherwise if you write it as for all X and Y for all Y P X Y that is going to be false so let us talk about the existential quantifier is represented as there exists some X usually if you write it in the sense existential quantifier is usually considered as a disjunction whereas universal quantifier is considered to be a conjunction of all the formulas that means at least one particular kind of thing is false the entire conjunction is going to be false so in the case of existential quantifier even if one disjunct is false if at least one can one disjunct is true it is enough for us to say that a particular kind of formula is true so the symbol it is represented by a symbol there exists some X sometimes it you represent it as for some X etc existential quantification P X is considered to be a statement which needs to be read like this a particular kind of property P X which holds for some values of X in the universe I mean some values of X means if there is at least one X which satisfies this particular kind of property then that will serve our purpose or equivalently you can also say that there exists a value for X such that that particular kind of P X is going to be true so in the last example P P X Y where X plus Y is equal to 10 that is going to hold at least for some values of X and Y there exists some X there exists some Y P X Y that is going to be true but the same formula may not be true for all the values of X and for all values of Y for example if you take X as 7 and then Y as 5 then 7 plus 5 is equal to 12 which is not equal into 10 which doesn't satisfy that P X Y is equal to X plus Y so that formula can only be written as there exists X there exists some Y P X Y X plus Y is equal to 10 if P X is considered to be true for at least one element in the domain then there exists some X P X is going to be true otherwise it is going to be false in the case of for all X it has to be true for all the elements of the domain otherwise it is going to be false so that is a difference between existential and the universal quantifiers. So now this is the way to interpret the quantifiers for all X A X that is going to be true in V if that means a domain some kind of domain if all the individuals in V satisfies A X that particular kind of property A X in the same way there exists some X A X is going to be true in a domain D or V if and only if at least one particular kind of one of the individuals in L or V satisfies your property that A X is the case A X holds for some at least one value then you call it as there exists some X A X is true. So now in a very informal way we discussed about the truth values of quantifiers etc and all are a given formula and we just indicated that the same formula is going to be true with respect to can be interpreted in different ways that means same formula sometimes it can be true in some domain like if you take only natural numbers into considering it might be true or in some other cases if you take the real numbers into consideration that means all the whole numbers etc and all then the same formula might be false as. How to formally talk about how we can formally express truth of a given formula in the first order logic first order logic is also called as predicate logic quantificational logic and all where the variables are ranging over individual sentences which are there in the domain it is not not we do not mean by predicates and functional symbols etc variables are not ranging over predicates functional symbols etc talk about those things you are talking about second order logic. So truth of a sentences in predicate logic it is determined by something called as modal we use this words interchangeable in all modal structure interpretation these three terms are one of the same. So some formula is going to be true with respect to a modal in the same way we discussed in the prepositional logic we discussed about a given formula with respect to a modal. So in the same way we can talk about the truth value of a given formula with respect to a modal structure. So we need to define what we mean by modal interpretation or structure now the semantics of predicate logic depends upon two important things are a important two important things which we need to note so they are first is the domain and the second one is the interpretation function I it depends upon the domain of individuals and semantic values of the constants predicates variables etc that is going to take. So now a modal consists of obviously the objects in the domain and the relationship between these objects in within the domain and interpretation function. So first of all what constitutes a domain domain constitute of the objects which are there in the domain for example set of people of all inanimate things etc and all for example those which does not have life etc. Chakpis, dusts, stables, chaves etc and all are set of trees for example it constitutes some kind of set of plants for example all the trees etc are come under that kind of category. So a modal consists of an objects and relations among them and then we have an interpretation function which defines references for this particular kind of symbols. So what are the three things which are there in the predicate logic constants, predicate symbols and functional symbols and variables. So now constants and variable symbols should find out some kind of object in the domain and predicate symbols have some kind of relations in the domain D and the functional symbols have some kind of corresponding thing in the domain that is functions from an object to another kind of object. So it takes these values and all now the definition of interpretation is like this. So interpretation means giving assigning some kind of truth values to a given formula in the case of propositional logic. So it is not as simple as in the case of propositional logic in the predicate logic when you say that interpretation need to be taken into consideration what values that we have variables etc constants and functional symbols are going to take that is also we need to take into consideration. An interpretation for an expression in a predicate logic consists of following things first to start with we need to have a domain of the interpretation which must include at least one particular kind of object. Sometimes the domain can also be empty. So in the empty domain suppose if property such as Px for all x Px is going to be true is going to be vacuously true whereas if you talk about there exists some x Px with respect to empty domain that is going to be false we are going to see in a while from now the difference between these things. So in general if you talk about domain it is usually taken into consideration that the domain is non-empty you do not talk about domain such as set of suppose if you are talking about particular kind of formula that all men are mortal so that is man so that is mortal and we do not we do mean by saying that at least some kind of objects exist in the domain that means you need to take into consideration some kind of domain which consists of some people at least. So if you do not talk about any kind of people and now if you talk about animals etc and all that does not make any sense to talk about this particular kind of all the formulas are going to be vacuously true that means all the universal formulas which are expressed by universal quantifiers are obviously going to be vacuously true. So now usually domain is considered to be usually non-empty at least one or some objects needs to be there in the domain and then an assignment of a property of the objects in the domain to each predicate in the expression and you need to have an assignment of a particular object in the domain to each constant symbol in that particular kind of expression then that constitutes what we call it as interpretation for example if you say that there is a formula such as there exists some x or x y now the same formula in some domains is going to be true and some other domains it might be false let us consider a domain of individuals to be set of all people and then we are trying to evaluate this formula there exists some x or x y we need to talk about what we mean by r x y also the set of all people who live who have ever lived in this world who has not lived in the world does not make any sense to talk about this particular kind of formula and then we are also taking into consideration r x y that means the relation between x and y it is like this x is considered to be a parent of y. So now if you take y to be Mahatma Gandhi then usually we call him as father of the nation etc father of everyone so x is a parent of y in that sense there exists of course we are not talking about for all x r x away we are just talking about there exists some x such that x can be Ravi or something like x can be Mahatma Gandhi and x r x y stands for x is a parent of that particular kind of y y can be treated as Mahatma Gandhi and what it essentially says is that every person who is who existed in this world have at least father and all so x is considered to be parent of y in that sense there exists some x r x y is going to be true so now if you take y to be Mahatma Gandhi then the sentence is obviously going to be true anything which you put it put it for why everyone has a parent so that is why there exists some x r x y is obviously going to be true suppose if you take further for the sake of fun we can take into consideration Adam you etc and all we do not know whether Adam Adam and you have parents etc and all so the same formula there exists some x r x y in that particular kind of domain where you have these objects Adam you etc and all that sentence may probably be false so what I am essentially trying to say is that the same formula have different interpretations so in sums depending upon the domain and the interpretation so now let us formally define what we mean by a structure or interpretation or model etc all these things terms are one of the same this is somewhat technically a little bit complicated kind of definition this definition is usually taken from taskies work taskies come up with this particular kind of definition which is been changed into our concern and this definition is like this a structure a or a model which consists essentially consists of a domain and set of an interpretation function I a structure a for a language L that is a language of predicate logic consist of non-empty domain I mean domain has to be a non-empty it is one particular kind of objects would exist in the domain and an assignment that is interpretation function which assigns to each nary predicate symbol r of L that particular kind of predicate logic of an actual predicate or a on the n tuples these are the terms a 1 a 2 to a n from a and they go this going to be an assignment to each constant symbol C of L to an element C to the power of a of the particular kind of domain a and to each nary function symbols L there is an nary function f a from d to the power of n that is a domain need the power of n to be so what essentially we are trying to say here is like this so we have this particular kind of thing and all there is a domain this structure consist of domain and interpretation function so this domain has to be non-empty I mean at least some set of some objects needs to be there in the domain it can also be we can also take into consciousness domain as empty but in general we take non-empty domain and all and interpretation function is represented as I so now so what we are essentially saying is we have constants we have variables we have functional symbols and we have predicates so now these predicates have to be mapped to something in the domain which has which has to be either true or false let us say in the case of two sets that we have seen px where x is even that particular kind of px has to be true it is going to be true when you take all the even numbers and all and if you take all our numbers px is going to be false so that has to be mapped to something such as T and F so now you have constants this is a domain nary functional symbols we have seen what we mean by this constants nary functional symbols and nary predicates and each one is mapped to some kind of individual in the domain that means we are assigning some kind of values to this one constants nary functions and nary predicates in nary functional symbols for each nary function that you that exist in the domain you have corresponding nary functional symbol in the domain and each nary predicate symbols you have nary predicate symbol symbol to function function that is d and 2 it maps to some kind of there are only two entities here as it has to be true or it has to be false px is false or px is true that particular kind of thing you know usually the interpretation over domain is considered to be an assignment of entities of D to each of the constant variables predicates functional symbols and the predicate calculus expressions such that here what we are trying to do is each constant is assigned to some kind of element in the domain D that is we are basically assigning some kind of entities to constants variables functional symbols and the predicate symbols that means we are assigning this some kind of values to these things. So now each variable XYZ etc is assigned to a non-empty set of domain where these are the allowable substitutions for that particular kind of variable for example XYZ it can be substituted by Socrates or Ravi Raju Rajesh etc and all and they all should exist in the domain that particular kind of domain. So now each function f of arity m nary function is defined on m assignments of D and defines some kind of mapping from D n to the power of m to D that is m stands for the number of arguments it is power of m maps to D. So if you have 0 arguments it will be D of 0 to D so each predicate p of arity n is defined as arguments from D and defines a mapping from D n to some kind of set of values D and f. So now what we are essentially doing here is like this to each constant we assign some kind of element in the domain D and also to each n place functional symbol we assign a mapping from D to the power of n where n is considered to be number of arguments n depends upon n plus functional symbol and where D n is considered to be 1 into n etc D1 D2 D3 Dn and to each n place predicate symbol we assign a mapping from D to the power of n to some kind of value 0 and 1. So it means the predicate is going to take some kind of value either 0 or 1. So now this is what we have done so far D assigns to the quantifier for all x a non-empty set D which is called as the domain of the universe first thing and structure a assigns to each n place predicate symbol r and a relation r a is a subset of Dn where Dn is considered to be tuples of members of the universe and a assigns to each constant symbol c a member of c the power of a of the universe or the domain a assigns to each nary function symbol is which you have been discussing for n nary operations f to the power of a on D this is much more formal way of saying the same thing. Essentially what we are trying to do is we need to have a domain and we need to have some kind of interpretation function which interpretation mapping or something which assigns some kind of values to variables constants predicates and functional symbols. So in this context let us talk about interpretation of ground terms a term that contains no free way no variable is considered to be a ground term and each constant term sees names that particular kind of element as c to the power of a a is considered to be the structure the domain if the terms t1 to tn of L name the elements such as t1 to the power of a t to the power of n a of the domain D and f is nary function symbol L then the term f of f of t1 to tn that is also considered to be term which names the element f of t1 to tn to the power of a as f to the power of a t1 rise to the power of a to tn to the power of a of a domain a. So now just let us consider one particular kind of example and then we will close this lecture then we will talk about some more examples a little bit later. So now let us consider predicate p x y in a language L now take the domain to be natural numbers n and we have a functional symbol that is less than here C and D are considered to be constants for example then we can assign to elements c to the power of a d to the power of a as follows c to the power of a is considered to be 0 and d to the power of a is considered to be 1. Now if you take the rational numbers with q which is again represented as less than then in that context the constants are represented in this one divided by 2 d to the power of a stands for 2 by 3 etc and if you take the integers into consideration with the relation a functional symbol greater than we have constants represented as c to the power of a as 0 d to the power of a is minus 2. So in this lecture what we have done is that we just talked about what we mean by a structure or a model and we have said that depending upon the model structure a given predicate logical expression will find its meaning we find its meaning in with respect to a model the same kind of formula can be true in some structures same kind of formula can be true falls in the same kind of structure. So what matters to us the most is the domain that you are trying to take into consideration same formula can be true with respect to natural numbers but it can be falls with respect to integers with respect to some other kinds of things. So in this lecture we define what we mean by giving interpretation or structure or model for a given predicate logical expression. So in the next class we will be considering some more examples and then we will be talking about some of the important decision procedure methods in the predicate logic and to start with we use semantic tableaux method because this which occupies the central position in our course. So we will be talking essentially about the semantic tableaux method and in that context we will be talking about different logical properties such as when a given formula is valid when particular formula is considered to be consistent satisfiable all these things which we will be talking about in greater detail in the next few lectures.