 Welcome back in continuation to the last lecture where we discussed primarily about syntax of predicate logic where we discussed about some of the building blocks of predicate logic. So predicate logic was one of the important building blocks of predicate logic are variables constants functional symbols and predicates these are the four things that we have discussed in greater detail and we discussed about the meanings of these things and then we discussed about one of the two important operators which is which is which makes predicate logic distinct from the propositional logic. So they are quantifies so we require these quantifies especially to talk about certain things such as if you want to say that all birds are black for example you need to have a quantifies and then in that context we introduce two quantifies first is for all x and the second one is there exists some x and then let later we discussed about the scope of a quantifier and when a given formula in a predicate logic is considered to be having free variables etc and when we say that a given formula is a closed formula or when a given formula has some kind of ground terms etc and all these things which we have discussed in greater detail in the last class and also we also discussed about we also discussed about some of the important formulas some of the important properties of quantifies especially whether or not quantifies distributes etc and all these are things which we have seen and in that context we have seen that when the quantifies are the same for example if you have for all x for all y and then p x y it is same as for all y and for all x p x y so the order does not make any big difference especially when you have the same quantifies so now in continuation with the syntax of predicate logic so what we will be doing now is we will be discussing something about some of the formation trees of a given well-formed formula in a predicate logic so just as in the case of propositional logic we have something called unique formation tree with respect to a given well-formed formula just like that in the case of a predicate logic as well we have for each given formula you have a corresponding tree a formation tree so the advantage of having formation tree is simply this that suppose if you are not given any parenthesis etc and all so once you draw the formation trees then there will not be much ambiguity with respect to the parenthesis that are concerning the given formula so now the formation trees just as in the case of our propositional logic we can make formation rules for the well-formed formulas in predicate logic explicitly and the definition of such terms as the definition of such terms as occurrences more precise by reformulating everything in terms of formation trees so what we mean by formation tree in the context of propositional logic here is the definition of formation tree so what I am trying to simply say is that each and every given formula will come up with its own unique kind of tree or a formation tree so suppose if you say for all x P x y implies Q x y for example that will have some certain specific kind of formation a unique kind of formation tree when you compare with another formula such as P x y implies not Q x y for example that will have his own formation tree so each formula comes up with this unique formation tree so now what is the definition of a formation tree a formation tree is a finite rooted dyadic tree where each node the one which is at the top most point that is considered to be the node the root point it is like a tree with a trunk and then your branches and leaves etc so a formation tree is a finite rooted dyadic tree where each node carries a formula that is a given formula will be the will be sitting at the node each non-atomic formula suppose if it is an atomic formula the tree ends there itself if it is considered to be non-atomic formula in the context of propositional logic P all individual letters such as propositional variables such as P Q R etc and all there are lot of make formulas in the context of predicate logic constants individuals etc all come under the category of atomic formulas so it carries a formula and each non-atomic formula branches into its immediate formulas and these formulas are called as sub formulas for instance if a is considered to be a formula then the formation tree for a is considered to be kind of unique formation tree which carries that particular kind of formula at its root so one important remark that I like to make here is is that if we can identify the formulas each and every given formula with respect to its formation trees in the abbreviated style then obviously there will be no confusion or ambiguity etc because we face this particular kind of problem earlier especially in the context that suppose if the parenthesis are not given then we follow certain kind of priority which we will be using on the logical operators that we are using in the given formula so in the context of propositional logic and the connective end is given the most important and then followed by that or and then of course negation is there and then implication and double implication that is the order that we follow in the case of repositional logic this is just a mere convention so one of the important thing which I like to point out here is that if you can have formation tree for a given formula and obviously there will be no confusion and all so that formula represent that unique tree represents a given kind of formula so that is the advantage of having the formation trees so for example in the context of propositional logic suppose if you are given a formula like this then it is corresponding formation tree will be like this for example if you have a formation tree like this p ? q and q ? r ? p r so this is one particular kind of string implies not q r s so this tells us that p ? q and q ? r ? this one and the whole thing implies this one so now the formation tree for this particular kind of formula this is considered to be a well-formed formula suppose if you write p q ? r this is not considered to be a well-formed formula that we have discussed in greater detail in the context of propositional logic so now this formula will come up with this unique formation tree so the first thing that we will be doing is this so this is considered to be a non-atomic formula if it is atomic formula it will end with only let us such as p q r etc not p etc this is not an atomic formula so then this will be we will be expanding the expanding this formula into the branches or leaves which you can call so that till to such an extent that we will end up with only atomic formulas so now this is going to be like this first so this branches into the first formula that you have on the left hand side is this thing p ? q ? r ? p r r is the first one and then it branches into not q r s so now this further changes into not q of s it will become q r s and then this q r s still it is not an atomic formula so this is considered to be complex molecular formula it consists of two atomic formulas and so when two atomic formulas combine and then it will become molecular formula then it further reduces to this q s so this is on the right hand side now this branch leads to p ? q and q r r and then the next formula that you will be reading is p r r so this is a way with in which you pass this formula or you will read this for particular kind of formula this is a mere kind of conventional which one will be following so this is still a non-atomic formula this reduces to p and r and then this reduces to p ? q q r r and this further reduces to p and q and q and r so these are the things so now we ended up with all atomic sentences and all and this completes our tree the formation tree for a given formula suppose if you change this particular kind of formula into some other thing and all not q or something like that then it will have its own formation tree and all so now till to this extent it is same now it will become not q r s suppose if you introduce this particular negation so this is completely different from the formula the formation tree that you have seen earlier so that means each and every formula in a prepositional logic comes up with his own formation tree and that has to be unique it cannot be the case that any two formulas will have the same kind of formation tree if they are if you have the same kind of formation tree then they are considered to be logically equivalent otherwise each and every formula comes up with his own formation tree so now this kind of idea we will be extending it to the predicate logic and then we will try to talk about what we mean by saying that a given well-formed formula in a predicate logic will have its own formation tree so why we are doing this particular kind of thing is just because of the case that each and every formula comes up with his own formation tree and that is the reason why there will be no ambiguity when you draw a diagram for formation tree for a given well-formed formula in a predicate logic then the ambiguity with respect to the parenthesis and the parenthesis will not arise when once you form a kind of formation tree so now in the context of predicate logic here is the definition of this thing term formation we have two different things one is term formation tree and the second one is atomic formula formation tree so in the predicate logic we have constants we have term we have constants variables predicates and functional symbols and then with that you can this constants will come in with the help of functions etc will form some kind of terms so now how we form this particular kind of terms so term for we have formation tree for terms first of all and it will be like this a term formation trees are usually considered to be ordered and they are considered to be finitely branching trees it will not go forever and ever and on ultimately tends up with some kind of atomic formulas it is labeled with terms satisfying the following conditions so what are these conditions the conditions are like this the leaves of the given term that is that is what is going to sit at the root of the formation tree and these are labeled with variables and constant symbols and each non-leaf node of T is labeled with a term of the form f and then t1 t tn are considered to be terms so now in node of T that is what is at the top of the tree is going to be labeled with a term of the form f t1 t2 tn that needs to be expanded now which has exactly some kind of successes n successes in the tree can be one it can be two etc so they are labeled with t1 to tn so a term formation tree is associated with the term with which its root node is labeled so so these are the things which we need to note every term T has his own unique formation tree associated with it just as in the case of prepositional logic a given well-formed formula will have his own unique tree just like that we are trying to construct a formation tree for a given term so that is going to be if you change some of the things in that term and that will have his own at formation tree so no two formulas will have same kind of unique formation tree unless until they are logically equivalent or they are the same kinds of terms you are talking about the same kinds of term they will have the same formation tree so the ground terms are in the context of this formation tree the ground terms are those terms whose formation trees will have no variables on their leaves so just imagine a tree which has its root at the node and it is like some kind of up upside down kind of tree it starts with the root that is a node that is where you start with the given term and then you will start it if it is it starts branching it out and it forms it goes to the leaves etc and then if you find there that there are no leaves etc then it is considered to be a kind of ground term so let us consider some examples so that we will get this point clearly for example if you have two terms like this then what is going to be the tree diagram for these kinds of terms so a term can be constituted by these things C is a constant G is a function and X and Y are variables so this is one particular kind of term for which we are trying to form a formation tree and the other one is something like f of DZ and GCA and then W so first we will talk about the formation tree this particular kind of thing so now this is what is sitting at the root and all or you can imagine that it is a trunk of a tree so now so the first term that you are trying to see is this one C and then you have G X and Y and then this can be further mentioned into X and Y so this formula it has it is not these are not considered to be ground terms because this formula consists of variables X and Y suppose if your formation tree has only ground terms only then they do not have any free variables then that is considered to be the ground term for example if you take the same formula like this f of C G of A B for example so this is considered to be term that we are trying to form a formation tree for this particular kind of term so what is important in predicate logics is this thing first you have variables X Y Z etc and then we have constants like A B C which are referring to some kind of individuals and then you have functional symbols like F G H etc and then you have predicates so predicates can be something like P X and Y etc where X and Y are related in such a way they have some kind of property like X is a father of Y X is a brother of Y etc so these are the things which are the building blocks of this thing and then we have to quantify for all X there exists some X so now this is considered to be a term because you have constants and you have a function and then you have followed by some kind of constants so this is considered to be term which we have defined the meaning of terms in the last class last few classes so we can go back to that particular kind of definition of the term so now what we are trying to do here is this thing we are trying to draw the formation tree for this one so this is going to be like this the first letter is going to see is C and then this is G of A so this is the way in which you read the formula so it starts with the letter C and then we will go on to the next one that is G A B that is the way computer reads out the things it goes from left to right that is a convention that we fall so now this further it is as to this one A and B so now we need to one needs to observe that in this particular kind of formula these are considered to be leaves of your root root of the tree so all these terms are constants they are not considered to be free variables so variables are like so XYZ etc. where anything you can substitute into XYZ etc so now in this formula all are considered to be constants only and the hen hence this particular kind of thing is called as ground term a ground term is a term which has only constants it does not have free variables at the leaves so you have C A B etc all these things are constants so that is why this is considered to be a ground term whereas this one is not considered to be a ground term because it has variables X and Y so anything we can substitute for X and Y depends upon what you substitute for X and Y and interpretation of this one changes so now coming back to this particular kind of formula so this is going to be this thing F B Z and then you have G C A and then you whatever you have so now this stops here itself you have term which consists of constant or something like that variable and now this changes to C and A and this goes to B and Z so this is the formation tree for this particular kind of term whereas the formation tree for this one is considered to be this one so that is the reason why you know these two terms are considered to be different because they have different kind of tree structure a tree formation tree so it is for this reason that every term will have its own unique formation tree so now we talked about how to draw formation tree for a term and then one important observation that we made is that if it is considered if at all it is considered to be a ground term it will have only constants it does not have any free variables at the leaves if it has free variables at the leaves at the part of leaves then it is not considered to be a ground term so now let us consider the definition of so now we have we talked about the formation tree for the terms now let us talk about something about atomic formula auxiliary formation trees so what are these atomic formula auxiliary formation trees they are the label ordered finitely branching trees again as is the case of the one which you have seen earlier of depth 1 it is depth 1 because it is all atomic formulas and whose root node is labeled with some kind of atomic formula and if the root node of such a tree is labeled with the nary relations like R of T 1 T 2 to T n then it has an immediate successes which are labeled in order with the terms T 1 to T n that is the first thing which you need to note the second one is that an atomic formula formation trees are finitely branching labeled order trees which are obtained from some kind of auxiliary trees by attaching at each leaf label with some kind of term T and the rest of the formation tree are associated with another term T such T is associated with the atomic formula with which the root is considered to be labeled. So now let us consider some examples of atomic formation trees so then we will understand this particular kind of the definition that we have discussed earlier so what I am exactly trying to say is that a given term will have its own corresponding formation tree and the same way an atomic formula will have its own corresponding formation tree let us consider an atomic formation tree and all like this C f of x y and g of let us say a z and w so this is considered to be an atomic formula so now the first one it has these three leaves so now the first one is C and then followed by that x and y and then g of a z w so now this ends here itself and then it goes to x and y and then you have three letters out of this one is some is some are constant some are variables and then a z w so this is considered to be the atomic formation tree for this particular kind of formula so these are atomic kind of well formed formulas and all so in the context of predicate logic we have three different things first is a given term will have its own corresponding formation tree and then the second one is atomic formation atomic formula which will have its own corresponding formation tree and the second one is anything which is considered to be a formula a formula in the predicate logic is simply like this that it will have at least one free variable if it has no free variable then it is considered to be closed kind of formula or it is also considered to be a sentence in the predicate logic all these definitions which we have considered in the last class so now so we discussed about the formation tree for a term and then when we have no free variables in that particular kind of formation tree then we called it as a ground term otherwise it is not a consider to be a ground term and then we discussed about atomic formulas and then so on corresponding trees and then let us talk about some kind of formulas which exist in the predicate logic in general it starts with there exists some x for all x etc and starts with the quantifies. So now the formula auxiliary formation trees are again considered to be a label ordered binary branching trees that trees considered labeled as T where the leaves of T are labeled with atomic formulas ultimately at the end of your formation tree all these things are considered to be atomic formulas. So now if you consider sigma as a non-leaf node that is what is going to extend to leaves node of T with one immediate successor so that is let us consider at sigma conjunction 0 which is labeled with a formula either it can be simply a formula 5 or a quantifier followed by a formula it can be existential quantifier or universal quantifier for some particular kind of variable V so that is what is the case in the first instance and suppose if sigma is considered to be a non-leaf node again T with two immediate successes is sigma conjunction 0 and sigma conjunction 1 that stands for two immediate successes then which are labeled with formulas let us say Phi and Psi then that sigma has to be labeled with either the conjunction of these two formulas or disjunction of these two formulas or implication or double implication so these are three things which we can have this just tells us that you know how to form this how to form a formation tree for a given kind of formula just we are giving some piece by the analysis for how to form this particular kind of formation tree. So the formula formation trees are again considered to be order label trees gotten from auxiliary ones by attaching to each leaf labeled with an atomic formula and the rest of its associated with the formation tree so each such tree is again associated with the formula with which it is marked which is labeled so let us consider some examples for this formula auxiliary trees and then we will see the difference between this thing so now one particular kind of example which we will be talking about which where we talk about formula formation tree so it starts with this thing auxiliary formula formation tree on usually it starts with there exists some x we started with terms and then atomic formulas and then we have these things a given formula in a predicate logic it will have its own formation tree so now quantified followed by some kind of atomic formula f of x and y and g of a z and w conjunction for all x let us consider another term such as r c f x y g a z w is a conjunction of these two formulas there exists some x r c f x y etc now for all x r c f x y etc first you need to note that it is not considered to be a close formula a close formulas will be like this for example for all x for all y p x and y so this is considered to be a closed formula because it does not have any free variables no free variables so that is why it is considered to be a closed kind of formula so now in this case with respect to x this y is free and even other things such as z etc they are also free here a formula in that context this is considered to be a sentence in predicate logic so whereas this particular kind of thing is considered to be formula in predicate logic because it has free variables y z and with respect to this is a conjunction with respect to this quantifier we have free variables such as x etc so in that context this is considered to be a formula but not a sentence in the predicate logic a sentence will have no free variables all the variables that exist in your formula are bounded by the given quantifiers for example x and y are both bounded in this particular kind of formula with respect to for all x and for all y with respect to these two quantifier x and y does not have any freedom so now we are trying to form a formation tree for this particular kind of formula so this will have quantifiers followed by atomic formulas and then quantified followed by some kind of atomic formulas so now this will have this thing first you need to write this thing or c f f x y g a z w and the second one is for all x or c f x y g a z w so now left hand side can be expanded to this thing so now we have you have to eliminate this quantifier and then this will be not eliminating so this is a way which which you form the reason x and y g of a z w will be expanding only the left hand side of this thing right hand side you can expand it by using same kind of thing so now it will be like this it will be further expanded to f of x and y and then g of a z w so this further expands to so an x and y and then g expands to a z w so in this way we can expand this particular we can construct a given formation tree for a given formula so only thing what one needs to notice is that each and every given formula will have its unique formation tree so with that particular kind of thing the important thing which will be noticing is that we can if you have a formation tree then you do not have to worry much about the parenthesis for example if you take particular kind of formula like this so usually this is the convention that we will be following we have considered some examples earlier and the preference ordering will be given like this first preference will be given to for all x and followed by that there exists some x and then negation and then followed by that you have disjunction convention disjunction and then implication and double implication for example if you have formula like this thing like you do not have any parenthesis or anything then let us see how to write this particular kind of formula so q x implies for all x p x and q x etc suppose if you have if you are given formula like this thing where there are no parenthesis which are given here you can consider r x also where p q r are all predicates and then for this there are variables now sometimes I will be writing like this and we can even write like this also in some textbooks q x is written in this sense I will be writing in this particular kind of way so now suppose if you are given a formula like this then where there are no parenthesis here there if there are no parenthesis the problem as in the case of prepositional logic for example if you have a formula like this p ? q r and this formula can be read in varieties of ways is it the case that p ? q r r is the case that this is one way of reading this particular kind of formula and the second way of reading the formula is this thing p implies q r r this is one way of reading it so there is a confusion of how to read this particular kind of formula now so for that we have we used some kind of convention with which you know we have come up with this thing so in the case in the context of prepositional logic we have this thing conjunction or disjunction implication and if and only if so now if you are given a formula like this p ? q r so now we have only disjunction and implication here so the first preference out of this implication and disjunction is this one so that is why we have put bracket here q r r and then you put bracket on this one the next one is for implication so this formula should be read as p ? q r r and you can draw formation tree for this one it will be like this p ? q r r so now it will be like p and then q r r so now it will be like this one q r r so this is the formation tree for this particular kind of thing suppose if you draw if you have a formula like this thing p ? q r r and the formation tree for this one is like this p ? q r p and q so this x and y the formation tree for these two things are looking quite different and all so that is why each and every formula will come up with its own formation tree in the same way for example in this particular kind of formula the predicate logic formula first we need to do what we need to do is we need to follow this particular kind of preferential kind of ordering for all x there exist some x negation and or implication and this one so now in this particular kind of formula so wherever for all x is there you have to put it in brackets that is the one first thing which we need to do and that the first step we will do this thing and then we will follow the second step later for all x p x and q x is the first step and the second step is that you need to give preference to there exist some x so now this will remain as it is except that you need to put bracket here and then there is no existential quantify in order to worry much now the next preference is given to end so now so this will be like this so for all x p x now for this one we need to put bracket here this will become p x and not q x and wherever end is there here you need to put bracket here p x and the whole formula next the formula which is closer to this particular kind of thing within the scope of this quantity you put brackets here so now this is what is going to be the case since there are no implication or implication is there here so now this formula will become like this for all x p x p x and there exist some x q x implies for all x p x and q x now it will have its own formation tree so I think we will end this lecture by saying that what we have done in this lecture is that each and every predicate logical formula comes up with his own formation tree to start with we have constructed a formation tree for term and then we said that if there are no free variables in that particular kind of term we said that it is considered to be a ground term and then we form we constructed well form we constructed formation trees for the atomic formulas and then obviously we have when atomic formulas are combined we will have compound kind of formula auxiliary kind of formula and then we talked about formula auxiliary formation trees by constructing for its own corresponding formation tree in the next class what we will be doing is that so what we will be doing is we will be talking about the semantics of predicate logic semantics of predicate logic is not as simple as the case of prepositional logic in the prepositional logic for example if you want to say that if you want to talk about the truth value of grass is green and moon is made up of green cheese what you will be doing is simply this that you have the semantics of conjunction so a conjunction is going to be true only when both the conjunctions are true and if you if you just know the truth value of these two individual prepositional variables this is a sentence which is represented by prepositional variables if you know the value of P what value P takes what you know what value of Q takes in all and based on that you can talk about the truth value of a given compound sentence whereas this is not as simple as in the case of prepositional logic in the case of predicate logic we have variables we have constants we have functional symbols and your predicates then what is required is we need to go into the details of deep structure of this particular kinds of things where we need to assign some kind of things to this variables constants etc and all constants predicates etc then in that context we will be talking about something called as interpretation interpretation is nothing but assigning some kind of values to some kind of things to this variables constants predicates and functional symbols within the domain and other important thing is that it does not make any sense to talk about truth value of a given predicate logic formula without referring to any domain so we need to talk about we need to fix the domain then only we can talk about truth value of a given predicate logical formula it does not make any sense to talk about whether or not a given formula is true without referring to any domain now so whatever is true with respect to suppose if you take the domain as people then if you take the domain as example real numbers or some other things it might change you know so the interpretation might change then we will be talking about the certain things which are considered to be true in all structures and we call it as tautology is now so all these things which we will be talking about in the next class where we will be dealing with the semantics of the predicate logic.