 Welcome back in the last class we discussed something about prepositional logic, minimal feature what are the important features of prepositional logic that is what we have discussed. We initiated our discussion with syntax and today we will be talking about syntax in greater detail. So prepositional logic is a study of logic of prepositions where prepositions are considered to be simple sentences in which you can clearly one can clearly speak them to be either true or false you know. So they are all considered to be declarative sentences you know. So these are the declarative sentences are only sentences which are within our per view you know. So prepositional logic in a in one sense it can be viewed as a language the formal language which has its own syntax and semantics you know just like English language ordinary English language has its own syntax that is alphabets etc and all and alphabets combine in certain way and form meaningful words and these words will combine in certain way and form meaningful sentences and all and then the grammar with the help of grammar we will be able to know which sentence is correct like cat is on the mat seems to be a better sentence and all appropriate sentence whereas if you one can talk about mat cat on and all so anyone who learns who knows grammar and all they will immediately come to the conclusion that it is not an appropriate sentence and all just like that just like in the case of English language in the case of formal languages like the language of prepositional logic we have our own syntax and semantics you know. So in this class I will be focusing on the syntax of prepositional logic in the last class we discussed in some detail about the syntax how what we mean by syntax etc and all in the last class so we will go into the details of what exactly we mean by syntax and how different kinds of formulas are generated with the help of some kind of syntactic rules. So just like ordinary English language has grammar and all grammar decides which sentence is appropriate meaningfully grammatically correct and grammatically incorrect sentence etc and all in prepositional logic we have what we call as well-formed formulas and all. So to start with the language of prepositional logic has some kind of alphabets which are considered to be prepositional variables which represent some kind of prepositions and all like for example if I say it is raining it is simply represented as the basic units that is the preposition so that is it can be represented as simple letter r and suppose if you say it is raining or it is not raining then you simply represent it as r not r and all. So we have to begin with we have prepositional variables we have n number of variables which are available to us to represent all the sentences that are generated in our prepositional language and in addition to that we have five logical connectives they are n r implies negation double implication and in addition to that just like ordinary English language has punctuation marks like full stop, etc and all in the same way in order to read the formulas properly we need to have some kind of punctuation marks punctuation marks in the case of prepositional logic or parenthesis left parenthesis right parenthesis and sometimes we use even, or full stop so all our convenience and all. So prepositional logic has I mean what we need to talk about when these meaningful strings combined together and form a kind of meaningful formula not all kinds of strings combined together and form some kind of formula in the prepositional logic. So for that we need to define what we mean by a well-formed formula and all for example if you say in English language mad cat on and all it is not well-formed kind of thing is at least grammatically incorrect in the same way an analogy in prepositional logic is that if there are many strings in all PQ and with logical connectives punctuation marks and all suppose if you have a sentence like this which is generated in this way so we have prepositional variables PQRs etc and all and then this is the language of prepositional logic and then we have logical connectives R and N implies N if and only if so this stands for N this stands for implies the material implication and then this stands for if and only if which is used for invoking necessary and sufficient conditions in addition to that what we have said is simple that is this is the left parenthesis and this is right parenthesis and then this is a punctuation mark which sometimes we use it. So sometimes suppose if you there are many formulas which can be generated from these things in all so like for example if you have if you write like this R implies Q etc and all so this is a string which is generated in the language of prepositional logic but it is not a well-formed formula not a well-formed formula so then what we mean by a well-formed formula so we require a definition for a well-formed formula for example if you write like this PRQ implies R so this is considered to be a well-formed formula whereas whatever is written above that is A QP implies and R implies Q etc and all this is as good as mad cat on etc and all so this is not considered to be a well-formed formula whereas this is considered to be a well-formed formula in prepositional logic so what is the definition with which we can say that the first one is a well-formed formula and the second one is a well-formed formula and all so here is the definition which we have so the definition of well-formed formula is like this every prepositional variable like P is Q is R etc and all suppose if you write like this just P is Q is R this stands for atomic sentences they are already well-formed formulas so that is the first thing and the second thing is that if something is a well-formed formula let us say P Q or anything and the negation of that one is also a well-formed formula so that means suppose if you have this P is Q is R etc they are all well-formed formulas and all atomic sentences are all automatically well-formed formulas if you put one symbol this is this stands for not and all if you write not P this is a well-formed formula in the same way not Q is a well-formed formula and whereas if you write P and negation this is not a well-formed formula and all so that is a way we defined it and all according to our definition the only case in which the negation comes the left hand side of this prepositional variable is considered to be well-formed formula whereas if I write like this P followed by that there is a negation then it is not a well-formed formula so this is all our convenience and all so first we need to define what we mean by well-formed formulas and all and we stick to the definition and then it makes our life simpler so that is why we follow this particular kind of definition and all so if something is a well-formed formula immediately the negation followed by that is also considered to be a well-formed formula and all and the third thing is that there are different clauses and all with which you can decide what is considered to be a well-formed formula etc and all just like we have so many grammatical rules to say that which sentence is grammatically correct which sentence is grammatically incorrect etc and all so we have very limited rules and all here not like grammatical rules are many and contextual etc and all but here we have only three rules simple rules with which you can say that a particular formula is a well-formed formula and all so why we are worried about this well-formed formulas first you generate the well-formed formulas and then you talk about what we mean by this well-formed formulas while providing the semantic so that is what we with which we will be occupying our attention in the next case so we will focus our attention on the syntax so how these strings are generated and out of all the strings which are generated only few strings are considered to be well-formed formulas whereas others are considered to be not well-formed formulas and all so out of these well-formed formulas some of them are valid some of them are invalid etc or sometimes you can even classify it into tautologies contradictions and contingent well-formed formulas and all so we will talk about it little bit later but we focus our attention on this particular kind of definition with which we can know we will be able to know whether a given formula is a well-formed formula or not so now the third clause is this that since these connectives are binary connectives and are implies if and only if they are all binary connectives that means at least they require two atomic sentences and these two atomic sentences are joined by this binary connectives it is in that sense and are implies if and only if they are considered to be binary connectives so now the third clause says that if A and B are well-formed formulas then whatever is written in the brackets A and B and ARB and A implies B and A if and only B are also considered to be well-formed formulas and all so and the fourth clause which is not explicitly written here which says that nothing is considered to be a well-formed formula which is not formed by using these three clauses so which is already implicit in it but you know if you want to state it explicitly you can state the fourth rule also nothing is a well-formed formula which does not follow these three rules so this one is considered to be not a well-formed formula because it doesn't follow any one of these three rules so whereas this one is considered to be a kind of a well-formed formula. So these are the four clauses which are considered to be important in judging whether or not a given formula is a well-formed formula or now so now once we identify that this is a well-formed formula then we can generate a unique tree structure for these well-formed formulas and then we say that every well-formed formula has is unique tree structure so before that we define this well-formed formulas in a kind of formal sense so it can be read as follows a string means any alphabets etc and all that means the propositional variables combined with the help of logical connectives and parenthesis and that will constitute a string you know a string is considered to be a well-formed formula exactly when there is a finite sequence let's say a 1 to a n which is considered to be a parsing sequence that means we will be parsing from left to right because one can pass it from right to left also but we use usually a condition is that we follow from left to right we go from left to right so you have a sequence a 1 a n etc and on it's like a formula in which the first letter a 1 may be taking care of p or may be a n is considered to be q or something like that in between there are some logical connectives so it is such that the nth nth letter that is a n is nothing but a only so that is considered to be you know we said in the beginning that every propositional variable p is a well-formed formula you know if the nth variable is itself is a only that means it's an atomic sentence p o so that is considered to be obviously well-formed formula you know we are exactly saying the same thing with the help of some kind of formal definition so if a n is equivalent to a and for each one less than or equivalent to i less than or equivalent to n so this particular kind of ai which is that has to be a propositional variable like simple p q or etc and all first one and the second one is that for some j less than i if this ai is equivalent to aj that has to be the case or in the third case if j and k less than i then that particular ai has to be a combination of these two things in all aj and ak and all and this star indicates any logical connective apart from the negation and all it can be or it can be and it can be implies it can be if and only if we are just exactly stating the same thing as we are done earlier so you know the first ones takes care of the first condition takes care of the first one which we have defined in the well-formed formula that is every proportional variable p is a well-formed formula the second one is taken care by the second condition here that is j less than i ai is equivalent to aj and the third one is taken care by the third condition so that is what it essentially says so now so this is the we can recursively use this particular kind of definition of well-formed formula in this we have three clauses in all one is the base clause which says that any statement constant or a propositional variable is a well-formed formula which is the first condition second one is a recursive clause that is if p and q are well-formed formulas the following things are also considered to be well-formed formulas p or q anything which is which joins these things any binary connective which joins this propositional variables and all is also considered to be a well-formed formula of course brackets are not there here we can insert it appropriately to for to read the formulas in a better way we need to know parenthesis which will come to it little bit later so now the closer clause is this that nothing will count as a well-formed formula unless and until it is constructed according to clauses 1 2 and all of course three clauses are there earlier but you have only two clauses here because negation is also incorporated in the second one so these are the three clauses which are important to judge whether a given formula is considered to be a well-formed formula or not so this definition is considered to be a recursive definition we can repetitively use these things for n number of variables and all and it is a generative definition because it tells us exactly how to generate instances of things that we are trying to define so this is considered to be a recursive definition of well-formed formula so now so we identified what we mean by a well-formed formula just like cat is on the mat it seems to be a meaning grammatically correct sentence whereas mad cat on doesn't seem to be appropriate for us anyone who knows the minimal grammar rules they will immediately come to the conclusion that there is not a grammatically correct sentence in all so in the same way in the case of propositional logic it's a formal language in this language it has its own syntax in which we came with a definition with which we will come to know whether a given formula we came to know whether it's a well-formed formula or not so now coming back to this readability question and all so there will be some kind of confusion to read this propositional logical formulas in all for example if we have P implies Q R R so how to read this particular kind of formula it can be read in two different ways in all first P implies Q R R it can be read in this way or there will be obviously confusion all if there is no parenthesis which is given here then it can be written as P and Q and then followed by that it is a letter R so these are the two versions which are possible for this thing two ways one can read the same formula in all so that adds us to some kind of it gives us some kind of confusion some kind of confusion arises in the process of reading this formula in all so in order to avoid this particular kind of confusion so what we will be doing is we follow some kind of convention again this convention is comes out of practice in all logicians in many textbooks follows this particular kind of convention the first convention is that you read formulas from left to right in all so we don't write we don't read formulas from right to left in all of course one can do it but your definition of well-formed formula and all other things changes in all but our convention is like just like convention that you will be following left hand side traffic and all so like that you know we will be following meticulously religiously we are following will be following some kind of conventions in although logic is considered to be rigorous in all it also binded by some kind of conventions and all like this it is only it makes our life simpler in all otherwise there is lot of difference between P implies QRR and P implies QRR and all these two are two totally different sentences in all they mean two different things but in the case of syntax how do we know that these two formulas are syntactically different in all symmetrically we know that it's they are different and all they mean different things in all so it is P if QRR is the case is the first one the second one P if then Q or there is another condition like R they are totally two different things in all how do we know that these two are different formulas in all although it is generated from the same kind of formula so logicians have come up with some kind of tree structures for this particular kind of formulas and then they use the definition of well-formed formulas that means the clauses that we have discussed earlier in making it in making these formulas distinct in all for example if we have the first formula P implies QRR so in this one the first thing that we will be using is this one P and QRR so how did we get this one suppose if P QR well-formed formulas P implies Q is also well-formed formula that is what we have said it in I think in clause 3 so this is the one which we have discussed in the last slide so we will be drawing three structures for these formulas till such till to such an extent where you will end up with only atomic sentences and the end point and all so since this is a complex sentence and all it further reduces to Q and R so this is the tree structure which is generated for this formula one now the tree structure for this second formula is like this now this formula says that first we have P and then this is joined this is combined with QRR so first P QRR and then QRR again we need to state these roles in all clause so this is clause 3 clause 3 is that P and if P is a well-formed formula QRR is a well-formed formula then P implies QRR is also well-formed formula so QRR again we apply clause 3 and then we can say that Q has to be well-formed formula and R has to be well-formed formula now this is the tree structure of this one so now at the end point we have only atomic variables in all so now coming back to this particular kind of thing P implies Q QRR this is second formula so what I am trying to say simply is this that every well-formed formula has its unique tree structure and no two formulas have the same kind of tree structure and all so unless until they are logically identical and all suppose if you write the same thing I will state about those things little bit later so now this is another kind of well-formed formula this can be read in this way first one we have drawn tree structure for this one the second one is like this so now you apply the same definition of well-formed formula then it reduces to this one this is again clause 3 so now this can be further reduced to P and Q so now if you see the structure of tree diagrams for this particular kind of thing then these two are totally different and all on the left hand side you have only one atomic variable here you have two atomic variables here and you have two atomic variables here in the right hand side but you have only one here these two are totally different these two have different structures in all so that makes this formulas P implies Q or R is totally different from P implies Q or R so you have to note that we did not invoke any meaning or anything for these formulas you have just drawn three structures according to the definition of well formed formulas and then we have stated that the first one is having a different tree structure compared to the second one and all that makes these two formulas distinct to each other so in this way we can distinguish these formulas to be different and all syntactically because they have we have we came we have we come up with a preposition presupposition that every tree every formula every well-formed formula has a unique tree structure so this formula has a unique tree structure like this and this formula has a unique tree structure like this so with this you can even know how to read this formula first you read this one P implies Q then R then this P and Q are red and all so there is another way of with this tree structure you can also come to know how to read this particular kind of formula so now so this is a confusion that arises in all suppose if you write a formula P implies Q or R then it leads to two different versions so how to avoid this particular kind of confusion and all so here is the convention that we follow and what comes to our rescue is how to use this parenthesis and all so these are some of the standard rules which you will find it in the standard textbooks and all in logic of course one can come up with once own kind of rules and all but you have to be it has to be uniform throughout the thing and all so the standard rules are like this when it comes to identifying suppose if nothing is given and all suppose a formula is there like this so in the textbook or somebody gives you a formula like this so how to put parenthesis and all so that you avoid this particular kind of confusion so now here are the here are some of the rules that we follow so apply the connectives and inserting the parenthesis if needed in the following preferential order so we have said that there are five logical connectives and all and R implies W if and only if negation etc and all and now among this five logical connectives preference should be given in the following order in the first preference should be given to negation and the second one it applies to the shortest preposition to its right and whatever preposition which is immediately following the negation within which it has a scope and all and and connective applies to the shortest preposition on each side of it because it is a binary connective because it connects both the things in or applies to the shortest preposition again and the each side of it in the same way implies double implies etc they are all binary connectives because of the binary name binary connectiveness so by it is because it is binary connective so it connects the immediate sentences on the left and right hand side of the right hand side of it so if at any time you are with the repeats of the same connective and all group them and you work them from left to right and all we follow some kind particular kind of convention and we group them in certain way and all like in this example A or B or C the repetition of R connective here and we move from left to right and all and you put brackets appropriately and all so we will try to see some examples and all and we will apply this particular kind of thing this is the most important thing otherwise there will be confusions like this one it should be it will be read as P ? Q or R or it can be even read as P ? Q R so we need to follow some particular kind of convention and then first we will talk about the simple example and then we will move on to the complex example so what is that we are trying to say first preference should be given to negation and then conjunction disjunction implication if and only if so now we have to apply this particular kind of thing since negation does not appear here so we need not have to worry much about it so now the next connective next preference should be given to and that is also not there so you do not have to worry about it and we will take some examples which involve these things also now the next immediate preference should be given to the R connective so now what we need to do is what we have said here this is R connective combines the shortest preposition on the left hand side of it so that is Q is the shortest kind of preposition which is on the left hand side and R is the preposition which is the shortest kind of preposition which is on the right hand side of it so we will put bracket like this so now this is taken care of so now the next in the order we have implies so now we need to put another bracket and all so P implies Q or R so now we need to put brackets for P and the whole formula so in order to distinguish it we use even you can use square brackets also to separate it otherwise you know it becomes impossible for us to read the things in all so we can one can use even square brackets or maybe some other thing as well so now the whole thing now implication should be given priority so now this is going to be the correct parenthesis of this one so one need not have to sometimes one can omit this parenthesis as well so even if you omit this particular kind of thing in all it doesn't make a big difference in all because you can read the same thing it means P implies Q or R so that is what we mean by the formula which we have written before so there is another way in which it tells us in all so in a given well-formed formula the last preference that we have given is if and only if and all so whenever you have this particular kind of connective that particular kind of connective which comes at the end is going to be the main logical connective so it will be definitely useful especially when you are evaluating a given well-formed formula to be true or false tautology or contradiction etc using truth table and all one needs to know what we mean by a main what you mean by the main logical connective and all usually the main logical connective here is this one so this connects the whole formula and all main logical connective means it is the connective which combines the maximum number of propositional variables so now here here is a formula which connects only two variables in all Q and R so this is a sub formula so this is considered to be a major major connective and all because it connects 1 P Q and R also so three variables it is connecting in all so that's why it is considered to be the major connective so this is a simple example in which you put parenthesis in all so now one can talk about some kind of complex example in which it has all this connective so just I am writing whatever comes to my mind and all then you will see what we mean by etc so here is a formula well-formed formula like this suppose if the parenthesis is not there and all it can be written thousand maybe four or five different ways etc now ultimately you may not reach any consensus to what we mean by this particular formula and all so this convention is very important to judge whether how to read this particular kind of formula so now we apply this kind of preferential kind of ordering and then the first preference should be given to negation and all so in the first step what we do is this one so there is no negation here so this is the one which we have so negation of P n so the first one is taken care this is the first step so now in the second step the preference should be given to end connective so now in the case of this one what you need to do is you have to follow some kind of convention and all either you move from left to right you move from right to left and all usual convention is that you move from this to that then you come across conjunction here and here also repetitively you are using this particular kind of thing and all so two times you will find this conjunction and all that means you need to move from left to right so now in this case we are applying this preferential order for conjunction and all so it will be like this so conjunction rule is this that I mean the shortest preposition on the left hand side and the shortest preposition variable on the right hand side all it connects this one so you need to put parenthesis like this and then this is as it is now again you need to put this particular kind of thing and all so the shortest preposition which connects this conjunction is P and the left hand right hand side it is Q and all now it is R so this is step number 2 so now step number 3 so now we have given preference to end and all we have taken care of end so now coming back to R and all the next preference should be given to R so I will use a different color chalk piece and all so that you know we will come to know how to write this particular kind of thing so now here you do not have any R and all so now the only R connect to this is R connect you know so now this has to be bracketed the immediate preposition which connects these two things on the right hand side we have R the shortest kind of preposition formula which is on the left hand side of this particular kind of connective is P and Q you should not take all the preposition into consideration but this is the considered to be this is considered to be the shortest kind of preposition so now you put brackets like this so now there is no R here so now we have taken care of this one so now coming back to implication so now again you apply the same rule and all then you say that whatever connects this one on the left hand side the shortest preposition is this only negation of P on the right hand side you have P and Q so now you put so now implication is taken care of so now till it is not it over and all so now there is one more connective that is implication double implication and all which stands for if and only if so now the shortest preposition on the left hand side is the whole thing and all and the shortest preposition on the right hand side is the whole thing P and Q and R because is already a kind of complex preposition so now the whole the parenthesis now becomes like this so now this is the final kind of thing and all so it should be read as not P implies P and Q if and only if P and Q or R is the case so in this way we can decide so what we how to read this particular kind of formula if you do not have this convention and all there are many ways which you can read this formula for example not P implies P and Q implies P and then Q or R that is one way of reading it or you can read it in this way not P implies P and Q if and only if P and Q or R that is one way of reading there are several ways you can read it and all in order to avoid the confusion and all we follow parenthesis and all like this so now once you have done this particular kind of thing so now we can safely omit some of the parenthesis and all because excessive information also causes some kind of unnecessary occupation of information and all in computer science language they are all characters and all if unnecessarily you put so many brackets in all like for example P implies Q you put it in several brackets like this it is a waste of energy and although it is considered to be same in all three brackets here of course four one two three four etc so left and right brackets parenthesis matches in all you can write P implies Q like this and all you can omit all these brackets and simply say that P implies Q itself so it saves our space and all so in this way the main logical connective here is this one if and only if and all so although we give least preference to this particular kind of thing but this is going to decide whether this formula is going to be true or not we will talk about it a little bit later this is also considered to be the major logical connective in the sense that it connects maximum number of prepositional formulas and all so now in this case you can omit parenthesis in this way not P you do not require any kind of brackets here now you can even omit this particular kind of thing also so now you can write P and Q and put one bracket here implies P and Q and this whole bracket need not have to be put in all you can remove this particular bracket and you can simply read in this particular kind of thing and all so these brackets should match and all otherwise computer will show some kind of syntactical error and all so now what we discussed so far is simply like this that we can generate n number of meaningful n number of strings and all but not all strings are considered to be well-formed formulas that is the first thing and the second thing that we have discussed in detail with some examples is that when the parenthesis are not given how to decide how to read a particular kind of well-formed formula and all it leads often leads to confusion and all but if you follow this particular kind of convention then it will simplify our thing and all if this is one way of reading this particular kind of thing and all so usually in good standard textbooks and all this convention is already given and all usually parenthesis is already given and all if you do not give it and all then if I follow some convention I read this formula in this way so what happens if someone comes someone follows a different kind of convention and all like the reverse the same thing for example first reference they will give it to if and only if then implies R and then double negation it might lead to some other kind of formula and all so that's the reason why it's always good to state it is important to state this parenthesis appropriately and all otherwise there will be confusion of reading this particular kind of formula and all so when there are groups of connectives like this if you find it then there is a way again convention is that you move from left to right and all so they are all conventions only but one can follow one's own convention to find out what we mean by a given formula not what we mean by what exactly how this formula is well-formed and all so these are some of the things which we have discussed in greater detail already and the convention here is that we can omit the use of parenthesis by assigning some kind of decreasing ranks to the propositional connectives as follows so if you find if and only if you can safely omit the parenthesis and all like the one which we have done there and next immediate thing is this particular kind of thing and all in this case also P implies q or R you can put one bracket here but you can omit this particular kind of parenthesis and all you can just simply state P implies q or R can be simply read in this sense and the next in the decreasing order of rank it is P and Q and then next one is R and negation so the connectivity with greater rank always reaches further and all greater rank in a sense that if and only if so it reaches further and all it connects as many sentences as possible and all propositional formulas as possible so in the examples that are there here in the slide for example if you have a sentence like P implies q and R or yes this can be written in this particular kind of thing I will just write it and then implies Q P implies Q and R or yes suppose if you have a formula like this so now again what we have done is we have to follow this particular kind of order implies and if and only so the first thing which we need to take care of is connective and so you put a get like this and then followed by that you need to give importance to this particular kind of connective and then you put like it like this in the second step so this is taking care of R and now what we what is left is P implication so now this should be taken care of so this is considered the formula in this P implies Q and R or S and all so now even if you omit this parenthesis and all this one is not going to make it different and all it is same as this one P implies Q and R or yes so in this way you can omit the paragraph parenthesis and all in order to save some kind of space and all so likewise you can draw three structures and you can say that these two formulas are different and all so use we need to use some kind of conventions to eliminate as many parenthesis as possible and all the convention that we have mentioned earlier unnecessary if you put too many brackets and all it will occupy unnecessary space and all so better to omit this parenthesis by using that decreasing rank kind of thing and all so now there are some definitions which we need to note so what we have discussed so far is what we mean by well-formed formula how to form a well-formed formula there are some rules for it and then how to read a well-formed formula and all so now there are some kind of definitions which we need to know usually P is Qs Rs etc are considered to be atomic prepositions when they combine with some kind of logical connectives they form compound sentences that means a sentence is considered to be compound if it logically contains another complete sentence as a component and all like P R Q and all it consists of another sentence Q and all so when these two sentences combined together will form some kind of compound sentence and all it is like atoms combined together will form molecules molecules combined together will form compounds etc so like this in analogy you can have it here in the case of propositional logic here we have started with basic units that is atomic sentences which stands for P Qs Rs etc and then they combine with some kind of logical connectives and then form compound sentences a sentence is which is considered to be simple if it is not compound like you know P Qs Rs etc and all there are simple sentences so one sentence is a component of another sentence if and only if whenever the first sentence is replaced by another declarative sentence the result is still considered to be a grammatical sentence for example what it essentially says is this particular kind of thing you know suppose if you have a compound sentence like this P and Q so now you replace Q with P and P with Q and all so this will become Q and P and all so these two are logically identical to each other which will talk about little bit later will the structure of this two sentences are same and all so it has the same tree structure and all like this P and Q and all so we will talk about little bit later but what I am trying to say is this that so if you replace P by Q and Q by P and all and it will become Q and P if you use commutativity property P and Q is same as Q and P this may not be same this may not be may not apply to day-to-day discourse and all one simple example could be like this that usually you know you when you become sick you will go to the doctor and all so this sentence can be put in this way in a complex compound sentence like this I became sick and I went to the doctor that is P and Q if you say the same thing Q and P I went to the doctor and I became sick nobody goes to the doctor to become sick and all so these two sentences we mean two different things and all in day-to-day discourse but in the case of prepositional logic they mean the same because P and Q is same as Q and P they are same similar tree structure and all they are identical to each other so it is in that sense they are logically identical to each other so a sentential operator is an expression containing some kind of blanks such that when these blanks are filled with complete sentences the result is also considered to be a kind of sentence in a complete sentence and all so now what we mean by a major main logical connective and all which we have defined already in one of the examples the main logical operator in the compound statement compound statement in the sense that it can be P implies Q or P or Q or it can be a mixture of all these things in all a big compound kind of sentence which is generated by R and implies etc and all is the one that governs the largest component or components of a compound statement in a minor logical operator governs only smaller components and all for example if you have in this particular kind of formula which is already there here in this one so the major logical connective now we are trying to find out what is the major connective here and minor connectives first of all what are the connectives connectives are these things negation or and implies so this stands for not of course I will talk about this thing when we talk about semantics in greater detail but in this moment even if you know this particular kind of thing it is enough and implies and if and only if which is used to invoke necessary and sufficient condition or equivalence relation so the major logical connective is a one which connects as many propositional variables as possible and all so now in this one so this is the this is considered to be the major logical connective because it connects not P P Q and on the other hand P Q and R at least three connectives on the left hand side on the right hand side and two connectives on the left hand side so it connects as many propositional variables as possible and all so that that's the reason why so this is considered to be the major connective so later it will be very useful this concept is very useful in the sense that so what we will be doing is in order to judge whether a given formula is there are three types of sentences which occur in propositional logic tautology is contradictions and contingent states statements and all under this major logical connective if you get all T is etc and all that is considered to be tautology under this major logical connective whatever connective that is there here it is if and only if under this if you get only false and all if you evaluate the truth value of this particular sentence and under this major logical connective if you get all f and all it is considered to be a contradiction if you get T is f etc it is considered to be a contingent sentence and all it is for this reason we need to find out what we mean by what exactly is the major logical connective so now the minor logical connectives are the ones which connects as many a few propositional variables as possible and all so now here this is a major minor logical connective here this is also they are all sub formulas and all whatever connective that occurs or figures out in the sub formulas is considered to be the minor logical connective here is the one and here is another one and all the sub formulas whatever connective is there that is considered to be the minor connective so these are all minor connectives in this particular kind of formula and all it's not you know this is the major minor and all in general but with respect to this particular kind of formula this form this symbol if and only if is considered to be the major logical connective and all other things are minor logical connectives so exactly the reverse order follows in all here so that is whenever you have this particular kind of thing usually that will serve as the major logical connective next possible thing is this one and then and are of same ground and all and then negation for example if you have a formula like this P implies Q implies R and all so you have something like this so now in this one the major logical connective is this one major one and this is the minor kind of connective so now the reverse order follows in this particular kind of thing and all so now we have identified what we mean by what we mean we have seen what is a major logical and minor logical connective this we use it in the semantics in particular that is evaluating the truth value of particular kind of sentences so now coming back to we have defined what we mean by well form formulas and how these well form forms are generated and we came to know how to read this well form formulas etc and all and then we also came up with some kind of convention with which in case you are not given parenthesis and all how to insert this parenthesis etc or how to omit this parenthesis unnecessary parenthesis etc needs to be omitted to save some kind of information etc all these things which we have discussed so far and we also identified what we mean by what is considered to be the major connective and the minor connective and all so now we will move on to different kind of thing which is called as semantics of the prepositional logic that means we are trying to define what we mean by this formulas and all so it is like the analogy is like this syntax is like producing things in the case of in the language of marketing language we have just you start producing the things and all without knowing the implications etc and all just the production is considered to be syntax just like that is what we have done here we have some kind of alphabets we combine them in certain way and then we are saying that it is a well form formula and some other strings are not well form formulas etc and all and in the market language we have distribution so distribution corresponds to semantics and all so we need to know how to distribute what you have produced and all so for that we need to know what we mean by what you have produced and all so that is what is taken care by semantics and all so now the connectives that we are talking about there only five connectives not or and implies double implication etc and all so these connectives are considered to be truth functional connectives and all so why they are truth functional connectives it is because of this thing the truth value of a compound statement that they form that means p in plus q or p or p and q etc it can be solely determined by truth value of its individual components and all so this is the main idea proposed by Fregge particularly in this principle of compositional compositionality according to which a compound formula gets this meaning only if you can evaluate the truth value of its individual constituents and all if the truth value of a compound sentence is solely determined by the truth value of its constituents then those truth functional connectives are also called as extension so this means that there will be a rule telling the rule which tells us exactly what the value of the compound must be for each combination of values for the components and all so what all you need is the truth values of the individual constituents and all with which you can judge whether a compound formula which is generated by this atomic repositions with the help of this logical connectives is true or false and all so we will postpone this discussion the next class because we will be dealing with semantics in greater detail so what we discussed in this class is simply like this that you know we presented a kind of minimal language for the propositional logic so that is a formal kind of language in which it has its own syntax so how we generated the syntax because it is we generated syntax in this way we started with the language which consists of propositional variables and then logical connectives and parenthesis and these logical connectives and propositional variables combined in certain way it will form some kind of strings but we said that we have said that not all strings are well-formed formulas and all only strings which are generated generated by means of definition of well-formed formula is considered to be what we are calling it as well-formed formula and all once we generated the well-formed formulas we have seen that how to read this well-formed formulas so there is a convention which we followed and then we gave some kind of reference to these connectives negation etc and all so that you know it makes our life simpler and all otherwise there will be confusion of reading the same formula a different way and all and other thing which we noted in this class is that every well-formed formula has its own unique structure and all suppose if the tree structure of two formulas are same and all that means here you are talking about the same formula like P and Q and Q and P it has the same tree structure and all in no way different and all but in day-to-day discourse we mean totally two different things and all P and Q for example I went to the doctor and I became sick suddenly different from I became sick and I went to the doctor and all we mean totally different things in the next class we will go into the details of what we mean by this well-formed formulas and all that is taken care by semantics of prepositional logic.