 Welcome back in the last lecture we discussed began with the semantics of predicate logic where we discussed about when do we say that a given well formed formula in the predicate logic is true when it is going to be false etc. In continuation with the last discussion of the last lecture we will be continuing and then we will be talking about some more examples so that we can get this idea in a better way. So we will try to talk about the semantics of the predicate logic in greater detail with some examples in this particular kind of lecture. So what is important for the semantics of the predicate logic is there are two things one is the domain it does not make any sense to talk about the truth value of a given predicate logic formula without respect to some kind of domain. So we need to fix a domain it can be natural number it can be real number it can be set up people say rivers etc and all we need to fix the domain and then you need to have an interpretation function I and that constitutes D and I constitutes what we call it as a model or a structure etc. So now in this particular kind of context we defined what we mean by we provided formal definition of a structure so essentially what it talks about is that in the predicate logic we have variables constants predicates and functional symbols each one when you assign some kind of some values to these things it has to find some kind of an entity in the domain where the predicates are mapped to 0 and 1 that means the property whether or not it holds or not is the one which we are going to see and then each individual constant should have a member in the domain D etc each function symbol finds another kind of energy functional symbol in the domain etc. So now how can we define truth the truth of a sentence Phi a given formula Phi in an L with respect to some kind of structure which consists of domain and the interpretation function structure a in which an element a belongs to a is named by a ground term of L and is defined by means of some induction like this for atomic sentences are T1 to Tn in a given structure a that is going to hold that means our T1 to Tn is going to be true with respect to a structure a if and only if if you have R to the power of a T1 rise to the power of a to T to the Tn rise to the power of a so that means the relation R a on a n assigned to R holds of the elements named by the terms T1 to Tn otherwise it is going to be false. So we will give some examples to talk more about the definition of truth with respect to the formulas in the predicate logic in the context of propositional logic we have seen what we mean by saying that a particular formula is true or false with respect to a structure a a model a for example not Phi is going to be true in a structure or model a obviously when it is not the case that Phi follows from a actually it should be written in this particular kind of sense so a does not belong to Phi if that is the case then not Phi is considered to be true in this particular kind of model a in the same way Phi R Psi is going to be true in a model a if either Phi is true in a model a and Psi is true in a model a that is a standard definition and the conjunction is going to be true when both conjuncts are true that is taken care by the third one and implication that is going to be true only when if you have premises true and the conclusion falls in the same way Phi if and only if Psi these are the things which are exactly same as the case of propositional logic so the additional things that we have in the case of predicate logic are some of the truth values with respect to quantifies what is extra in predicate logic are two more operators they are considered to be quantifies first is there exists some V Phi V that is true with respect to a structure a if that is going to be true at least for one ground term T that means if you substitute one ground term T then this there exists some V Phi of V holds then that particular kind of formula is true with respect to a structure a that means for some ground term T Phi of T that means X is V is substituted by T and then Phi T has to be true in a structure a if that is the case for at least one of the ground term T then obviously that is called as there exists some V Phi of V is true with respect to structure a and for all V Phi V is going to be true in a structure a if it happens for if Phi of T is going to be true for all the values of T whatever value that you are going to take into consideration for T as in all these cases Phi of T has to be true in that sense we call it as for all V Phi V is going to be true with respect to a model a we can talk about some other important logical properties such as satisfiability validity in this context. So a sentence Phi in a predicate logic first of all what is considered to be sentence in predicate logic we does not have any free variables then obviously it is called as a sentence otherwise it is going to be formula in the predicate logic so that sentence Phi of L the language of predicate is considered to be valid which is usually represented as models and Phi especially if it is true in all structures for L if that is going to be true in all kinds of structures whatever interpretation that you are in all interpretation that is going to be true it is considered to be topology so that is what we mean by validity truth in all structures is considered to be what we mean by validity and given a set of sentences Sigma which consists of P1 to Pn we say that P1 is considered to be logical consequence of Sigma if and only if P1 is true in every structure in which that particular kind of structure all the members of Sigma are also going to be true in all the members of Sigma are true P1 also has to be true if that is a case then we say that P1 is a logical consequence of Sigma and third important thing is that a set of sentences let us say P1 to Pn is going to be satisfiable if there is at least one structure a in which all the members of Sigma are true where P1 is also true in such a structure is also called as model of a given set of formula Sigma if Sigma has no model then obviously that means there is no interpretation in which X is true it means you are not able to find out at least one interpretation in which your X is true then it is called as unsatisfiable. Let us consider some examples let us take this into consideration FB connection of the following three formulas they are F1, F2, F3 the first one is represented as red as for all X there exists some Y or XY now our XY means here that X is less than Y suppose if you take one obviously it is less than two if you take the natural numbers into consideration if you take pick up X as one and Y as two and obviously one is less than two that is what we mean by RXY and then you have F which is a conjunction of all these things F1, F2, F3 now we are going to show that it is going to be satisfiable in the domain of natural numbers it might be false with respect to real numbers some other numbers etc but we are going to say that it is going to be satisfiable when you say that this conjunction of formulas are going to be satisfiable at least in one interpretation in which this particular kind of property F1 and F2, F3 is going to be true then it is called as satisfiable otherwise it is going to be unsatisfiable. So now F1 is the formula which is represented in the sense for all X there exists some kind of Y or XY in the context of natural numbers for all X whatever number that you have taken to consideration in the domain of natural numbers there always exists some kind of Y where that particular kind of X is always X is less than Y. So for example if you take into consideration one to be the particular kind of thing there always exists there exists some kind of Y2 which is less than sorry if you take X to be greater than 1 then all the elements all the numbers greater than 1 2, 3, 4, 5, 6 etc and all for all those numbers obviously one is less than those particular kind of numbers. The second one is there does not exist X such that our X relation is X is less than X and for all X for all Y for all Z the third one is stating that if X is less than Y and Y is less than Z then obviously X has to be less than Z it happens to all XYZ in that holds otherwise it is going to be false for example if you take three numbers 1, 2, 3 etc and all one is less than 2, 2 is less than 3 obviously one has to be less than 3 in the same way 2, 3, 4 if you take into consideration in order 2 is less than 3, 3 is less than 4 that means 2 obviously has to be less than 4 so now analysis is like this so now in at least one particular kind of case where it this property holds in particular then F1 is going to be true F2 is going to be true F3 is satisfiable so then each F1, F2, F3 are satisfiable then F is obviously considered to be satisfiable so the first one for any natural number X there is always a number Y number Y there is a number Y such that X is less than Y you always find some kind of arrangement like this for any number that you take into consideration X that always exists some kind of Y where X is less than Y natural numbers it consists of 1, 2, infinity and all for example if you take any number such as let us say 25 you take into consideration then always there exists some kind of number which is greater than that one which is less than another number set 26, 29, 30, 35 etc and all so at least one such kind of situation it happens so that is why for all X whatever number that you are taken into consideration in the natural numbers there always exists some kind of Y where X is always less than Y so that holds that satisfies. Now the second thing is there does not exist X or X so it is written as for all X it is not the case that R X is there is no number of X is less than itself which is obviously the case in terms of natural number suppose if you add 0 to it then things will change or minus if you add integers to it this may not hold for in the case of natural numbers if you take 234 anything into consideration that 2 cannot be less than its own number that is 2 it has to be equal to 2 it is definitely not less than 2 so that also holds and the third one for any numbers you take any natural numbers into consideration in some kind of order if X is less than Y this holds and Y is less than Z and then obviously X is obviously considered to be less than Z if you take 234 etc and all two is less than three three is less than four obviously two is less than four. So now let us consider some interesting formula that is stated as stated in this way for all X P X implies there exists X P X this formula is going to be valid in all non empty domains when you say that domain is non empty at least it has at least domain is domain has some kind of objects otherwise the domain is considered to be empty for example if you talk about set of people at least some kind of people have to be there in that domain otherwise if there are no people etc and all only animals non living beings etc and all the domain is considered to be empty. So if every element X has a property P then of course there is at least one X in it having that particular kind of property for example if you say that all human beings die in some day or other if at least one human being has that particular kind of property I mean everyone has to die in some day or other. So then it means by saying that some XYZ if you take it arbitrarily from the domain of people that they also have they also satisfy this particular kind of property P so obviously the formula seems to be certainly valid in case of non empty domain that means the domain consists of set of people in that if it happens for all the things for example if you take into consideration set of birds for example birds crows in particular if all crows are black most of the crows are black and all then if it holds for all the crows and all then you take any two or three birds into consideration which are considered to be crows which are also considered to be black obviously. So for all XPX if it holds then there exists some XPX also holds so this happens only with respect to a non empty domain but what happens if you take into consideration empty domain like like for example unicorns devils demons etc all empty domains it does not exist so in this case what happens is that for all XPX is going to be true for any choice of P because empty set is set of all the sets so in that sense for all XPX is going to be true of any choice of P but the consequent in this conditional that is there exists some XPX that is going to be false because that leads to the existence of X and so for all XPX does not we do not have any commitment that that particular X has to exist in the universe with respect to empty domain for all XPX is going to be true and with respect to empty domain there exists some XPX is going to be false. So for any interpretation I mean any structure that you are taking to consider which has domain and interpretation function etc where the antecedent is true here I mean the forward XPX is going to be true whereas the consequent is going to be false here there exists some XPX is false hence the given well-formed formula is going to be false and hence this formula is going to be invalid with respect to empty domain. So in general when we try to evaluate the well-formed formulas that means evaluating the truth conditions of given well-formed formula in a predicated logic we usually take into consideration that the domain is non-empty you can also take into consideration the empty domain then in that case only universal quantifiers the formulas which begin with universal quantifiers are going to be true and others the property px with universal quantifier is going to be true and existential quantifier there exists some XPX is going to be false. So these are some of the things which we need to talk about in the context of semantics of predicate logic. So now a formula Phi of a language L which consists of free variables v1 to vn is considered to be valid in a structure a for L which is represented as Phi models a Phi is a semantic consequence of a structure a if the universal closure of Phi so that is the sentences for all v1 to vn to Phi which you got it by putting for all vi in front of Phi for every free variable vi in that exist in Phi and that if it happens to be true in a for all vi that the formula is going to be true in that structure a then obviously Phi is true in that particular kind of form structure a so a formula Phi of L is considered to be valid if it is valid in every structure for L otherwise it is considered to be an invalid formula. So now let us consider some more examples considered a language which is specified by some kind of binary relation symbol or which relates to objects in some way it can be plus it can be greater than minus etc and all there are binary operations and we have some constant C0 C1 to Cn and we can talk about two possible structures in the context of the formal definition of validity a structure that we are given earlier now let us talk about a domain which consists of here D sometimes you write it as a D etc it consists of natural numbers and let R a be usually the relation or with respect to structure a that is usual relation we take into consideration less than and then there are some constants which find some kind of members in the domain you write it as C0 rise to the power of a that is when it is the case it is 0 and if you take 1 into consideration C1 it takes the value 1 and now in that context the sentence for all x there exists some y or x y says that in the context of natural members we are taken the domain domain as D and now we assigned some kind of values to the constants and we have a relation which is a function between this thing and now the sentence for all x there exists some y or x y states that states like this for every natural number there is a larger one that R x y stands for this thing for all x there exists some y means for every natural number that is for all x there exists some y means there is a larger one y so obviously that formula for all x there exists some y x y where x is less than that particular kind of for example if I take a number as 25 25 etc and all 41 there always there will be number 42 which is this 41 is always less than 42 you always come across with a number which is greater than 41 so if R a is considered to be usual relation greater than then this particular kind of sentence is going to be false for example if I take 1 and 2 into consideration 2 and 1 into consideration then for all x there exists some y that is there exists at least one y that is one which is less than greater than that one one is not greater than 2 so that is why this sentence is going to be false so depending upon how you define your function that R x y and the domain that matters to us so your interpretation also changes so now let us consider domain a to be rational numbers q to be q0 to qn and r a to be again it is taken as a relation less than and constants are represented in this sense we see the power of C0 a is 0 and C1 a is 1 you are taking into consideration to constants 0 and 1 now the sentence for all x for all y R x y implies R there exists some x R x z and R z y that is going to be true in this structure it says that usually rationals are dense however the same thing is going to be false with respect to this is going to be true with respect to q but the same thing is going to be false with respect to natural numbers so what essentially I am trying to say is that same formula is going to be true with respect to some kind of domain of natural numbers same thing when you take real numbers into consideration the same formula here in this case for all x for all y R x y implies so and so the formula is going to be false so now let us consider some more examples two formulas for all x px and there exists some x not px now let an interpretation be as follows you have a domain D which consists of two numbers usually natural numbers one and two and you have an assignment for P so whenever you have P P to the power of one is going to be T when it is to that formula is going to be false and now we need to show whether the following formulas are true under this particular kind of interpretation so now the first formula for all x px this kind of property px is going to be false when it takes a value to so it is not true for all the values of x so that is why for all x px is going to be false because px is not true with respect not true for both it is true for x is equal to one but definitely it is not true for x is equal to two because we said that P of two is false it is not true for all the things is true for only one particular kind of thing only there exists some x px holds rather than for all x px so now if you take the second thing into consideration there exists some x not px which is going to be true in this interpretation because not of P to obviously is going to be true in this particular kind of interpretation so if it is satisfied by at least one particular kind of interpretation then there exists some x not px is going to be satisfiable otherwise it is going to be unsatisfiable if it is true in all the interpretation it is unsatisfiable if it is false in all the interpretations it is considered to be unsatisfiable so here there exists some x not px is true in this particular kind of then at least one interpretation in which the formula is going to be true and that will serve over purpose so now let us consider important theorem which is which is stated in the sense is stated as follows let ? be an open formula of a predicate logic that means quantified free I mean it is at least free variables in that kind of thing it is a formula and we may view ? as a formula ? of prepositional logic by regarding every atomics of formula ? as the prepositional letter. So what is theorem essentially says is that you have some kind of tautologies in prepositional logic and if you substituted with some kind of instances when you have a substitution instance which are formulas in the predicate logic that are also going to be tautologies for example in this case p ? p is considered to be a tautology or px ? px so now you substituted it like this thing for all x px implies for all x px for example if you say that thing it is obviously going to be a tautology so in this case for all x px implies there exists some x px that is going to be go going to be true in the non-empty domain but definitely is going to be false that is what we have seen earlier that is this formula is going to be false so now if you have if you have a formula PC if something holds for some particular kind of entity then you can say that there is at least one kind of entity which has this particular kind of property if at least one chalk piece is white in color and you can say that there exists some chalk piece such that this chalk piece is white in color so the always holds so it is considered to be a tautology. So in the same way for all x we have we know that p implies p if and only if not not p is true then you replace it with px in this particular kind of formula an instance of preposition logic is a substitution instance of tautology in the preposition logic and that is also considered to be a tautology. So now let us consider some more examples so that we will understand this particular kind of things the semantics of the predicate logic in a better way. So let us consider one simple example and we will stop here. So the problem states like this what do the following formulas mean meaning of a formula means giving the truth conditions that is what we mean by that are they true are they true or false so now we are taking into consideration few examples simple examples so for a predicate logical formula to be true or false we need to have a domain first of all and then you need to have an interpretation right. So now this is the formula which we have x2 greater than 0 so now where the universe of discourse or a domain is like this is set of real numbers this universe of discourse is considered to be set of real numbers which are represented as this R so what are real numbers we have all these things natural numbers 1 to infinity and then we have whole numbers that is 0 1 to all the natural numbers together with 0 whole numbers and then we have race integers and it is just like minus 1 minus 2 minus 3 etc this is minus infinity and then plus infinity 1 2 3 etc and then you have rational numbers 1 by 2 2 by 3 etc and all so all these things are considered to be real numbers so now if you take this particular kind of formula into consideration with respect to real numbers now we want to see whether this particular kind of formula is going to be true or not so now if you take you know universe of discourse to be only natural numbers so now for example if you take natural numbers into consideration to take x into x as 1 then it says that 1 square is less than 0 obviously it is less than 0 so for natural numbers it seems to be the case that whatever value that you substitute for x is going to hold 2 square which is 4 which is obviously greater than or equivalent to 0 which is greater than 0 so now this particular kind of formula for all x the x square is greater than or equivalent to 0 for every real number x we have this particular kind of thing x square is greater than 0 is the case so that is why this is going to be T that means in all these situations even if you take into consideration minus 2 or minus 1 etc and all minus 1 whole square is equal to 1 obviously 1 is greater than 0 so this formula holds for the real numbers so hence that is that formula is going to be true so now let us consider another example for all x x square is greater than 0 you remove this this thing x square greater than 0 but here real numbers also consist of this whole numbers also example if you substitute 0 square then definitely 0 is not greater than 0 and all but 0 is greater than or equivalent to 0 so now if you take x square greater than 0 now if you take this into consideration and that is going to be false in at least one instance this formula is going to be false then this does not hold for all x x square is greater than 0 does not hold so that is why this formula is going to be false whereas this particular kind of formula going to hold because if you take 0 into consideration this formula is telling us that at least one x for all x for example if you take 0 into consideration 0 square is 0 only that is greater than or equivalent to 0 the second condition holds and all that is 0 is equivalent to 0 but in this case it is strictly stating that 0 is greater than 0 which is considered to be false so this formula does not hold in particular for the real numbers so now if you take another kind of formula so now let us consider the domain to be real numbers only that is the domain which is written in this sense so now if you take another example such as there exists some x square plus 1 is equivalent to 0 so now in this case so is there any real number which satisfies this particular kind of property for example if you take 1 2 3 etc and all natural numbers then suppose if you take 1 square plus 1 is equal to 2 is not equivalent to 0 it does not satisfy this particular kind of thing or you take 2 or anything into consideration any natural number that you are going to take into consideration is always it is not equivalent to 0 so now coming back to the whole numbers if you take 0 into consideration 0 square plus 1 which is obviously equivalent to 1 so there also it is not going to satisfy the whole numbers also I mean it is not true in any domain so now let us consider integers it consists of even negative numbers also suppose if you take minus 1 minus 2 whole square for example it is considered to be 4 4 plus 1 5 which is not equivalent to 0 even that also it will not hold and then this is the integers and even if you take into consideration rational numbers and this is not going to be equivalent to 0 and that means that this formula x square plus 1 is equal to 0 it does not hold in any structure and all so the formula which does not hold in any structure is considered to be contradiction so x square plus 1 is equal to 0 for example if we just talk about only x square plus 1 is equal to x square plus 1 is equal to 0 and usually we write it as square is equal to minus 1 and x is equal to something like pressure minus i x is a complex number and all which is different from the real numbers so there is no model or no structure which satisfies this particular kind of formula that means this formula has to be contradiction and now if you take into consideration some other examples such as there exists some x x square plus x minus 2 is equal to 0 whether this is going to hold in some cases are not is the one which you are trying to see whether if you take natural numbers into consideration if you substitute one for it what will happen 1 plus 1 2 and 2 minus 2 is equal to 0 that means it holds in at least in the case of natural numbers at least one instance this formula is going to be true then this is going to be the whole formula is going to be T so that means this formula is going to be true that is true it holds for all at least natural numbers then that particular kind of formula is obviously true. So in this lecture what we have seen is that we started with the semantics of predicate logic the definitions and then we have seen with some examples when a given formula is true and when a given formula is considered to be false the same formula is considered to be true of some kind of domain which is considered to be false in some other kind of domains. So in the next lecture what we will be talking about is some important decision procedure method which is called as which we have been using it in the case of the context of prepositional logic that is the semantic tableaux method and using semantic tableaux method will be dealing with some of the important logical properties such as when group of statements are satisfiable with us in the predicate logic when a given formula is considered to be a tautology when a given formula is contradiction etc all these things we will be talking about in the next class.