 Welcome back now we entered into the third phase of this course introduction to logic. So in this module we will be talking about some of the basic concepts of predicate logic. So far we have discussed we started with basic concepts of logic where we discussed about what we mean by argumentation etc we talked about various kinds of arguments and where these arguments occur etc then we moved on to the logic of prepositions where we the basic units of our logical system is or the prepositions a preposition is considered to be a sentence which is which can be spoken as either true or false. So we have cleverly chosen our sentences in such a way that we can clearly draw a line between let us say mortal non-motor etc and so a sentence can be spoken as either true or false all the sentences are represented by simply by means of sentential letters that is what was the case of prepositional logic. So preposition logic is also considered to be the logic of the minimal logic of connectives with these connectives are like this negation and are implies and if and only if so these are the minimal kind of logics with which you can represent our knowledge. So now what we are going to say in this class in this lecture is that so we introduce prepositional logic we will discuss the rational for introducing this predicate logics what are the limitations of prepositional logic and then we will talk about some of the important basic constituents of the predicate logics that are predicates terms etc. So in this lecture I will be talking about these things so why after all we need to move to predicate logic if you have if you already have prepositional logics with us. So in the case of prepositional logic the world is described in terms of basic units and these basic units are considered to be atomic prepositions an atomic preposition is considered to be a sentence which can be clearly spoken as either true or false. And all they can also be treated as declarative sentences so if you are given any sentence in all you see for example if you have a sentence like it is raining you simply represent it as P or not Q etc. So a proposition is considered to be a statement that is either true or false so the basic units in the prepositional logics are sentences and these sentences are represented by sentential variables like P is Q is R etc and we have finitely many number of such kind of variables with which we can represent our prepositional prepositions. So the most important problems that are associated with the prepositional logic is that they fail to deal with singular terms and then when it comes to complex terms they fail to deal with the complex terms and even when it comes to the relational terms prepositional logics fail I will talk about these things with some examples little bit later but when it comes to dealing with individual terms etc prepositional logics fail. And other thing is that we have no way in prepositional logic of talking about individuals of asset which may which may have or fail to have certain properties etc and there is no way of quantifying over the individuals that is the main reason why we will be augmenting our prepositional logic with two more quantifies these quantifies are for all x and there exists some x. So there is no way of which you can talk about individuals in a set are all members of an array etc and all so it is all they are all represented with only some simple kind of preposition P. So these are some of the limitations of prepositional logic and that is the reason why in order to explain incorporate all these things we move on to predicate logics. So to consider a simple example let us consider a simple example such as the one which is in the right color which appears in the slide so for every number x there is a number y such that x is less than y for example you have a natural numbers 1, 2, 3, 4, 5, 6 etc and all for any number that you take into condition let us say you take two there always be one number bigger than that one that number is smaller than the other number it is successful two is always less than three three is always less than four etc. So now these kinds of sentences which we need to invoke some kind of relations between the individual terms x y etc and all for this if you represented in terms of prepositional logic it may not capture the deep structure of this particular kind of sentences. So this is simply represented as some kind of sentence P but prepositional logics are the most simplistic kind of logics that are in which usually propositions are represented by simple variables and all. So they are also important in the sense that in prepositional logic we have some nice features such as prepositional logics are complete prepositional logics are said to be consistent and even they are considered to be sound. So they are the wonderful logical features that you will find it in the case of prepositional logics but if you want to explain the entire mathematical reasoning this prepositional logics fall short of many things mathematics requires lot of relations etc and all like in this example we need to go into the in depth of this particular kind of sentence then we need to analyze this particular then only we will come to know whether or not this particular sentence is true now for example this particular kind of sentence for every number x there is a number y such that x is less than y is represented as symbolically as this for all x there exists some y such that x is less than y. So this is self is not sufficient for the truth of this particular kind of sentence but for that we need to have some kind of domain. So we need are we talking about natural numbers we are talking about integers we are talking about rational numbers etc but all these things needs to be stated explicitly so that we can talk about truth of this particular kind of sentence. So we are going into the depth of this particular kind of sentence and then we are talking about we are invoking relationships etc and all so that predicates usually will take care of this relationships. So these are some of the things which are also considered to be the limitations of propositional logic so the objects in elementary objects in propositional logics are elementary statements their properties and relations are not explicitly represented in propositional logic for example in the last sentence x is x less than y that simple sentence is represented as one particular kind of thing p just one letter p so that is not sufficient in enough for us especially when you are talking about the relationship between two individual objects etc. So objects in objects in propositional logics are simply considered to be elementary statements and the statements for groups of objects especially like the one which we have discussed earlier require some kind of enumeration like for example if you want to say all men are mortal you cannot simply represent this sentence as just p or q etc. So we need to quantify so that is for all x if x is a man and x is considered to be a mortal consider to be mortal x is a human being x is considered to be mortal so we need to quantify quantifiers so for all x there exists some x are considered to be the two main quantifiers that will be using it in the predicate logic. So in a nutshell predicate logic is considered to be an extension of the propositional logic with these two quantifiers is for all x and there exists some x so predicate logic is also considered to be an area of logic that deals with basically predicates which talks about the relationship between the objects whether or not an object possesses that particular kind of property etc and all for example if you say all men are mortal mortality is the property which is attributed to the human beings so mortality is considered to be the predicate so predicates given much more importance in the predicate logic. So the other term for predicate logic is what we call it as first order logic first order logic means is the predicate logic plus propositional logic is already there sitting at the background so one of the advantages of first order logic or predicate logic is that it permits quantification over variables like all men are mortal etc that permits us that mortality is quantified over all the human beings so and if you take into consideration higher-order logics then it permits quantification over functions and predicates if the quantification happen over only variables then it is called as first order logic if the quantification happen over the functions and predicates etc and all it is called as higher-order logic this is one of the important difference between differences between first order logic and the higher-order logic but we will be restricting our attention on quantification over variables so now let us consider a simple example and we will see why prepositional logics fail to explain particular kinds of arguments like this. So we need to note that when we introduced Aristotelian logics there is also considered to be a kind of predicate logics but it has its own limitations so because not all the sentences can be put in the form of the four categorical propositions and then you can talk about validity of a syllogism and then Aristotelian logics which are also considered as traditional logic which has a limitation that it talks about terms a group terms which are represented to a group rather than individual terms individual beings individual things there are some of the limitations which I will talk about it in a while from now so let us consider one simple example that is this thing the famous example all men are mortal all IITK community are men so all members of IITK community are considered to be mortal so these are the three sentences that we have suppose if you represented with the help of prepositional logic when the first one all men are mortal is represented by some kind of a letter P and all IITK community are men it simply treats it as a anatomic proposition so that is represented as just simply Q and then we have the final thing that is all members of IITK community are mortal is simply represented as some kind of another atomic proposition are so from P Q R follows from P Q we can easily come up with a counter example in which P and Q can be true but R can be false so that means the argument will be invalid if you do not look into if you do not go into the deep structure of these sentences which are expressed as premises in the same way all circles are figures therefore whosoever draws a circle draws a figure so suppose if you talk about this particular kind of argument again the first sentence is represented as simply a letter P in the third second sentence is simply represented as letter Q that means Q is reduced from P and you can easily come up with a counter example in which P is true and Q is false that makes this argument invalid but the actually our intuitive intuitively we know that this argument is valid so all circles are figures and whosoever draws a circle is said to have drawn a figure is intuitively or commonsensically we can say that that indeed follows from this thing but if you take only preposition logic into consideration where each and every sentence is expressed as simple prepositions like all circles are figures is represented as P and whosoever draws a circle draws a figure is represented as Q then P leads to Q N so we know that these sentences are valid kind of sentences and all but how to know how do we show that these arguments are valid so validity here is not merely a matter of how these simple statements are related by means of some kind of propositional connectives what are the propositional connectives and are implies if and only if negation etc. So there are many cases in which these prepositional logics works and all but in many cases where wherever you find this relationships etc and all this we need to go into the depth of the sentences and we need to look into how these objects are related to each other and then only we can talk about truth of a particular kind of sentence so validity is not simply how these sentences are going to be pieced together etc and all but it also depends upon the inner structure of the simple statements so the inner structure could be the predicates terms etc how these predicates are related to each other etc objects are related to each other all these things which we should know or we need to talk about truth with respect to some kind of domain etc so that means the same sentence can be true with respect to natural numbers the same sentence can be false with respect to integers etc. So we need to talk about domain first and then we need to talk about some of the building blocks which are related to the deep structure of this statements so for example in this case all men are mortal socrates is man socrates is mortal it is simply represented as first sentence is represented as P and second sentence as Q and P and Q leads to R so you can easily come up with an assignment where your valuation P and valuation of Q true and valuation of R false means you can have premises true and false conclusion that makes this argument invalid but actually this argument is a valid argument so all men are mortal socrates is man there is no way in which socrates cannot be mortal socrates is mortal necessarily follows from all men are mortal socrates is a man to analyze that this argument as being valid we need to break inside these prepositions and to capture more of the information that they convey etc and we also need to analyze prepositions into predicates and arguments and also deal with the quantification what is the quantification involved here all men are mortal motorities attributed to all human beings and there is a person called socrates is considered to be human being and then there exists some X such that socrates is that X is considered to be mortal so the first sentence is represented as you knew with the universal quantifier and the next two sentences are represented by means of existential quantifiers so we need to we need to have apart from the simple logical connectives and the relationship between simple relationship between the sentences we need to have quantifiers etc so predicate logic in this sense extends the prepositional logic with predicate latest piece Qs Rs etc. capital P capital Q or etc that are interpreted as relations on a particular kind of domain this domain can be for example if you are talking about arithmetic the domain could be natural numbers so if you are talking about some Indians that means all the people who reside in India etc that can be called as some kind of domain the same thing can be falls in something which is true of natural numbers cannot be true of all in all the domains and all like the domain in which we have only integers etc. So why predicate logic the axioms and theorems of mathematics are defined on arbitrary sets such as sets of integers etc where always we invoke some kind of relationship between any two set of statements so we need to be able to write and manipulate logical formulas that contain relations on values from arbitrary set let us take one simple example that is that R be an energy relation on the domain D where D is considered to be set of it can be natural numbers it can be real numbers or it can be rational numbers so that R is a subset of the domain DN where you consider that the property that you are trying to invoke is the prime number so you are saying that X is a prime number and which is a subset of N that is a set of natural numbers if that is the case then all these things such as 2 3 5 7 11 17 etc 13 etc all these are considered to be prime numbers for example if I take a number such as 4 then of course that belongs to N and all belongs to natural numbers but it is not considered to be it does not belong to this particular kind of set so this is 4 is not a prime number the same kind of property that is prime of X is a subset of N is interpreted in this particular kind of that is going to be true only when it falls within this particular kind of set if it is if it does not belong to this particular kind of set then that statement prime of X does not belong even though it belongs to N but that sentence is going to be false so now we said in the beginning of this lecture that predicate logic phase as preposition logic fails to explain the inability it shows inability to explain singular terms complex terms and also the relational terms so what we mean by this singular terms complex terms etc so usually singular terms or words or phrases that represent individual things such as Ravi Dester pen moon Kanpur etc I take a library all individual things so singular terms are more commonly called as names of individual things any name of anything is considered to be a singular term so we need to note that even classical logics traditional logics such as Aristotelian logics also fails to explain the singular terms it could not include explain these singular terms in a proper way so now the complex terms are like the sentences of our everyday language usually contains obviously complex descriptive phrases usually to represent group of things but in the case of traditional logic that means Aristotelian logic does not distinguish between simple general terms and complex general terms there is no way in which you can distinguish between simple general terms and complex general terms which is the distinction is very important for us to make so with the with one small exception that it distinguishes definitely between affirmative terms and the negative terms that classical traditional logics succeed in making such kind of distinction but it fails to distinguish between singular terms such as Ravi Santa Moon Kanpur etc with some kind of complex phrases that we use in our day-to-day discourse so there is no way in which you can discuss distinguish between singular and complex terms and there are lots of examples which one can give the propositional logic failed to explain its particular kinds of things let us say for example if you say some pink birds are long leg and all birds are having wings therefore you say that some pink things that is considered to be a singular term are long leg and winged so if you represent it in terms of simple propositional logic it will be like two sentences like the first one is represented as P second one is represented as Q and therefore the entire statement some pink things are long leg and winged is represented as our so that will not serve our purpose so we need to talk about how these sentences are related to each other that is expressed by predicates etc and then we need to go into the deeper structure of these sentences and then only we can talk about validity of sentences arguments like the ones which are showing it here so the singular terms are usually represented by a unique letters as A B C D etc and all but XYZ are usually used for variables you know suppose if you are sure that the person is Ravi or Raju or Ramesh then usually represented as A B C D etc and all suppose if you want to represent it as X X is a human being that human being can be anything it can be Raul Gandhi or it can be anyone Ravi or anyone so you represent it as variables XYZ so now we are trying to talk about some of the basic building blocks of the predicate logic first we talked about the limitations of propositional logic there are certain things which propositional logic fails to explain now these things needs to be we need to be in a position to explain all these things with the help of by augmenting the propositional logics with quantifies such as for all X and there exists some X so predicate logics can in some in a certain sense it can also be viewed as a study of this quantifies quantifies so now these are the single singular terms are usually represented by constants individual constants A B C etc and simple predicates such as mortality etc and all in all beings are mortal mortality is considerably the property which is attributed to all human beings that is considered to be the predicate in any a simple grammatical sentence we have a subject and we have predicate so predicates takes the central position in predicate logic all simple general terms are symbolized by unique capital letters usually we represent predicate letters as capital letters A B C etc and all so we also call this as simple predicates so these terms represent the properties that I think so for example if I say all human beings are mortal mortality is a property which is attributed to the human beings so that property which is attributed to some kind of objects is called as a predicate so there are sentences sentences are in this way for example if you say Ravi is a painter this is analyzed in two parts first is the name that is Ravi and next one next sentence next thing which follows that is is a painter so the first part is a name and refers to some kind of individual thing singular term that is represented by some kind of constants such as ABC etc and the predicate is represented by some kind of capital letter so that sentence is piece has two parts first is the name sentence that is Ravi and is a painter represents the predicate the second part is a simple predicate that identifies some kind of characteristic the characteristic is that being a painter is a characteristic of that particular individual being Ravi so this is the way we represent sentences in the predicate logic so now this sentence is represented as suppose being a painter is taken as P P and Ravi is considered to be a small R then it will the sentence is represented as P subscript R so now what are complex name sentences a basic rule of analysis for the new language that is a predicate logic is that complex expressions must be broken up into some single and simple predications each of which is applied to are attributed to the subject so each idea gets one kind of sentence if you have a complex sentence you break it into some simple sentences and then each kind of property is attributed to both of the subjects that you that occurs here for example let us consider a simple example if you say Ravi is a painter but not a magician so now you write it in this way first sentence is first sentence is represented as Ravi is a painter is represented as RP but not means you have to use conjunction and it is not the case that are M where M stands for magician for example if you want to say that one Moun Singh is a good Prime Minister of India etc. So now let us say the first sentence S stands for one Moun Singh so now you need to represent it in this way there are two things which are there here two things which you can attribute to one Moun Singh in this case sneaky is one property which can be attributed to one Moun Singh and the Prime Minister of India is can also be attributed to him so that is the reason why we wrote it as actually should be the other way around SM and SP where P stands for Prime Minister ship and M stands for one Moun Singh here so SM and PM that is what you can represent it you can represent this particular sentence in this way so what the idea here is that if you have a complex sentence it needs to be broken into simple sentences and then we need to represent these sentences in a particular kind of order. So now let us try to discuss in the predicate logics we begin with the syntax of predicate logic it tells us what kind of things that we are using it in our language of predicate logic and then we move on to semantics of predicate logic where we discuss about what do you mean by saying that a particular sentence in a predicate logic is considered to be true or false that is what we discuss about it and then we will talk about some of the important decision procedure methods with which you can check the validity of a given formula that exist in a given preposition predicate logics and then in that we will discuss at least one or three important methods to start with we will talk about simple semantic tableaux method and then we also talk about resolution refutation method and then in natural direction in the context of predicate logic etc then there is another method with which you can talk about validity of a given formula that is reducing the given predicate logic formula into its corresponding normal form so instead of talking about conjunctive and degenerative normal forms here we talk about prenex normal forms. So if you can reduce any given formula into prenex normal forms then you can talk about validity of a given formula that exist in the predicate logic and then we will talk about then we move on and we talk about identity relations etc and all and we will talk about something about definite descriptions etc so these are the things which are there in the agenda of this predicate logics to start with we need to have some kind of language to begin with so the language of predicate logic consist of these things it is not just sentences and the sentences are combined together with the help of logical connectives and then form complex kind of sentences as is the case of propositional logic but here we need to go into the deeper structure of the sentences so in your syntax the language of your predicate logic we have individual constants they refer to some kind of names or individual things etc chair table etc and all you are referring to that particular kind of table the singular kind of things they refer to they are referred by individual constants a b c can be Ravi Ramu Raju etc and all if you are talking about human beings or if it is talking if you are talking about some table means you are talking about a specific table or you are talking about some kind of monkey or table or any other thing they are talking about a specific kind of monkey so now we have individual variables for example if you want to say we do not know clearly who that person is suppose if any property is attributed to some kind of human beings let us say if you want to refer to some itk students are very bright so the brightness is attributed to some kind of human beings so that some can involve it can be many there may be at least one or it can be a few number of people and so we don't we are not sure who they are exactly so we represent it as some kind of variable X X can be anything it can be this student of that student etc who all come under the category of bright students and then we have predicate let us such as P Q R they refer in general usually we represent these predicate let us by capital let us P Q R 6 so in addition to that we have quantifies so there are usually two quantifies that are used one is the universal quantifier if you are referring to the entire class etc group then you require universal quantifier like in the case of all human beings are mortal mortality is attributed to the whole set of class as class of human beings and existential quantifier is represented as simply there exists some X and as usual we have these connectives which are already there in the case of propositional logic so we need to note that we are just augmenting the propositional logic with two more usually with the quantifiers so that is in order to express this quantifies and all we need to have all these predicates terms etc and all so predicate logics are in a way extension of the prepositioning logics in a sense that you are augmenting this preposition logics with two quantifiers that are there for all X and there exists some X so these are the things that that are there in our language and we have some functional symbols F G H etc and all if it is 0 array kind of thing it is usually represented as 0 G 0 etc and all so now here it is clear that we do not have any propositional kind of force let us that exist here in the predicate logic because 0 array symbols are usually in this in the predicate logics are treated as constants like a b c etc and all so we do not have individual let us such as p q s r etc and all as is the case of propositional logic because they are all 0 array symbols but 0 array symbols are here represented as constants but we have functional symbols with arity at least 1 2 2 n so we do not have propositional letters that means 0 because propositional letters are usually represented as 0 array predicates arity is 0 so it is in that sense we do not require this propositional letters so we just simply use if there are 0 array kind of symbols which exist they are simply treated as constants and as usual we need to have this punctuation box, bracket square brackets etc to avoid disambu to avoid ambiguities in the well-formed formulas so the syntax of predicate logic to in continuation with that we so we use individual constants to formalize names such as a b c etc and all and individual variables like x y z to refer to individual variable words like this man that man etc and all we are not saying which man actually is then predicate symbol variables such as some kind of properties which the object is object possesses which are considered predicate expressions and we use two quantifies for all x and there exists some x to formalize the quantification of expressions expressions now just as in the case of propositional logic we can construct some kind of well-formed formulas within the predicate logic PL stands for predicate logic just as in the case of propositional logic not any kind of string will combine and form a formula well-formed formula in the propositional logic just like that so here also the formulas combine in certain way and then form some kind of well-formed formulas and you need to note that just like in the case of propositional logic whatever formula that you come up with that is not considered a tautology are valid formula so they can be infinitely many strings that you can generate by using the symbols that occurs in that particular kind of language with the help of logical connectives but not all are considered to be valid formulas just like that even in the predicate logic also you can generate some well-formed formulas but not all generated well-formed formulas are considered to be valid formulas valid formula is also considered to be total so we need to have some kind of definition with which you can formulate you can form these well-formed formulas in the predicate logic so these are some of the important rules that one employs in finding out whether a given formula is a well-formed formula or not so if a is an nary predicate later in the vocabulary of your predicate logic and each of the terms t1 to t2 to tn is a constant or that means ABC or it can be a variable it it can be like Px or it can be PC etc so where P is considered to be a property which is attributed to C or even to individual variable like X then a T1 to Tn is considered to be a formula of L so that means in that sense it simply says that we have a predicate P let us say these are all predicates PQ R X and then we have so these are individual constants and all and then there are some variables which refers to this man that man etc so now it says that simply if you write like this that is considered to be a well-formed formula PA or if you simply write this thing that is also considered to be a well-formed formula so now what we have in our language are all these things so this represents predicates it presents individual constants referring to individual things and these refers to individual variables and then we have these two things there exist some X for all X and then as usual as in the case of preposition logic we have all these connectives already so this is implication by implication conjunction and disjunction the set of logical connectives which are already there so now if you simply write like this PA that is considered to be a well-formed formula or if you can even write Px that is what the rule number one tells us so now the second rule is that if P is considered to be well-formed formula and not P is also considered to be well-formed as is the case of prepositional logic the only thing which is extra is the first one so this is what one we can write it in this particular kind of way this is a unary kind of predicate suppose if you want to write this way so if you write in this way X is related to Y in a certain way at some properties attributed to X and Y where X and Y are related in certain kind of way so there are binary predicates and all so the same thing which can be written in this particular kind of thing some test books it can be it is written in this particular kind of sense X Y is in some kind of order X is related to Y in a certain way and they have the proper P so it can be like for example there are two objects such as X and Y for example X is considered to be father of Y then you put like this you write in this way so this means there is some kind of order which is there in this one X is considered to be father of Y so this is different from F Y and X so now we need to replace this with son of this thing so F of X Y if you change order that is not equivalent to its corresponding form so there is some kind of order which is there in this one so here in this formula this can be even written as F X and Y in some test books they prefer to write it in this way X Y and this is a predicate and these are individual variables is also considered to be a well-formed formula in some other test books you will find it in this way F X followed by Y example if you want to invoke this particular kind of relation that is relationship between X and Y is father being a father so that comes first and then followed by that this particular kind of order so this should be read as X is a father of Y so X is a father of Y suppose if you this is different from Y and X X is father of Y does not mean that Y is a father of X so these two are different kind of formulas so as in the case of prepositional logics we have suppose if any variable anything is considered to be well-formed formula not piece also considered to be well-formed formula and P and Q are well-formed formulas then the conjunction disjunction implication and by implication they are all binary connectives if you can combine with any one of this individual variables with these things that is also considered to be a well-formed formula means all the well-formed formulas of prepositional logic are already retained in the predicate logic it is in that sense predicate logic is considered to be an extension of prepositional logic so the fourth rule is that if Phi is a well-formed formula in L and X is a variable then these are the two other things which we have so they are like this so the two extra things that you will find it in the prepositional logics is this suppose if anything like Phi is a formula well-formed formula just a constant or a simple individual variable is also considered to be well-formed and these two combined and then form another kind of well-formed formula so now in addition to that if Phi is also already considered to be well-formed formula then this is also well-formed formula but if you write like this then this is not a well-formed formula so we define it in such a way that first there is quantified followed by that there is a sentence in the same way if something Phi is a well-formed formula even there exists some X such that Phi is also well-formed so these are the additional things which you will find it in the case of predicate logic so now finally only which that which can be generated by using one of these four things is considered to be a well-formed formula should not be in a position to derive any other kind of thing apart from these particular kinds of rules so so these this is what we mean by well-formed formulas in the case of predicate logic the only thing which extra thing which you will find it here is this particular kind of thing so that is Phi is a well-formed formula then for all X Phi is also considered to be well-formed formula and Phi is a well-formed formula there exists some X Phi is also considered to be a well-formed formula so now let us talk something about some of the basic building blocks of predicate logic and with this and this particular kind of lecture so just like in the case of propositional logic we have the basic building blocks for the propositional logics are the propositions the propositions are the sentences which can be clearly spoken as a true or false but in the case of predicate logic there are there are objects that means the things in the world such as dust table chase etc and all individual things I can give these names such as a person John Ravi earth etc and all individual things so they are considered to be the building blocks of predicate logic and then in addition to that we have relations such as properties relations between objects such as example Ravi is related to Sita in such a way that Sita is considered to be wife of Ravi the other way round you should write it here and we also have functions such as functions talks about types of relation that maps an input into some kind of value so now these are some of the building blocks we just now we talked about that particular kind of thing to start with since the predicate logic is all about study of predicates so predicates are occupies the central position here so with this I will end this lecture so I will talk some I will talk something about the predicates so usually we discuss we represent these predicates with capital letters and the predicate letter will usually be associated with list of at least one particular kind of variable for example if there is only one variable here it is called as a unary predicate and if you have two letters X and Y it is a binary predicate if you have three letters such as XYZ for example I will talk about the example in a while from now for example if you say Ravi is a bright student so you simply write it as BR so where B is considered to be predicate that is being bright is considered to be a predicate and then the constant that is here we are referring to R so that is why it is represented as BR that is considered to be unary predicate for example if you want to invoke relation between X and Y on like X is a wife of why then you write W X Y W is W stands for being wife of someone and you need to follow some kind of order so that is a binary predicate suppose if you want if you are trying to talk about the heights of three people in all XYZ suppose X is less than Y less height X X height is less than Y or vice height is less than Z etc so you need to require relationship between three people so that is considered to be tertiary predicate in all so like this based on the number of variables we have unary binary and tertiary kind of or may be nary kind of predicates so predicate is usually used to represent property of his variable that is for example if you say all men are mortal motority is a pretty predicate which is attributed to the variable that is all human beings or a relationship between its variables like for example X is wife of Y etc so now we have the connectives and are implies if and only if etc and these are same connectives that are used in the propositional calculus any addition to that we have LX LX X etc there all LX is considered to be one place predicates because X is related to X only LX Y is considered to be a two place predicate and if you want to invoke the relationship between three group three people for example then you need for three place predicates so in this lecture we just introduced some of the limitations of propositional logic so we discussed that not all the things can be represented in terms of simple relationship between the sentences that is what was done in the case of propositional logic although propositional logics were considered to be sound complete and consistent etc wonderful features are there in that one but that is not sufficient enough to capture many parts of your mathematical reason so the basically our goal was to capture the mathematical reasoning with the help of this first order logic first order logic I mean it is a it is a combination of propositional logic and the predicate in this class we peripherally we discussed about some of the important building blocks of the predicate logics so in the next class I will be talking about what exactly we mean by this predicates terms objects the functions etc and all so when I talk in greater detail about the syntax of predicate logics I will deal with some of this important concepts are the basic building blocks of predicate logic then we will move on to when do we say that a given sentence is true or given sentences false except in the we will continue with this thing in the next class.