 Welcome back in the last lecture we presented Hilbert Ackermann axiomatic system and before that we presented one of the important axiomatic system that you will see it in Principia Mathematica that is another axiomatic system which is due to Russell and Whitehead. In the Russell and Whitehead axiomatic system what you will see is that there is a choice of taking the primitive logical symbols Russell and Whitehead took into consideration the negation and disjunction whereas Hilbert and Ackermann took implication and negation as the primitive symbols. So in the axiomatic propositional logic our goal was is that we have some kind of valid formulas and then we are trying to come up with theorem we are trying to come up with proofs of those theorems. So how you prove the theorems you state the axioms explicitly so they are absolutely true and then we have a set of transformation rules which preserves the truth of a given formula and then you have a simple rule of inference that is the rule of detachment which is also called as modus ponens principle and using these three things we transformed the axioms we trimmed these axioms in such a way that we derived minimal things such as law of identity law of excluded middle and of contraposition etc. So now in this class what I will be discussing is this that whether this principle of Mathematic or Hilbert Ackermann system axiomatic system are these consistent in a sense that it is not the case that you derive both X and not X in that sense it is consistent or these systems are said to be complete a strongly complete a weakly complete etc all these things which we will be talking about in some detail in this lecture. So one of the important advantages of knowing this particular kinds of theorems is this that suppose if your axiomatic system is complete in a sense that whatever what all you prove are valid and all valid formulas are also provable then instead of checking for example if you if your proof is very hard to come by then you can use the completeness theorem and then you can use the completeness theorem and you can say that you can show that instead of proving the theorem you can simply show that a given well-formed formula is valid so that is one of the advantages of having this completeness theorem in particular. So the outline of this talk is that each principle of Mathematica the one which we have presented in the last class is it consistent is it complete or that means all or all axiomatic systems the one which we spoke earlier are they free from contradictions that means within your given axiomatic system you should not be in a position to derive contradictions so as we have seen earlier if you have contradictions if you start with the contradictions you can derive anything or from impossibilities you can derive anything so that is why we have shown in the last few classes. So then we will be talking about the relationship between probability and validity so that is what is completeness establishes so if all the provable things are also considered to be valid that means true tautology validity are one of the same in the propositional logic so all tautologies are obviously valid statements if whatever you have you have come up with a proof of some theorem and then it so happened that it is also turned out to be valid. But is the case that all provable theorems in your axiomatic system are considered to be valid in the case of propositional logic this is the case all what all you can prove are obviously considered to be valid because these step of your proof is a result of applying either an axiom or theorem which is obviously true or some kind of transformation rule which preserves the truth or the rule of detachment which also preserves the truth so each step is considered to be true so the final step of your proof which is considered to be a theorem that is also considered to be true. So this consistency gives us guarantee against some kind of triviality results such as x and not x if you derive it it leads to some kind of trivialities whereas completeness guarantees some kind of adequacy so if it makes your systems adequate so these are the two things which it does consistency guarantees that it ensures that there are no trivialities in your axiomatic system and completeness is it will give us guarantee of your formal guarantees adequacy of your formal system. So there are first to start with we will talk with consistency consistency is the one which we have already seen earlier so whenever we have for example two groups of statements one and then suppose if you have another statement like x implies y etc these are the two statements that we have so these two statements are consistent to each other especially when when you construct a tree for this one using semantic tableaux method at least some of the branches opens so now you construct a tree for this one not x and y is the first one we checked it and then this is x y x y so this branch closes but all the other branches opens that means this particular kind of assignment satisfies this particular kind of formula when not x is T and y is T then it satisfies this particular kind of formula whereas in the when both x and y are takes value T that also satisfies this particular kind of form so using the semantic tableaux method one can find out when a group of formulas are said to be consistent if at least one branch is open and that means that is said to satisfy this particular kind of formulas so that means that makes these two formulas true so it is in that sense we usually call it as consistency so now there are three kinds of consistency one can talk about the first one is consistency with respect to the main logical operator negation so what do you mean by consistency with respect to negation if a system any axiomatic system will be said to be consistent in the sense that if there is no thesis this is means it can be an axiom it can be a theorem if there is no thesis x such that not x is also considered to be thesis if that is a case then system is said to be called as consistency with respect to negation so that is what we have seen on the board suppose if you have a formula like x and y and not y so for example this is these are the two groups of statements that we are taking into consideration so now again if you construct a semantic tableaux method then x and y can be written in this sense and not y is like this is y and not y there is a contradiction so it closes that means these two statements are inconsistent to each other so this is what we mean by consistency with respect to negation so it tells us that either it should be in a position to derive x or it should be in a position to derive not x but it should not be the case that both y and not y should be there in your proof so that means either it should be in a position to derive only x or it should be in a position to derive not y but it should not be sorry not x but it should not be the case that x and not x if both x and not x you derive that system is called as a trivial kind of system so it is in that sense there is no thesis if you consider any formula well from formula x if you derive x and if you derive not x also then your system is considered to be trivial kind of system and that system is considered to be inconsistent inconsistency with respect to negation so a system which is consistent with respect to negation is usually free from contradictions as you see here in this case you constructed you are checking whether this these two formulas are consistent to each other in this one suppose if you constructed a tree and then all the branches closes and all that means it leads to unsatisfiable unsatisfiability leads to inconsistency but here it is not the case in the first in the first one that is not the case but in the second one you have this contradiction so all the branches closes so consistency guarantees that there are no such kind of trivial things which are present in your axiomatic system triviality how triviality results in because you have x and not x so it ensures us that your system is free from contradictions one of the problems with this contradiction is this that which we have seen already using classical logics you can derive anything for example if you start with this particular kind of thing x and not x which is considered to be inconsistent then this is the first step and then you assume this one x and then not x now since x is true even x or y is also true why because so this is law of addition so if x is already true our semantics allows us that x or not why is also true so this we can add it here now this is law of addition so this is already hypothesis or assumption or anything which you can take into consideration so now 3 and 4 modus ponens lead to y sorry this is not most one distance to syllogism leads to y so now from x and not x you can derive y but in the same way you can produce the same kind of proof and then you can derive even not y also so it is like this again you start with the x and not x then to you assume the same things x and not x they are all assumptions or hypothesis sometimes so now on the fourth step since not x is already considered to be sorry x is already considered to be true then you can add any other stage kind of thing even not why also because this is already true this ensures the truth of the whole disjunction irrespective of whether not y is true or not why is false it doesn't matter because it is already true so this makes the whole disjunction true so this is law of addition so here we have added y here we have added not why so then this leads to so now 3 and 4 disjunction to syllogism leads to not why so now in our system let us say you have set up formulas gamma and then you have taken one inconsistency that is the contradiction x and not x then you derived why and you have also came up with same kind of rules etc and all you have also come up with not why that means in your system you have x and not x so this is nothing but a trivial kind of system a system in which both x and not x are proved is considered to be a trivial kind of system so this is what we mean by consistency with respect to negation so as far as possible your axiomatic system should be free from these contradictions like this so there is another way of defining consistency which you will find it in the literature logic literature so that is absolute consistency so what we mean by an absolute consistency a system is said to be absolutely consistent if not every well-formed formula of the system is a thesis so that this means that let us say that you have a formal axiomatic system and then there are so many well-formed formulas in all so but not all well-formed formulas are valid kind of statements whatever formula that you take into consideration will not be a kind of valid kind of formula so that means not a true state so if it so happened that your system is said to be your system is said to be absolutely consistent if in only if there is not every formula of the system is considered to be a theorem or a tautology only selective kinds of things are considered to be either tautologies or axioms so we started with some axioms and then we proved some theorems in all that theorems are set of some kind of well-formed formulas so out of this well-formed formulas some are tautologies some are contingent statements some are also considered to be contradictions in all so your set of well-formed formulas are big enough so in that only few formulas are considered to be valid formulas so if you can ensure that not any kind of thing is considered to be a valid kind of statement or a thesis then your system is considered to be absolutely consistent that means you should ensure that the only statement that you have are only tautologies if the only statements that you have are only tautologies what about contradictions and contingent statements in all if you if you are built your system in such a way that you allowed for only tautology is not possible but if it so happened that if your system has only tautology is nothing else then that is not called as absolute consistency but you can build a such kind of system where you can come up with only there is no way in which you can you can come up with usually with a system in which your system is considered to be absolutely consistent because it is very difficult for us to construct only it is difficult to visualise a system in which there are only tautologies it is not quite possible so your propositional logical system is usually considered to be absolutely consistent in the sense that not every well-formed formula is considered to be a valid formula or a thesis so third this is another kind of consistency which discussed in greater length by Emily post another important logician is also responsible for the truth tables etc following Wittgenstein so according to Emily post a system will be said to be consistent in this sense if there is no thesis of the system which consists of a single propositional variable so it is applicable only if the system contains some class of variables identifiable as at least propositional variables so what happens here is that let us assume that you have constructed a grand axiomatic system in that system if there is no thesis of the system which consists of a single kind of propositional variable if that is the case then also it is considered to be consistent so you should ensure that you do not have this single well-formed formula that is P or Q or something like that which turns out to be a well kind of thesis then so that is not considered to be consistent in the sense of Emily post so as far as possible you should avoid this particular kind of situation so that is a system will be said to be consistent in the sense that if there is no thesis of the system which consists of a single propositional variable so it is like this particular kind of so we can have this these things at this is an all like we have seen in their cell white head axiomatic system etc P ? Q ? B does not make any problem at all but if it so happened that only this particular kind of thing is considered to be a thesis in your system then your system is not considered to be consistent if this also is viewed as this thing thesis then your system is considered to be inconsistent so according to Emily post we should ensure that these particular kind of formulas like P Q symbol single propositional variables should not be a part of your thesis what is this is this is either axiom or a theorem so it is in that sense your system is considered to be consistent so now these are some of the important theorems related to the axiomatic system due to the cell white head is in principle a mathematic so the theorem number one tells us that if X is a thesis of principia Mathematica that is Russell white head axiomatic system that means X is also considered to be valid so how do we know that this particular thing is the case X is the thesis means X can be either axiom or X can be it is obtained by means of applying some kind of transformation rule or we could have got this particular kind of thing through modus ponens and all so if X is a thesis of PM this principia Mathematica X has to be valid one example some examples which you can take into consideration so suppose X is considered to be thesis that is it can be either axiom or it can be a theorem so now let us consider this particular kind of thing qRp this is permutation axiom in Russell white head axiomatic system so now we have a method with which we can check whether this particular formula is considered to be valid or not so that is the semantic tableaux method suppose if you take this X as this one not X will be not of P or Q implies qRp that it needs to be close proper so now if you construct a tree for this particular kind of thing and this will become PRQ and qRp so now if you elaborate it a little bit then it will be not Q and not P and then this if you simplified it will you will get this one and then PRQ needs to be written here so now you have not P here and P here leads to contradiction and then not Q and Q is to contradiction so what is that we have showed so we showed that not of X is unsatisfiable because if you take the negation of the given well-formed which is usually considered to be an axiom in Russell white head axiomatic system that is considered to be a thesis so if you negate that thesis you leads to contradiction that means not of X is unsatisfiable that means X has to be valid so now like this any theorem that you can you can take into consideration and you can use semantic tableaux method and you can establish the validity of those given formulas so it will have the same thing will hold even for a for the case of a theorem if anything is considered with theorem in any axiomatic system if you apply semantic tableaux method that means if you deny the well-formed formula not of X and construct a tree by using alpha beta rules which we have seen in the case of semantic tableaux method then you will see that not of X is going to be unsatisfiable if not X is unsatisfiable X has to be valid so that means you are not able to come up with a counter example in which you are like in the case of argument you are not able to come up with a counter example in which you have true premises and a false conclusion that means the original conclusion holds so this is what we mean by saying that if X is a thesis of Principia Mathematica that is Russell weighted axiomatic system that has to be valid because it is obviously tautology and then hopefully all tautologies are obviously valid now theorem number 2 tells us that Principia Mathematica is said to be consistent with respect to negation so what is consistency with respect to negation a system will be said to be consistent in a sense that if there is no thesis X such that both X and not X is also part of your not X is also a thesis either X has to be thesis or not X has to be thesis that means X has to be theorem or not X has to be a theorem but not both the things if that is the case then it is called as consistency with respect to negation so how do we prove these things let us assume any kind of well-formed formula X in this axiomatic system then X and not X cannot be both valid it is obviously the case because it is a contradiction so obviously that X and not X is going to be false it cannot be true so therefore theorem 4 which we will be talking about they cannot be both thesis of Principia Russell voided axiomatic system because combining both of these things leads to contradiction at least one of these things should be a thesis of that one either X has to be a thesis of your axiomatic system or not X has to be a thesis of your axiomatic system so it is in that sense Principia Mathematica is consistent with respect to negation so it is straight forward pretty straight forward that you will not be in a position to derive both X and not X so either X has to be part of your thesis or not X has to be part of this is but not definitely not both the things X and not X if you have X and not X if you allow for this particular kind of thing it leads to trivialities so one can also show that Principia Mathematica that is Russell voided axiomatic system is considered to be absolutely consistent so what is absolutely consistent a system is said to be absolutely consistent if not every formula of your system is considered to be a theorem or axiom or valid kind of statement if you ensure that whatever arbitrary formula that you take into consideration is not going to be a valid formula then the system is considered to be absolutely consistent that means it your system has other things as well that is contradictions and even contingent statements also so when we discussed about group of statements that you commonly a that commonly occurs in the preposition logic we have seen that there are three kinds of statements which usually you will see in the prepositional logic that is on the bottom we have contradictions on the top tautology is occupies top most position so statements which are always true and in between that there are some contingent kind of statements suppose if you can ensure that not every kind of formula that you take into consideration how did we construct this every kind of formula so by using some kind of formation rules you construct a kind of well-formed formula so that doesn't mean that whatever formula that you come up with that is going to be a theorem and all so that is not the case in that sense it is called as absolutely consistent so now we are trying to show that Russell weighted axiomatic system that is principia mathematical is absolutely consistent so how do we show that that is a case you select any axiom whether it is even or a 2 or a 3 suppose if you take even then not even is a well-formed formula of principia mathematical which by theorem is not a thesis of principia mathematical so therefore principia mathematical is considered to be absolutely consistent so what it essentially says is this particular kind of suppose if you take any particular kind of axiom q or q is p or q so this is axiom number tool of addition so you take any such kind of random axiom and all so what we what essentially we are trying to show is that any formula that you are going to take you are going to take into consideration that is not considered should not be considered as a theorem suppose if this is the formula that we have and I will take the negation of this particular kind of formula so this this is already a thesis of thesis because already in axiom thesis means as an axiom or theorem so now I take the negation of that one and I will show that this is not part of your axiomatic system so now if you take the negation of this one again using semantic tablox method you can clearly show that not of X is going to be unsatisfiable that means is going to be an invalid formula so how it results you take the you expand this you apply the tree method for this one then you will have this particular kind of thing then not p and not q since not q and q are there it closes that means not of X is unsatisfiable that means this is not a thesis of your axiomatic system what we established any kind of thing which you pick it randomly that is not going to be a thesis of your axiomatic system that itself will be enough for us to show say that your system is considered to be absolutely consistent it ensures that not any kind of formula is going to be a theorem if that is again the system is obviously considered to be absolutely consistent so now principia mathematical is also consistent in the sense of Emily post so what is what is considered to be consistency according to the famous logician Emily post it is like the system will be said to be consistent in the sense of Emily post if there is no thesis of the system which consists of only a single prepositional variable so you say that you know suppose if you have a q and you say that that is a thesis of your axiomatic system if that means that the system is not considered to be consistent so if proportional variable simple prepositional variables also serve as a thesis in all then the system is not considered to be consistent so we can show that even this kind of consistency also holds for this famous axiomatic system so how do we show that let X be any well-formed formula consisting of only sing single prepositional variable that is let us say X then a single prepositional formula cannot be valid or invalid you know so if your statement is true we can only talk about truth of a prepositional variable if suppose if I say that this is a tester and the negation of that one is this is not a dexter but only you can talk about validity only when you combine with another kind of variable that is it is a dexter or it is not a dexter that is X or not X that is going to be valid it is a tautology but not a single prepositional variable can be taken as a valid kind of state you know only be true or false so in that sense X is not considered to be valid in all in that sense anything which is not a well for valid formula should not be a thesis of your axiomatic system so in that sense principia mathematical is said to be consistent even with respect to the consistency that Emily post talks about that means in your axiomatic system there is no way in which you can have a single prepositional variable as your thesis that is the axiom or you only have that particular kind of thing so that is not permitted and all so if that is there then it is not consistent with respect to Emily post so then now we just discussed three kinds of consistencies in all so mostly we will be using consistency in the form with respect to negation that means any axiomatic system you should not be in a position to derive both X and not X if you derive it then it is considered to be a trivial kind of axiomatic system so consistency ensures that there are no contradictions in your system once you have contradictions you can prove anything I can prove X and you can prove not X and you can prove any other strange kind of propositions so now let us move on to completeness so far we discussed about consistency and we showed that principia mathematical consistency with this consistent with respect to negation consistency with respect to absolute consistency and even consistent with respect to whatever Emily post talks about so what do you mean by completeness for every valid well form formula of a given axiomatic system is considered to be a thesis of your axiomatic system so now you have all the well form formulas in all so they are all considered to be thesis of your axiomatic system that means as it should be a theorem or if it is not a theme it has to be an axiom so that is a thing then it is it shows that all the true formulas are can be shown to be provable that means all valid formula should find a proof if that is a case then it is usually it is called as completeness suppose if you say that all provable things are true it is sound and then all the true formulas are also find a proof if not today or tomorrow then it is considered to be complete so axiomatic basis is sufficient for the generation of set of all its well form formulas so we know that if you have some solid foundational solid foundations based on axioms what are these axioms they are considered to be self-evident truths which are obviously considered to be absolutely true so there itself is sufficient enough for us to say that since the axioms are absolutely true they are also considered to be well form formulas you can use semantic tableaus method or any other decision procedure method and you can check this particular kind of thing so usually in general axioms does not require any proof suppose if your axiomatic basis is sufficient for the generation of set of all its well form formulas which is usually called as weekly complete if the axiomatic system cannot be made more powerful without inconsistency resulting then the system is called as weekly complete the in the first sense it is called as strong completeness and all written in a wrong way here so an axiomatic just your axiomatic system is itself is sufficient for the generation of all the well form formulas and all that means what what essentially it means is that you have an axiomatic system which consists of some set of axioms and transformation rules and modus ponens etc that is all you need to generate all kinds of well form formulas that exist either in any given field you are talking about arithmetic or geometry or anything all the truths of arithmetic and geometry should be should come as an outcome of just these axioms it is in that sense mathematics can reduce logic or you talk about all the mathematical concepts in terms of the concepts of logic using only conjunction disjunction in some set of axioms etc so either it should be in that sense or your axiomatic system cannot be made more powerful like in the case of the first case which is considered to be strong completeness without some kind of inconsistency resulting in the given system in that system is considered to be weekly complete so this is what we what we mean by the difference between what we mean by strong completeness and weak completeness so the first case is considered to be a strong completeness and the second one is considered to be weakly complete. So now we need to show that Principia Mathematica is weakly complete weakly complete in the sense that the second thing if the axiomatic system cannot be made more powerful without some kind of inconsistency resulting then the system is called as weakly complete so these are some of the theorems which I will just go into the I will just give you a brief idea of this particular kinds of theorems but all the proofs are already there in any either in the book any important book that you read it in the formal logic so there are some references given at the end of this slides and all so in those books you will find proofs of all these theorems but what we need to get is the central idea of central idea behind these theorems so this theorem tells us that if X is a valid well-formed formula of a Principia Mathematica it is a Russell-Voited axiomatic system then X has to be a thesis of Russell-Voited axiomatic system or Principia Mathematica so it is in that sense PM is weakly complete so now one corresponding lemma based on this thing is this particular kind of every well-formed formula X of Principia Mathematica as it is corresponding conjunctive normal form let us say X prime such that this formula is equivalent to its X prime so what essentially it says is that you have a thesis like this particular kind of thing let us say T implies T this is considered to be a thesis in Russell-Voited axiomatic system so now this is the formula X so now if something X is considered to be a thesis then it has his corresponding CNF so what is CNF it is conjunctions of disjunction so where it is it is a conjunction what is a conjunction of disjunctions D1 or D2 and D3 D3 D4 etc so each conjunct is each conjunct consists of disjunctions set of disjunctions so any given formula can be appropriately transformed into its corresponding conjunctive normal form so that is what this particular kind of theorem tells us in the same way one can transform a given thesis into disjunctive normal formals so there is a advantage of converting a given formula into conjunctive and disjunctive normal forms it is quite simple suppose if you have a formula X is C1 and C2 C3 and on if any one of these conjunct is false irrespective of whether for example C1 C2 C3 C99 even if they are all true the C hundredth one so that particular kind of disjunct I mean in that what are the elements you have only disjunctions if all the disjuncts are false that makes the whole disjunct falls and hence C hundred one is false so that makes even though 99 are true the hundred one is false so we have the semantics like this for and this is A and B so you have A and B T T F and F T F T F and it is going to be true only in this case when both the conjuncts are true then only it becomes true in all other cases it becomes false so that is why even if you have in your conjunct in you in your conjunct you normal form let us say there are hundred kinds of conjuncts like this if 99 conjuncts are true but 100 conjunct is false that is C hundred is false that makes the whole thing unsatisfiable I mean this makes the whole formula false the conjunct to normal formula becomes false means it is unsatisfiable means it is in value so this is this particular kind of lemma any given formula you can transform it into its corresponding conjunctive normal form so in order to show that the above lemma holds all that is needed to show is that we have sufficient kind of machinery that means we have all the rules of all the rules such as double negation demurgan laws etc and all then you can transform any given formula into its corresponding conjunct to normal form now lemma B tells us this every well-formed formula every valid well-formed formula which appears in the conjunct to normal formula normal form is also considered to be a thesis of principia mathematical so that means so when a given formula is going to be valid formula let us assume that you have a formula like this so this is C1 C2 C3 and all then only it will set to be in conjunct to normal form where each C1 let us say it is like this P1 or not P2 P3 and P2 I am selecting in a clever way such that you know each descent will automatically be true P3 and not P3 and P2 or not P2 or P4 or P5 etc so let us try to talk about only this thing so now this is in CNF so this is a conjunctive normal form conjunctions of disjunctions that is why it is in CNF so what this the lemma tells us that any such kind of conjunctive normal form which is which holds and all that means you can clearly see here that a literal and negation which appears in a given formula so that means this formula is obviously going to be true irrespective of whether P3 is true or not this is going to be true now you have P3 and you are not P3 is absolutely true and whether P2 is false or P2 is true it does not matter it is going to make this true in the same way here your P2 and you are not P2 here so that makes this whole formula true so that means you have shown that each conjunct is true C1 is true C2 is true C3 is true so that is why the whole formula is also going to be true according to the semantics of conjunction so it is in that sense any CNF which is considered to be valid should also be a thesis of your axiomatic system that is we are talking about principia mathematical so that should be part of your axiomatic system so we have this particular kind of thing which is related to the validity of any given CNF formula so this is like this a valid constituent disjunction in CNF in CNF what we have each it is a conjunct which consists of disjunctions so now if you observe the interior part of it that is the disjunctions of each conjunct we have only dyadic operators that is R is the usual sign which you will find which by use of commutative and associative laws can take this particular kind of form why are PKR it is a literal and it is negation is there in a given formula so PKR not PK is always going to be true so some kind of propositional variable PK and its corresponding negation is there in that then obviously it makes the disjunct true and in C1 also true if each C1 C2 C3 all are true then your cognitive normal form is also going to be true somehow your formula should be like why are PKR not PK so now in this case Y is considered to be this one and then all the other things are PKR not PK and here in this case Y is considered to be this formula and then this is considered to be PKR not PK except so like this each and every conjunct will have this particular kind of thing so that is why a given CNF is considered to be a valid kind of formula so the idea here is that in any given CNF you should ensure that you have a literal and it is negation present in the disjunctions of your each conjunct so now one of the another thing is that PM is there is a Principia metabolic is also considered to be complete with respect to negation that we have talked about an absolutely absolutely and in the even in the sense of Emily postals so these are some of the theorems which we can talk about with respect to Principia Mathematica and similar kind of things can be we can do it with respect to even Hilbert Ackerman axiomatic system as well just I will quickly go into the details of whether or not Hilbert Ackerman axiomatic system is consistent complete and sound exit by now so now we have we presented our axiomatic system earlier in the last class so we are now saying that the Hilbert axiomatic system is considered to be sound so a system is said to be sound especially when you proved something that is a is probable and that means a has to be valid something you proved that whatever you have proved is considered to be a valid state so this can be done by using I do not want to go into the details of the proof and all but this proof can be done by means of some structural induction so what we show here is that the axioms are considered to be obviously valid and all because you can check with the semantic tableaux method and you can check that all the axioms are going to be obviously valid because there is no way in which you deny the axiom and then you will it leads to unsatisfiable it leads to satisfiability and all so all axioms are obviously considered to be valid and that if the premises of another thing important thing is that the other things rule that we have used is the modus ponens that is P P M plus Q and Q so that means that also it should be that rule also should be truth preserving so now if the premises of your modus ponens that is P P M plus Q are going to be true and obviously the conclusion also have to be true there is no way in which P P M plus Q is absolutely true and then Q is false Q is false so that makes the argument invalid but that is we cannot come up with a counter example which can establish that modus ponens is wrong so it also we can establish so we can take in the same way we can take any axiom into consideration and Helbert Ackerman axiomatic system like P M plus Q M plus P and then take the negation of that one obviously negation of this particular kind of thesis that is axiom number one leads to unsatisfiability means not x is invalid that means x is considered to be valid or x has to be true so like this you know you can check all the things that you have proved to be of absolutely to be true so it is in that sense whatever you proved that is a singleton style A that is whatever is provable is of also turn out to be true at the end of the rate also turn out to be true as you can see clearly you can use your common that you can we can see from the proof itself what is considered to be a proof each step of your proof is obviously considered to be true so that is why the final step of your proof that is a theorem which is obviously considered to be true so in that sense you can establish that Helbert Ackerman axiomatic system if something is provable in the axiomatic system that has to be true state that is it has to be valid formula so now we can as far as axioms are concerned there is no way in which you can show that they are wrong and all they are absolutely true we can use semantic tableaux method you can check all the axioms to be true there are three axioms which you can check them to be true using semantic tableaux method which we have already or you can use any decision procedure method like truth table or anything then you can check the validity of a given formula or whatever you have now the next thing which is important is that we have used modus phonens also as one of the important things in our axiomatic system so how do we know that that modus phonens is true so now suppose that modus phonens is not sound that means let us say P P and Q it does not lead to Q then there would be set of formulas like this particular kind of a A ? B and B such that the first two a A ? B are true but B is false so now since B is false then there is an interpretation in which we such that V of B is going to be false that is what we mean by B is false is the way we write this particular kind of now since A and A ? B are obvious already true for any interpretation in particular V the same interpretation we have VA and VA ? B that is to be true so from this we can deduce that whenever you have VA equivalent to VA ? B the valuation of A ? B is true then valuation of B also have to be true there is no way in which a valuation of B can be false because we know that we valuation of A ? B is also true if it is false then it is a valuation of A ? B will become may become false so there is no way in which you can get a valuation of B to be false so we get only valuation of B to be true but we started with valuation of B to be false so valuation of B is equal to T is in contradictory with valuation of B that is false that is that is what we began with so it is contradicting our choice our choice what was our choice in the beginning valuation of B is false that means valuation of B should not be fall but it should be T so there is no way in which you can question the modus ponens in this way that is modus ponens is also considered to be that is also truth preserving rule which is also considered to be sound so that means you can prove the modus ponens rule but that also turned out to be a valid kind of formula this truth preserving kind of formula as far as completeness with respect to Hilbert Ackerman axiomatic system is concerned this is what we mean by completeness which we already discussed in the case of Principia Mathematica if we can discuss with one system and all then it is as good as same as other systems as well so Hilbert Ackerman system I think with this I will end it Hilbert Ackerman system H is also considered to be complete in the sense that a valid formula is also probable so this is a beautiful thing about propositional logic that is all the probable things are obviously true I mean soundness ensures that they are all true and all the valid formulas are true prepositions are also provable if that is the case then whenever we can use this theorems in proving in checking whether or not a given system is complete or etc now suppose if you are asked to prove a complex kind of statement and all preposition in the complex well formed formula and all then instead of proving that thing using axiomatic system etc and all you can invoke the completeness property and assuming that you know preposition logic is considered to be complete if it is complete then it is as good as checking the validity of a given formula rather than check finding a proof sometimes proofs might be very difficult to come by so one can use one can employ semantic tableaux method and you can check the validity of a given formula how do we check the validity of a given formula you negate the formula and look for the unsatisfiable if you can establish the unsatisfiable that means in if you constructed tree and all the branches closes then that is considered to be not x is going to be unsatisfiable that means x has to be valid. So we have the following theorem that is in the case of semantic tableaux method if something is a valid statement if and only if it is provable in the natural deduction system or another system which we actually did not discuss but is more or less similar to natural deduction system there is again gents natural deduction system so something is valid that has that is also provable we know that that is a case in the case of natural deduction system so a is considered to be valid if not he is unsatisfiable so if and only if there is some kind of close semantic tableaux for not a and if and only if there is a proof of a either in natural deduction system or in the Gensen system and all this is what we have already discussed so that is indeed the case so we have a correspondence between natural deduction system and of course this Hilbert Ackerman system or even the principle of Mathematica so any proof of natural deduction system can be appropriately transformed into a proof in the Hilbert Ackerman system so if we can do that thing since whatever all the valid formulas are obviously provable in the case of natural deduction in the same way we have a corresponding kind of proof in the Hilbert Ackerman system corresponding to the natural deduction proof even that in that case also all the valid formulas are also provable even in this case that means Hilbert Ackerman system so finally we can talk about consistency with respect to Hilbert Ackerman system that is x is considered to be inconsistent if and only if for all a if a is reduced from x whatever be the case is reduced from that particular kind of thing then it is said to be inconsistent so it is we are talking about in this case absolute consistency so the proof can be like this you take any arbitrary formula a b and arbitrary formula and since x is inconsistent for some kind of formula b we have both the things b is derived from x and not b is also derived from x so we have another theorem such as this is a thing we have a ? b ? a but instead of a we substituted we have already this particular kind of rule b ? not b ? a so now this is like this thing so what it essentially says is this that so let us assume that your system is consistent not b and we already have a thesis which is like this not b b ? not b ? b ? not b ? e so this is already a thesis or you can check whether it is valid or invalid so now first time when you apply modus ponens on these things one and three you will get not b ? e so now you apply modus ponens again then you will get not be and not be here so you get this thing so now how did you get this one one and three modus ponens and two and four modus ponens you will get this a so using modus ponens you will get a as a single preposition variable as a thesis so that if you can come across this particular kind of thing it is inconsistent in the sense of Emily post that what we have established it here the converse of this one is that if a is deduced from x then that x is inconsistent that seems to be a little bit trivial so so this is a one of the important corollaries of this one is this if x is consistent if and only if for some a is not a consequence of that particular kind of x so another important theorem is that if a is deduced from x even only if x union if I add not here to it so that system is that will become inconsistent so with this I think we have discussed all the important theorems in all there is another important theorem which I will discuss it in the context of when I discuss about predicate logic it is also considered to be one of the important theorems that is what is called as compactness the compactness tells us that in roughly I will talk about this thing so if let us say s be a countably infinite set of formulas x 1 x 2 x 3 like that which are some kind of formulas and suppose that every finite subset of s is satisfiable then s is going to be satisfiable so that means for example we have some kind of statements like quickly I will end this one a or b a or not b and then a ? b a ? b b ? a etc. So now these are the four statements that we have so now the compactness properties one of the wonderful properties that will happen in the case of classical logics that the prepositional logic so instead of checking all the statements to be consistent to each other so what you do here is that suppose if you take into consideration this is the set which consists of these three propositions 1 2 3 4 etc. So now the compactness property ensures that you take any two any two statements and all if you can establish that these two are consistent to each other that is good enough to show that your whole set is considered to be consistent. So this is the finite set suppose if you take single out only these two things 2 and 3 only so this is a subset of let us say a and this is b b is a subset of a if you can show by taking only 2 and 3 to be consistent then that is good enough to show that the whole set a is also considered to be consistent this is what we mean by compactness so with this I think I will stop here so what we discussed in this lecture is simply like this that we presented Principia Mathematica and Ilbert Ackerman system in the last few classes now we questioned couple of interesting questions they are like this is Principia Mathematica complete or is Principia Mathematica consistent etc or is it sound etc now we showed that Principia Mathematica or Russell Wighted axiomatic system or you take any axiomatic system into consideration Ilbert Ackerman or some other axiomatic system which follows so they are considered to be complete consistent and considered to be sound. So one of the advantages of having your system complete is this that instead of checking a formula to be instead of checking instead of providing a proof for a given formula you can check whether a given formula is considered to be valid because all the valid formulas according to the completeness theorem should find a proof so that simplifies our tasks in particular in the sense that you know if your proofs are very hard to come by then you can check the validity of a given formula and say that so that will have a particular kind of proof. So in the next class we will be talking about we will be entering into the third module of this course that is we will be talking about the predicate logic so I will be focusing my attention on the predicate logics which whatever preposition logics could not achieve so we try to fix some of the problems related to preposition logic by using the predicate logics.