Welcome back in the last lecture we discussed one important decision procedure method that is the semantic tab-lux method, semantic tab-lux method phase better than the other semantic methods that we have that is the truth table method, truth table method will become quite difficult especially when the number of variables increases to 4 or 5 I may be more than that a computer can easily do it but as humans it is very difficult to process that much of information you know. So today what we will be doing is we will be talking about one syntactic method where we will be basically studying about some of the proofs of some important theorems in the propositional logic. So we started our journey with some kind of well-formed formulas and out of this well-formed formulas some are considered to be tautologies which are always true and some are considered to be always false they are considered to be contradictions and there are some other well-formed formulas which are sometimes true sometimes false these are called as contingent kind of statements. So what we will be doing today is we will be presenting a different kind of method which also serves as kind of decision procedure method with which you will come to know that I mean how to prove certain kinds of theorems. So anything which is there is one important theorem in propositional logic which tells us that whatever is provable is obviously true and whatever is true is also provable. So it is in that sense if we can prove that something is a theorem then you can already said to have shown that it is a tautology. So in this lecture we will be trying to study about the natural deduction and then we will talk about some of the rules of inference for proving certain theorems. This is considered with outline of this lecture first we will be talking about what we mean by natural deduction and then we will consider some examples of some of the important proofs which are proofs of some of the theorems. So then we will talk about conjunctive and disjunctive normal forms which we will talk about it little bit later but first we will focus our attention on natural deduction. So how did it come into existence? So it is introduced by two logicians in two different papers they were working independently to each other they are not aware of each of the works and all. So parallel is two papers were presented and each one is not knowing about others work and all. So the first is due to it is attributed to Gerhard Kensen and Jacos is a Polish logician they were working at the same time and then they came up with more or less similar kind of results and all. So it is in the year 1934. So what there of the view is this thing? This discovery is as important as other important discoveries such as discovery of resolution by Robinson which is there in 19 just come up little bit later in 1965 there are some of the important results that are very important in the prepositional logic. So these are like this are the discovery of logistical method by Gottlob Frege in the year 1879 and even I mean there are some discoveries like path breaking discoveries by the Greek philosopher such as Aristotle in the 4th century BC. So this natural deduction method is equally as important as one of these path breaking kind of decision procedure methods that are discovered in the history of logic. So why it is called as natural deduction? So logicians call this way of reasoning a natural deduction because it is meant to come as close as possible to the human reasoning which they use in the day to day discourse. So as far as possibility is closer to the intuitive reasoning or the reasoning that is employed in day to day discourse by us. So that is why it is considered to be a natural deduction method. See natural deductive approach is designed more for the ease of use and the closeness to intuitive methods of reasoning. So that is why it has got some kind of prominence in the literature of in the history of logic. So what is that we are going to do in this natural deduction method? So the founder himself says like this Gerhard Genshin remarks like this. My starting point was this the formulation of logical deduction especially as it has been developed by Fregge Russell and Hilbert is rather far removed from the forms of deduction used in practice of in practice in mathematical proofs. So we are going to see little bit later about some of the some of these proofs which are which we will be doing today in the context of axiomatic systems. And you yourself will note that how difficult it would be to prove simple theorems such as p ?p or law of excluded middle p or not p etc. So these things will become little bit simpler in natural deduction method. So there of the this view that mathematicians especially when they prove certain theorems in all they may not be following clear the method that is adopted by Russell and Hilbert. So they went on to say this thing that considerable formal advantages are achieved in written in contrast I intended first the Gerhard Genshin intended first to set up a formal system which comes as close as possible to the actual reasoning. So actual reasoning is the way we reason actually in day to day discourse it is a kind of some kind of linear process etc. So the result was this thing that a calculus of natural deduction. So it is called as Nj if you are an inclusionist you name it as Nj otherwise Nk especially the classical logicians belong to classical predicate logic we call it as Nk. It does not matter in what way you call it. So this natural deduction method is closer to the actual reasoning that we that a human being employs in day to day discourse. Then he went on to say that about this natural deduction that so there are two methods with the one method which we will be talking about little bit later that is the axiomatic method. So the difference between natural deduction method and the axiomatic method according to Genshin is as follows that the essential difference between natural deduction derivations and the derivations in the systems of Russell, Hilbert, Hitting etc. Hitting is an inclusionistic logician is the following. In the later systems that means in the Russell, Hilbert, Hitting axiomatic systems true formulas are true formulas in a sense they are theorems true formulas are derived from a sequence of basic logical formulas usually they are considered to be axioms by means of few forms of inference rule and so what they achieved in the axiomatic system in case of Russell, Hilbert, Hitting is this. They started with the fundamental first evident self-evident truths and all they are called as axioms which cannot be questioned and all they are obviously true and they used very few rules of inference. The only rule of inference mostly they use is the modus ponens principle that is if A in place B is the case A and then B follows from that. Only that particular kind of inference rule very few inference rules they use and then rest of the things are only axioms and axioms are also very few in number. Sometimes in Fregge there are 5 axioms in Russell Wighthead there are 5 axioms whereas Hilbert Ackerman has 3 axioms to begin with and then they use modus ponens and then all the theorems I mean those statements which are obviously considered to be tautologies are all derived within that formal axiomatic system using some kind of transformation rules etc. On the other hand natural deduction however does not in general start from basic logical propositions they do not start with the fundamental principles or self-evident truths such as axioms but rather from the assumptions to which the logical directions are applied. So what they do is they start with there are some obvious rules of inference in used in logic like modus ponens, modus tolens, constructive dilemma and all these things law of conjunction, law of addition. So these are the principle they start with they begin with these assumptions truth preserving rules plus there are some assumptions which comes from the given formula and from that they will deduce some kind of theorems. So by means of a later inference the result is then again made independent of the assumptions there are no assumptions etc are used in the formal axiomatic system used by Russell Wighthead etc except that they used some kind of axioms which are considered to be self-evident truths which are obviously true etc. So now we will be considering some of the important proofs within the axiomatic system. So before that we need to the motivation for this thing for doing this natural deduction method is as follows. So far we have seen semantic tablox method truth table method etc in the truth table methods so they are very simple and they are mechanical and they are large proofs of validity for example if there are n propositions we have 2 to the power of n rows in the truth table. If n becomes large then it becomes difficult for us to handle because for checking the validity of a given formula for example if there are if n is 6 then that means 2 to the power of 6 entries will be there in the truth table that is around 64 entries you need to check all these 64 entries and then you need to check for a row in which your premises are true whether your premises are true and the conclusion is false if that is the case then you will say that the argument is invalid instead of checking all the 64 rows and then those rows in which you have two premises and false conclusion etc for proving the invalidity. So it becomes a little bit difficult for us to handle when n becomes large usually it is unsuited for human use because the once the number of rows increases then it thinks will complexity increases. So it is usually unsuited for the human kind of reasoning especially when n is large the entries in the truth table are large then computer can process it but it is very difficult for us to handle. So there are some other effective methods which we have introduced earlier that is the semantic tableaux method which is considered to be elegant method so far and then now we are going to see the natural deduction method it is suitable for human reasoning because it is in the beginning we said that it is closer to the actual reasoning the way we actually reason in day-to-day discourse it comes closer to that if not it is completely same as that one but it comes as far as possible it is closer to the human reasoning that we employ in day-to-day discourse. It often allows short proofs of validity for validity in particular we can prove it and that is considered to be a true statement and then all the true statements are obviously valid formulas. So it allows for shorter proofs of validity which are going to see how we are going to achieve this thing and it is also arguably too flexible for machines use it is not mechanical a large number of rules it involves large number of rules but it occupies it also occupies infinite such space is another limitation of natural deduction method. So what essentially we do in the natural deduction method is as follows we start with simple truth preserving rules like modus ponens modus tolens and the constructive dilemma they are all valid principles of reasoning in that we employ in logic. So we begin with these rules of inference and then we add these rules of inference to assumptions that we get from the formula which we are trying to prove and then from that we will get the desired kind of formula that we are going to derive. So natural deduction is used for constructing proofs of theorems theorems means there are obviously true formulas they are also considered to be tautologies etc. They are different from axioms axioms are considered to be self evident rules etc. So theorems will all be valid sentential formulas and all valid sentential formulas will obviously theorems of that particular kind of system any formal system you begin with all the valid kind of sentential formulas are obviously true tautologies etc. So all such valid formulas constitutes what we are calling it as sentential calculus. So what is that essentially we are saying is this thing you began with well formed formulas and then out of that there are some tautologies all these tautologies are obviously valid formulas and then this all these valid formulas constitutes what we mean by a sentential calculus. So it becomes a particular system the system sentential calculus is it can be viewed as a natural deduction system as it makes use of rules without any axioms. So we are going to see how we are going to achieve this particular kind of thing. So the derivations in natural deduction corresponds to how we actually are naturally construct these theorems. So in the tree method especially we what we have done is simply for establishing the validity of a given formula well formed formula what we have done is simply this thing we listed out all the premises and we included the negation of the conclusion and then considering the negation of the conclusion leads to the branch closure that means it is unsatisfiable then we have said that negation of the conclusion is false that means the actual conclusion is true. So we list the premises but not the conclusion in the natural deduction method we do it in a different way we list out all the premises and then using all the truth preserving rules which we are going to talk about in a while from now we use those rules and we have premises which serves as assumptions and from that you will derive whatever you wanted to derive. So what we will do here is that we list out all the premises but not the conclusion is the one which you are trying to derive from these premises then we apply natural deduction rules which are true preserving and that will lead us to our destination that is the conclusion that is what we are trying to prove. So we are able to write the conclusion of the argument based on the assumptions plus natural deduction rules. The natural deduction rules are usually truth preserving thus we are able to construct the conclusion by applying them to premises and we know that the truth of the conclusion is entailed by the truth of the premises that is what we mean by a deductive argument the deductive argument is a one in which it is impossible for the premises to be true in the conclusion is false but here all the rules are all the steps of your proof are all truth preserving. So based on that there is some kind of truth preservation is achieved in this particular kind of proofs and obviously each step of your proof is true and ultimately the final step which is considered to be a theorem which is obviously true and all the true formulas are obviously they are considered to be tautologies and all tautologies are obviously valid formulas but the problem with this method is we talk at the end of this lecture is this that sometimes we may fail to derive the conclusion using these particular kinds of rules your derivation go on and on and on so then that sense it may not be a kind of effective method. So for that what we will do is we will follow another kind of method within the natural deduction that is called as reductio ad absurdum. So what we will do is we list out all the premises and we negate the conclusion and then we will derive a contradiction in sense that a and not a and that is the case then you cannot deny the conclusion that means the conclusion denial of the conclusion leads to the contradiction that means the actual conclusion is true. So now before doing natural deduction there are some rules which we will be following these rules are like this. So these are basically two different methods that we will be using first one is RCP conditional proof so what we will do here is this thing. So we have some kind of assumptions a which usually comes from the given formula and we obtained a line where you found B. So now B is a tautological consequence of A if that is the case then A ? B is considered to be a tautology. So what have what actually happens here is this thing so you have A here and then obviously after few steps after applying natural deduction rules you got B so that means A is derived from B is derived from A if that is the case then you can simply write like this A ? B you draw a line like this and then say that A ? B is derived from some kind of assumptions and natural deduction rules which you employed that helps us in moving from A to B that means A ? B is the one which you have derived so this is called as conditional proof.  B because this says that B is derived from A so that means if this goes to the right hand side then this becomes A ? B suppose if I say this is a single turn style this means that it is a theorem this should be read as like this it is a theorem that A ? B A ? B is a theorem within that formal this is that system that is a natural deduction system so this is what is called as RCP that is conditional proof. So now the second thing which we will be using in the natural deduction method is this thing which is called as reductio and absurdum. So in this reductio and absurdum you started with some kind of assumptions it can be premises it can be hypothesis you can call it as a hypothesis and all the way down here in your proof you got some kind of contradiction so usually it is mentioned in this way so there are two symbols which we are trying to use this is the symbol which we use it for a formula which is always true and there is something which is called as a same formula which is always false this is top and this is what this is different from T and F for truth values. So now what happens here is that from a all the way down somewhere else here you got a contradiction so that means a implies this one so from a you got contradiction and all so that means if a implies contradiction then that means it is not it is not simply a and all it is it should be not so this particular kind of thing is called as reductio and absurdum and all so it essentially says that given some assumptions which leads to contradiction that means let us say if your assumption is a and that led to contradiction then the actual assumption is false that means it should be not a suppose if you began with not a and led to contradiction then that should be a so this can be also written in this way from not a it led to contradiction then it should be not not a that is so these are the two important methods that we will be using in the natural deduction so the idea of reductio and absurdum is this that if you derive a contradiction from a given assumption then that assumption is false that means it has to be not a so that is what we all said here. So these are the things which we follow but the actual founder Genshin he proves theorems in a totally different way and all so mostly only in few logic textbooks you will find the proofs which are given by Genshin but usually we follow just go see ski's proofs in particular which are which seems to be a little bit simpler and simpler to understand but both the proofs can be used so now what are the rules that we will be using in this natural deduction method so this these this can be compared with playing a game and on example let us say you are playing cricket you do not have to know how how and when it is going to existence etc you do not have to know the foundational principles etc what you exactly need to know is the rules of the game you have to know about white ball you have to know about no ball etc now so once you know these rules then things would be relatively easier so then you there will not be any defects in your in our game so are mistakes in the mistakes which can happen in the same way natural deduction you start with the assumptions and then you have some kind of rules and these rules are like this particular kind of thing so this some rules are considered to be discharging some kind of assumptions and then in some of the rules you will introduce some kind of implication etc and so these are some of the rules that we will be using in the natural deduction method you might ask do I need to remember all these rules etc and all it comes you know through practice will not be much to remember and all these things obviously follows so these are some of the rules that we will be using in the natural deduction method so first let me list out these rules and then we will talk about something about these rules and then we will move on to some other I mean improving certain kinds of theorems so these are some let us say ? is some set of assumptions that we have and from this let us say union there is another formula B and from this you go so this is a and then from this you got B this says that ? is something like set of formulas which are sitting at the background so it can be some kind of natural deduction rule a principle or whatever it is and these things which you are adding it to the assumption that is there which occurs in a given formula and from this you got B so now what you will do is your assumptions are a and B and all here now you discharge these assumptions and then you will write simply this thing from ? you obtain a ? B so now earlier it was only a proposition a and proposition B and all now you discharge those assumptions and all instead of saying that it is a B and all you will say now a ? B is the case so now you discharge these assumptions and all so this is one of the rules of which come under the category of discharging the assumptions and all in the same way if you have a formula like this thing not a so you might ask what is this ? and all ? involves all this truth preserving principles and all it can be this thing a ? B which I will talk about it a little bit later so they are all truth preserving principles and all you can be considered as a kind of tautology you know but it is not axioms in this case or in the case of natural deduction method something which is true now from this let us say you got something called a contradiction so that means what you got is this thing now not a place contradiction so this is what you got so that means not a implies this is the case then it should be a so now you simply write a so this is one thing which we will be using there is something called reiteration rule which is some kind of redundant kind of rule and all suppose if you have a formula a and then you can simply say it is a nothing great about this particular kind of formula if you have a you can derive obviously a so this is a reiteration kind of rule so now there are some other kinds of rules which are there for conjunction disjunction etc so then I will talk about some kind of rational for these particular kinds of rules so in all these things gamma is already there means something which is true and all so now so this stands for conjunction introduction I stands for introduction so when you will do that particular kind of thing you have obtained a and you have obtained b in your proof let us say you have gamma and then all the way down in your proof you got a and you got b then you will simply write you will introduce the conjunction and then you will write a in plus b so now this rule is called as conjunction introduction so now conjunction elimination so each natural reduction rule comes up in some kind of pairs and all one is introduction another one is the elimination so each connect you will have this particular kind of rule for the conjunction if you have a b you have proved a and you have also proved b then a and b can also be proved from this particular kind of using particular kind of truth preserving rules conjunction elimination is like this if you have a formula a and b you can simply write a it is like it is raining and the grass is wet from that you can prove grass is wet or you can even prove it is raining so what we have done here is that we have eliminated this conjunction and then in your proof you are substituting it with simply a and in this way in this sense a and b b also can be derived so this is called as conjunction elimination rule so now this stands for disjunction disjunction introduction suppose if you have a formula a that means is true and you can safely add any other things without disturbing the truth value of this one so in your proof each and every step has to be true so without disturbing the truth value of your propositions you can safely add another b because if the b value even if it is true or false it is not going to affect the truth value of the compound statement even if b is t and this all become true even if b is false then a or b is again true so you have to ensure that in your proof all the steps are obviously true then only the final step is going to be true that is a theorem so in that sense that is going to be a tautology and obviously that formula is going to be valid so our effort is this that ultimately we have generated some kind of valid formulas and then we are trying to prove those valid formulas and for proving there is some kind of decision procedure method that is the natural deduction method so this is what is the case and disjunction elimination is like this so now you have shown that a or b is true and then from a independently you proved this thing see that is what you write it here and then from b also you got C I mean C is deduced from a and C is deduced from b and a or b is already there if that is the case then you remove this disjunction and you will simply state here so you simply say it is C so this is what is called as disjunction elimination so you have a or b which is already true and then from a independently you proved C and from be you proved C and that means from all these things you can prove C you can eliminate this disjunction and then show that this is the case suppose if these are not there then you cannot say that a or b C can be deduced so this is another rule the rational of this rule is that one which we have discussed so now as far as this implication is concerned again we have introduction rule so especially from a let us say and you prove b then what you will do is you will introduce this implication and you will say that a implies b is reduced from this thing so this is what we mean by introduction so there only assumptions a and assumption b is there and then now we are introducing this thing because b is reduced from a so that is why a ? b will come as a theorem of this one so now this is what is the introduction of implication and in the same way on the other side introduction elimination when I write e is stands for elimination I means introduction this will be like this for example if you have a formula like this thing this is usually modus ponens rule and all so now you will eliminate this implication and you will say that it is simply b so what is that we are trying to do is simply making use of some kind of rules so what are these two there are all truth preserving enough you see any one of these rules obviously for example if you take this one if a is true any ? b is also absolutely true 100% true then obviously b has to follow from this this is the principle of valid reasoning so instead of starting with axioms etc and all we start with only these rules but the problem here is that one has to remember these some of these rules of course with some kind of practice with some kind of strategies which we can use with that you can say you can usually remember these simple rules or if you do not know some of these rules are you can derive these rules from some other kinds of things which you have already know so this is one thing and then we are talking about conjunction disjunction and implication and the negation of this one is like this for example if you have a formula a then you will derive only this one and then negation elimination is like this if you have not not a then you will negate this elimination and you will simply write a in your proof so that is what is with respect to negation so now there are some other rules for example if you have not a and then from this you got a contradiction then this means it is not a it is not not a but it is a in the same way if you have a formula a and then you got contradiction contradiction in the sense that you got x and not x that is the case it is called as a contradiction then since a led to contradiction it should be not so that is a rational of this particular kind of rules it is just like you know in playing while playing cricket you will be discussing about what you mean by no ball what you mean by wide ball or when you will get five runs etc and all these things you will be discussing what you mean by a boundary or six or something like that all these things are some kind of rules which game in the game subsequently we evolved with this particular kind of rules just like that traditions while practicing with the number of theorems and all proving number of them they came up with this particular kind of rules they are all truth preserving rules there is no way in which you can say that if this is true this is true then it cannot be the case that a and b can be false enough if that is the case it is not a directive argument at all so these are some of the rules which we have and then there is one more thing which is important so whenever you have a and then you came up with not a then so this led to this contradiction you derived a and you derive not here also then obviously that is a contradiction you cannot simultaneously say that it is raining and it is not rain so then this leads to this one what is happening here is that you again you removed the conjunction and then it is actually a and not now we can simply remove this conjunction then you say that we simply the contradiction and in the same way if I this is what we have already discussed so these are some of the rules that we need to remember before solving some kind of problems and there are some other obvious rules such as modal stolons etc and all they are already there sitting at the background so so there are some other rules such as replacement the rules so what are these replacement rules so example if you have formula a or b by commutative property it will become b or and this happens for conjunction also so this double colon stands for equivalence and all this means these two are logically equivalent to each other it can be b and a suppose if you find a and b in your proof you can easily substituted with b and a so these are some kind of replacement rules they come in pairs and all conjunction and disjunction this is a commutative kind of rule so now we have another kind of thing a and b RC suppose if you find something like a and b RC then you replace it with this particular kind of thing this is a and b RC you can write it in this particular kind of way so in the same way if you have a R B and C and this is same as this particular kind of thing a or b associative law and then see that means you know whenever you have this particular kind of formula you can simply replace it with this thing that is what it says and then third set of rule is this thing suppose if you have a formula aim plus v and you can this is by definition is not a or b you can simply substitute for aim place b not a or b and then there are some kind of D Morgan rules which are obviously we know for example if we have a or b negation of a or b and you can simply substitute it as negation of a and negation of b because negation of disjunction is conjunction in the same way negation of a and b is same as negation of a and then negation of conjunction is disjunction now it is not here so like this we can list out all these rules and all another rule is not not a is there in your proof you substitute it as simply a and there are some other kinds of rules like a implies b implies c if you have this particular kind of thing you can simply substitute it as a and b implies that means whenever you come across a implies b implies c that can be replaced by this particular kind of thing that is why these are all called as replacement rules and finally without this thing alpha and not alpha a and b are c and this is same as this is a distribution law and all distribution law it says that a and b first one or it is a and c this is distribution law used on conjunction and distribution law in the same way it can be used for disjunction or disjunction and suppose if you have this thing and this is same as this particular kind of thing a or b and a and c a or c so now you might be asking written so many rules etc and all so now what is that we are going to do with this particular kind of thing so now we try to prove some simple kind of formulas and then we will try to see how we can prove these particular kinds of theorems using the rules of inference that we have seen so far so now this is the one which we are trying to we will try to begin with obvious things a implies b and b implies c and now this is what we know that this is the one which comes as an outcome a in place c so now what we will do in the natural deduction method is simply we write these things as assumptions and all this is assumption one and this is another kind of assumption one two so now this is what we are going to get a in place see so now what we will do here is this thing that you add principles of natural deduction the one which we have seen rules of natural deduction then ultimately you will generate a in place see so now how do we get this a in place see is the one which we are trying to see so now we start with these assumptions and you will work out till you generate this particular kind of thing using the principles of natural deduction so now what you will do here is you will assume the antecedent part of the conclusion so this is antecedent and this is the conclusion so now what you will do here you will assume the antecedent of a conclusion again this is an assumption in some test books it is assumption is written as premise in some other test books it is written as hypothesis does not make a big difference so now this is a thing so now we assume these things now we need to use rules of inference truth preserving rules so now observe one and three one and three modus ponens what is modus ponens if a in place b is the case and a is the case and then you can get b so that is the truth preserving rule which we will be using a in place v and a you you get b so this is one and three modus ponens so now observe two and four two and four modus ponens will give us C b in place C and b C is the case so now we have used some truth preserving rules now from a you got C so now what is that there is something called in the beginning we discussed about RCP rule of conditional proof rule of conditional proof this is what is RCP so from a suppose you got C that is what is the case here from me you got C then you can discharge these assumptions can give up this a and you can give up C and all and then ultimately you say that it is a in place so for that one what you will do here in the natural reduction method is that you draw a line like this from a to C and then the six step what you will say is from three to five RCP according to the rule of conditional proof which is there here then you will simply write a in place C so now a in place C is the one which we wanted to do that so now you draw another line here from the whole thing a in place b and b in place C you got a in place C so now what you will do here in the step to a in place B and B in place C and from this you got a in place C so this is what we have deduced so like this one can derive many formulas for example let us try to talk about some simple things in all like which will be quite difficult in the axiomatic system which we are going to see little bit later so now we are trying to prove P implies P so which this involves in any axiomatic system you take into consideration it involves at least four five steps but in natural reduction method it is quite simple so this looks very strange for us but actually this is considered to be the proof so now you started with this assumption P what you need to prove is P only so now you write again P only so why you have written this thing if P is there so you can deduce P automatically that is reiteration you can n number of times you can write the same P again and again does not make any sense and all but so this is reiteration suppose P is deduced to be true when P is derived and again P can also be derived now you draw a line like this and you will say that P implies P is derived and all this is the most simplistic kind of proof that one can use it seems does not make any sense and all but it involves but according to the rules of inference and natural reduction it involves only two methods two steps and the same P implies P in the given axiomatic system sometimes it might take eight steps or maybe nine steps and all if you start you can start with axioms and then you can use modus ponens and rules of transformation and then maybe after six seven steps you will get the desired thing P implies P in the same way if you want to prove this thing P implies not not P so if you want to derive P implies not not P again in the natural deduction method you will start with an assumption P now so to whenever you have P you can substitute this particular kind of thing in the same way not not P is there you can substitute P so now this is not not P 1 and double negation so DN stands for double negation so P is there you can substitute not not P I can even substitute four not also it is going to be P so now since you have derived P from not not P you draw a line like this and in the third step you will say that P implies not not P that is one two one to two conditional proof because not not P is reduced from P so that is why P implies not not P and then you will write it like this to say that this particular thing is considered to be a theorem so you must take care of this particular kind of justification these are the steps and all on the right hand side whatever you find it here is a justification for writing this particular step so how we are justifying this thing based on the two preserving rules that we already have so this says that one we have applied double negation to it and then this is what we got and then since not not P is reduced from P using the conditional proof you can say that it is P implies not not P so this is the derivation of a theorem that is P implies not not the same thing which you can do it in semantic tab looks method in semantic tab looks method what you will do is semantic tab looks method so in this what you will do is you will take this as X and then you negate the conclusion so that is P implies not not P and then you will see whether the branch closes are not so now this is P the first one this is not of X implies Y is simply X and not Y so now you write the same thing and then you substitute another not P so not of not not P this is not not P is P and negation of P so this is not P so now you are derived P and not P and all so hence this branch closes because of this particular kind of conflicting information is there your P and your not P so the branch closes so that means what we have said not of X is false are usually you write it in this way so that means X has to be T the original formula should be T that means it is a valid formula this is another way of showing it using semantic tab looks method but in the natural deduction method of course things are little bit simpler you just use one devil negation rule and then you got the answer but here little bit of some more rules which we are using of course this is also very efficient method which also involves 1 2 3 3 steps so these are considered to be effective kind of methods for proving that this particular kind of formula is a valid formula so how are we showing that this particular kind of thing is a valid formula because we started with the truth and then we used the principle of natural deduction which is obviously truth preserving kind of rule truth preserving truth preserving and ultimately conditional proof also preserves the truth and all so obviously the final step of your theorem is obviously true so that means so all the tautologies are obviously valid formulas so there is one important theorem which tells us that if something is probable like in this way in the natural deduction method and that has to be true that is a tautology and all tautologies are obviously valid formulas so completeness soundness etc takes care of this particular kind of thing propositional logics are all complete sound and even consistent as well these are some of the things which we will be doing and then some more proofs which will try to see then things will be little bit clear so the more and more you solve these problems the more and more today we will get expertise with this particular kind of method so we all know that this is a law of contra position from p ? q you will get not q ? not p so this is the one which we need to prove first you will write the assumption p ? q this is the assumption and then this can be written as in this one p ? q and then not q ? not p now you take the antecedent part of this conditional not q as your assumption this is also an assumption so now three there is a rule again in the natural deduction so that is if a ? b is the case and not b is the case then it should be not that means you deny the consequent here in the second step you have to deny the antecedent also this is called as modus Tolens root modus Tolens so now with these two things you can derive not p ? q and you are denied the consequent you have to deny this antecedent also so now this is what is the case so now you draw a line like this what did you get from q you got not p so not q and not p so now you need to write justification for this one one and two modus Tolens you will get this you have to write justification otherwise nobody will be able to understand this particular kind of step and usually that is not considered to be an effective proof and justification used to be given after all logic is used as a justificator a tool so now not q ? not p is the one which we got so now you draw another line like this and say that from p ? q so how did you get this one ? q ? not p 2 3 from two steps 2 to 3 conditional proof rule of conditional proof you got wrote this one and now you say that in the fifth one you write like this because it is a theorem so from p ? q you got this one from p ? q you got not q ? not p so this is what we wanted to prove so how did we get this one so 1 4 conditional proof or you can write rule of condition proof you got this one so this is the way in which you can show that law of contra position can be derived in your natural reduction method so now let us talk about law of excluded middles so in your formal axiomatic system you should ensure that at least you know all these laws fundamental laws such as p ? p that is a law of identity and p are not p is the law of excluded middle and law of contra position law of non-contradiction all these things should come as you should be in a position to derive these things first and then rest of the things obviously follow rest of the complex theorems etc will automatically follow now so in the next class we will be talking about some more complex proofs based on natural reduction and then we will see how this method is can be considered as an effective kind of method for proving certain theorems and all one of the advantages of this method is this that your proofs can be simpler and it might involve very few steps and all so in this lecture we have presented natural deduction method natural deduction method is based on some kind of truth preserving rules and we stated those two preserving rules and we discussed the rational for using this to preserving kind of rules and then we proved some simple theorems such as p ? p and the law of contra position etc in the next class we will begin with we will we will consider some more complex proofs and all based on natural deduction so that we will become well acquainted with this particular kind of method.


And it's time for our second hot topic, and this second hot topic has to do with our culture and history as Africans. The pride and identity of our people is our culture, and culture is passed down if the people have a good and respectable sense of history. What then happens when culture is jettisoned for something alien and history is no longer taught to our children? Well, we'll be looking at the value of knowing our history as Africans, and we've been joined by Aidee Soji Iginla, author of Africa Illuminator. He's joining us from the UK. Good morning to you. Good morning, good morning. So let's start with why this is of great concern to you. Okay, contrary to the fact that most of our books have been authored by Europeans, it means the legacy of colonialism would be watered down to the point where we don't sort of get an inference. And I will give you the entry point for me. It's chapter 20 of The Great Book by Chino Achebe. The conversation between Oberica and Okonkoa where he says, does the white man understand our custom? And Okonkoa's answer was, how can he? He says our ways are bad, and our people are also saying now that our ways are bad. So how do we build community? So this book is an entry point to help Africa illuminated. It's an entry point to help re-accustom ourselves with what is the hidden facts about those various African countries. I mean, the problem being that, yes, they were created largely by Europeans, but we inhabit those countries. And so we should actually own what the legacy is and reframe the narrative about those countries. What have you observed in our young ones, especially? I understand the book you've written, Africa Illuminated. Yes. Yes, you're targeting the teenagers, if I understand correctly. Correctly. What are some of the concerns you have that you've observed in teenagers today that's making them the focus of your attention? I know one of the things I have noticed is that some of our teenagers no longer speak our mother tongues. They do not speak our languages, our local languages. Oftentimes they just speak English. So what are some of the concerns you have that's made you target them in this book? Yes, it's a story closer to home in the sense that I have an aunt here who has teenagers and I often go and pay visits. And on one of those visits, I was having a conversation with the kids. And so the question was, oh, your mom is speaking to you about what you are answering in English. And the answer is, we don't, we understand, but we can't quite speak it. And I said, OK, fine. So where are you from? So they said, yeah, we're from Nigeria. I said, how many countries around Nigeria can you mention? One said Egypt, another one said South Africa. I said, well, I'll give it to you. They're both in the continent of Africa, but they're not exactly around Nigeria. So to which the aunt will then say, you see what the problem is? You have an older generation, my generation, that came through a system where at least you had a modicum of what it is you speak in Yoruba, which is, you know, in Yoruba, which is you do not point home with your left hand. So invariably, you don't lose track with where you come from. And so I said, OK, I said about writing this book. And again, it is important. It is very, very important. In this book, something new points to the point that when you speak somebody else's language, what happens is your memories, your cultural memories get replaced by the new language you've acquired. So if I start quoting Shakespeare now, everything I deem of value would be in terms of looking through the prism of what Shakespeare considers to be great and not the likes of Wale Shoika, Ngukubi Wafiungo, Mesera Mutgo, Chinoachebe. I mean, the list is endless, Flora Unwapa, Aima, Aitai Ado, Aikui Aoma, the whole host of Nigerian, the whole host of African authors. I think if we just delve into them and read about them, they would actually open our gaze into the world that we call Africa. Talk to us about the impact of knowing African history on our mindset and our identity. OK, with regards to mindset and identity, if we pose the question to the general public that how many waterfalls are there in Africa, the mind immediately goes to Victoria Falls. But Victoria Falls is not the name, but it's not the name given to you by the local. That's a name given to you by David Livingston, who when he set his side upon him, he says, oh, it is only as great as Queen Victoria. So I named D, Queen Victoria. But actually the name of the place is called Mose Intunya, which is the smoke that thunders. Because when you are nearby for miles around, you can actually hear the roar of the water dropping down and the magnificent scene that around you. So when you speak and see, and you can't quite name what is yours, or in our case, growing up, we're told, the Mongol Park discovered River Niger. How can that be? Because obviously, people were living on the river. The river stretches from Mali, runs through Nigeria. And so how is it that one European will come along and says, yes, I discover something that is already there? All right. Efforts, I guess, must also be made towards freeing and mobilizing black minds. And when I ask this question, what comes to mind is Redemption Song by Bob Malley, the late Bob Malley. What are these efforts that should be made towards liberating freeing African minds, black minds all over the world? OK, cultural is a very important thing. There is a building sitting at Igomu, Festak, National Arts Theatre. That's a legacy of a program that was Festak 77. In fact, Festak Town is a legacy of that cultural festival. And that brought around people from all over Africa and the diaspora, bringing them together so that they can exchange value, experience, cultural practices, and rekindle all ties. So when Bob Malley played that song, Redemption Song, Redemption in the sense that, yes, we can buy back what is ours. We can claim back what is ours. There are the pyramids and not only in Egypt. There are also pyramids in Sudan, because at that time, the greater Egypt went from the top now to the bottom, not across, as it currently says. So again, it's important that we delve into our history so that we can not only reclaim this because our story needs to be told by us. There's an African proverb which says, as long as the story, as long as the lions do not have a historian, the story of the haunt will continue to glorify the haunter. Meaning, when Europeans write about themselves with encountering Africa, they will write Africa from their point of view. Not from our point of view. Well, before we go, just give us a one-minute summary of the book you've written, Africa Illuminated. It's a recovering, remarkable fact about Africa, an entry point for anyone who wants to delve deeper into what Africa is to us, to me, to everybody that lives within that continent. I mean, it is important that people speak their own truth as supposed to be written about. When I say written about, it is incumbent on us, the responsibility. Like Chinua Chebe says, every generation has the duty that history has bestowed on them. Our history is to keep our legacy, our culture, our language, and memories alive. Thank you. Yeah, the value of knowing our history as Africans is what we just discussed right now with Addis Oji Iginla, who has joined us from the UK. He's also authored a book titled Africa Illuminated. You probably want to grab a copy of that book. Is it on Amazon? It's on Amazon, it's on Kindle. All right. Plans are free to bring it to Nigeria, but at the moment, it's on Amazon. All right. Thanks for being a part of the show this morning. Thank you for having me. Thank you. OK. You're still watching the Breakfast and Plus TV Africa. We'll be back to give you sports. Do stay with us.


Welcome back everybody to another stream. Hopefully you guys are doing great. Without further ado, let's get started Yes, they pretty much went as intended. I mean I had only like two times to sort of take care of other than that nothing sort of too major and those times being the 830 new cycle that we got and then followed by the Follow-up coming in at 2 o'clock EST Other than that, there was not really too much from yesterday Now we sort of see price coming back up again. So I'm collecting that sort of trophy of longs that we sort of like to see and we've got another sort of 830 start today and Other than that today doesn't really have too much news other than that 830 830 sort of PPE Jobless so other than the 830 coming in here. There's not really too much that will fit the chart. So looks like we'll have a sort of finalized set direction coming from today and tomorrow there's a Michigan sentiment at 10 o'clock. So We're probably finalized. I think the entirety of the direction here because next week. I think it's looking a bit more quieter in terms of news wise so Doesn't sort of see any sort of big directional changes from news coming news affecting it then So which makes us sort of Be happy aside because it means we won't have to sort of pay attention to All the BS that's going around around us and just focus on solely the charts Coming in now. You can see they've got they added a one plus one key on this high end. So Like we sort of talked about for normally it's just about doing whatever Is the inverse of Current sort of mark a chart So if you sort of break it down now give Think it does the exact same thing it does yesterday. You can see that yesterday. We built down Came in had that news release exploded high false move correct against that Building short. So ideally if we were to go down, we want to clear us out. You can see here We built that here cleared those guys out built a lot more So it stands to reason that we should like clear these guys out move into She I think they'll still be Some of these guys will still be in the market, but it's majority of guys would not Moving into this direction. You still see that big sort of Delta here of a lot of fires. So It does stand to reason to say, okay, we'll possibly come in and just hit these guys out and Potentially move up to that wall that we see there. So It's a little bit of a red herring that they did face that just before the news augmented 2d Coming into what the sort of spot is looking like We're in last week's one just this range and see that a lot of buying has occurred So which is usually a nice sort of Downward sign if we're getting buying see short term that they got that panic selling that occurred yesterday Just just sort of drive it home. So they might Injuice again these same walls that were in futures. They might come up to here. I just say Convince these guys to be buyers at some point. It's good. They see to go on my own. Okay all minus five. Oh That'll be interesting. Okay, we've got we've got that news coming just instantly came up. So we see how it tracks now So I mean sentiment was decently high Each yesterday you can see that massive little peak Whilst we're waiting for charts to move Further you can see that once it reached a high peak and it came back down again once people get comfortable in that it's just That's the reason they just we just want to make them uncomfortable again Uncomfortable as humanly possibly manage but Still don't rule out that you might just try and lull them into a felt for sense of security One final time before dropping that much further because sentiment can always drop Because it's still decently high high enough that It could definitely go low and best way to make it go low is have a little bit of a downside Seriously, nothing's happening so far Is there anything's happening on yes? I Mean a large spike up I mean Yeah, a large spike up on this or 15 minutes you see that big explosions Grabbing a lot of sort of trades here grabbing a lot of stops as well Again, just sort of carries a home that we want to potentially go back down. I mean This yes, they had a lot further sort of down didn't sort of consolidate like Bitcoin did At least didn't consolidate back to the same area instead of continue to go down. I mean Bitcoin a Bitcoin is up react. I mean you see that's what yesterday one big candle down this one just up and down just Decided to have have a decent time But from this perspective, we can definitely continue to go down a lot Lot more it's just Waiting for that. So right time Coming to flowers Looking at that large candle down Big quick on this area here came back down close the volume on open open down for the entire New York Came back up off Bitcoin just move sideways So there's where bitcoins volume is concentrated Sort of agree And you can see that the same thing did happen here, but it did move further down Holding off in this area after very large I move high Then sort of have the head has more break basically interesting reaction Wanted we wanted wanted numbers. I mean, but nectar numbers are nice mean sort of things are bearish Overview so might get that rate hike multi chart doesn't like it Just jobless claims rise to the highest levels of mid January. That's always a bad news again Major thing Major thing is fixing the labor market because fixing the labor market always means that there's more possible income to spend and a lot of less people sort of worrying about Saving so Fixing that it's gonna wait for 10. Yeah, probably 10 for this because this is really slow, especially for an 830 news day So maybe we're just delaying these orders. I mean Yes, did have Heropia at reaction very lackluster Very sort of lackluster. You can see that's that how a very high here this It was not really fast Still the volume is closing this and you see that there's the collection of Traps here that have been laying and you can see that people have been added adding here They've been sort of adding in the general vicinity of this area here. So it does stand to reason say, okay General outlook is probably bearish in the long run Look at the option. I mean option day is mostly on I mean we've got put so we might hang around for a bitch Because those clear out what tomorrow Then you've got an insane level of sort of puts coming made in January May and June so these might be our very bullish months these expiring Back into exactly where they are. That's expired today. So I mean hanging around young opens where we'd want to be January past I mean tomorrow you can see that Definitely want to probably go higher or lease Hang around a bit further Jet needs to be above four four eleven first 408 so And probably want to be around here or even here to week tomorrow to try and clear the puts out sort of negative For some sort of closing that even closing profits because anything below that these lines here are the other men profit There's not really too many Generally just this year Very large large, but so definitely looking to be higher tomorrow Maybe we'll set it up prior to launch so when we get everything Fiery and after expiring and then talk there ourselves back down That's what would we have again consumer index 10 o'clock tomorrow, and that's by itself 30 minutes Push the market higher Before sort of an inevitable crash in Speculating those nice draw conclusions, especially we have nothing that to do but what paint dry In spot and futures pretty much same thing collecting water selling and Now we're sort of building that potentially looking to go higher for today at least hold out before the Friday Have a lot of spires coming in on that Friday Small amounts of money is a big money as well Big retail traders Okay, so I mean this training view marked out. I mean we can see how we mark these times out There's two o'clock and yes, he does your 830 Nothing to hang on in this little outside ledge But here, I mean I'm still waiting for something to happen. That's why I'm apprehensive about going through yesterday's data Something might happen But nevertheless, uh, yeah, we talked about these walls here and how we just Obviously here by said normally Advice try and catch that you'd normally get a retrade back Then we saw these walls pop up as soon as we started moving down then your lunch gone next step When we're seeing that downwards is looking for a retracement at one of these points We've got that coming in at At 10 Coming at 915 here Then next step came down much lower here. So you could have either 915 a lot earlier We'll wait for the 10 o'clock in a lot safe. I guess So the safest step would have been sort of watch that clear it down a bit then it comes back This and then this is 10 o'clock And I got comes in those nicely with here one last spike and you see that entire green So I skate then finally sort of drops the ball And initially enough they sort of recover leading to the 10 and this is a nice little Recovery show you've seen this recover you've seen this recover Then they're like, okay, they're not gonna go higher. They're gonna go lower because they already recovering if they instead did here and went down like this and they would have Recovered into the higher territory here, but they just instead recovered That sort of two o'clock announcement made people sort of say are nice. It's not too bad numbers Numbers yesterday were rate of change is slowing down So that means I'll give people thinking that's not much then came in gave you that double bottom inevitably Then finally sort of pushed up and came back a century where we opened yesterday Still waiting for something to happen Any time now Any day now We've got our expires so I really want to sort of push these guys into Disarray, I mean, I don't mind it where it is right now and some tomfoolery Within the current training session probably sort of like fire by tomorrow So you're probably looking to end the day higher than I Mean If you get a low point if you're going lower, we're only going lower to go higher essentially I think that will be the sentiment today just because there's no way we're gonna come in and just have all of that All that sort of negative sort of thing being profit. So We want to be we want to make people hurt and that is that is the goal of training is to make people hurt So how can we hurt people the most making them cry? And we can do that by simply going into the opposite direction where they place the trade Yeah, I mean, it's a nice little build up. You mean you talk about again all the shorts So expect the high all the selling expect the high we want to see those buyers come in Once you get those buyers come in we can then come back down but Generally what we want to see so very heavy so Might even expect a little bit of Little bit of a trace because see with Bitcoin like even with this range is such a sort of heavy divide of web Where people will instantly just come in for a long and instantly sort of come in for a short so They're very much getting eaten by the truck I Enjoy is a good factor. Oh finally Again, if we start moving higher That gives you that force to say that wall that we saw in book map We hit that then we can come down and when we do come down, it's just okay Now we need to find a point where we can then safely safely go back up Because we're essentially only looking for it to just lay people out because We wanted to go higher for what Friday's open. I mean doesn't mean I'm saying we're gonna go higher in terms of trend wise What I'm saying is we're gonna open high just to liquidate Yeah, just to sort of liquidate the or close the options in a negative and The idea that it's just looking for a peak high and then it just comes back down And it has to if it does two sets of that Probably gagging for the last couple of days And even this consolidation here just taking its time and sorry so much tire So we're looking for probably a double top here So maybe we wherever the walls end up stopping guys where we want to sort of see See that and again, we've got a lot of time in our hands. So we want to sort of slow build up I mean we want to see that slow built up of buys Those buyers coming into the actual market needing us into sort of 3,500 is a nice little happy 450 In these shorts slowly becoming unprofitable now moving into sort of Where some territories only sort of cover is this sort of large you can see this Bunch of longs here They stop it out early if they just decide to say, okay, you know what this they just this is all we wanted So add more here and just sort of pushing because here's a decent amount of shorts all it did was Contract the move here the higher side. So these guys become these guys and These are new sort of sellers new shorts Which we're still we're in sort of close profits and We've got sort of these shorts added on top of these shorts So I've got a decent amount of shorts So we probably want to clear these guys out and again start of their stops will be up here and And it does probably sort of stands and reason if this wall is there then they expect the stops to be a much higher as well So maybe maximum pain is there So that's where we'd look to target very sort of disappointed with How it's Presented so I mean yes This is labor as your 15 candle that's kind of what I wanted for Bitcoin but We sort of mark yesterday and see Yes, they did. There's your 830 here and You Just to put this eight o'clock you can see that sort of same thing but a lot more clearer I mean this opened up much lower gave you a much safer trade Then they gave it started building up and a lot a lot a lot more reactive the same thing I'm getting that reactive So There's your 830 currently and then we'll look out tomorrow for 10 o'clock. It's our next level in terms of sentiment-wise and shifting shifting sort of short-term trend So it's very far out. So from here to here. We can pretty much have fun basically and The idea Okay, don't concern is five of this Is that we have options at so here exactly numbers are so Four or six or eleven Draw those out No, no one in the larger one. We want to be above 406 I really don't want to go below here again Insulting injury will be being above here. So doesn't look down moves don't really look to Down moves don't look Too aggressive Mailed out because we want to be close enough to Insularly sort of recover above this because this is again and look where we sort of coming in when we're sort of pushing below we don't want this guy to ever be too far into profit and He's expiring tomorrow So now we're sort of building our cake. We can just hang around and just hold up as long as above here by Tomorrow this guy Starts crying And it's what 1.1 mil in nuts not sort of One of the quote chump change An adequate amount Essentially to warrant price or shifting fire. That is how you use options to advantage Even if you don't trade them We've got these walls coming in now so now it's looking a lot more Advertising to try and at least target a bit higher These short changes coming in now you can just see these pressure walls that come in then instantly drop Just to change sort of our goals and our behavior still pretty mellow still pretty sort of tamed live data feed live data feed just closing out here so Our I mean it's got a large little spike People went long on the break here. They saw the breakout got punished now. They come back and say, why did you sell? You should bought your food and again need to hit them again. They're just pinning them up. So Relative we're looking to begin just above this area here above 10 so even if we sort of drop looking just for Worst case Coming around here or here sort of meet with these guys just smaller Tracement push it back up games try drive for that higher little price for the Friday go Friday open rather a bit kind of exact similar thing So we've got all this sort of volume up here all this long bearishness Using both this and book with that the longer term is just saying okay Here's where all your sort of breakout traders are here's where all your longs are every time we've got them instantly punish them Insistently sort of punish them. We sort of punish them each time But we wanted to be above this line as as for a while now, so we want to be above so Again, if we sort of push down, let's see it hold just above here probably take out these stops case If we can take out these stops, then maybe we can then push up higher again trying try to close I Mean well being pushed up by the spider Closing this a lot higher and potentially try grab moss alongs added to this area here Because again, even though we punish them there'll probably be a few stragglers They are you know what a hold off was no stop or whatever They ultimately end up doing they sort of believe in the potential breakout and they get justified each time It comes back to touch the price, but they never break even because they want to be in profit because they They sort of They spent time in the trade so now they invested in the trade and now they want to see profit And they won't close it for anything less than It's fixed profit that they have in their mindset arbitrarily So within the day sort of you're looking for potentially Swiping out. I mean we've got a limit in terms of how high we want to go Numbers and bookmark our bids and ask me add and remove. Yeah, it's a add-on. I don't think it's available for everybody liquidity marker there you go You can change Threshold and stuff like that. It's nice little add on you can see load this guy larger trades this Plus one K got removed And in favor of this and now we don't look this high now and As you can see we're slowly gradually going high high and higher Which will lead into that downwards move that we're looking for For the day because it just means that we're getting longs push the market down and now those guys who are Sitting pretty in Asia not really took it looking too hot now now the sort of Questioning life decisions saying oh wait, I should have taken profit or whatnot. What is it is London? Now the sort of sitting here and just say ah, I should have taken profit. It's a bit late now Let me just see what happens sort of scenarios We'll grab a few stops here Again depending on if they're on break here or not and you can see a lot of these a lot of this wall got removed I mean 20 of it still remains so maybe see if they try and iceberg the rest Or they just look that higher Still doing this a bit early for my liking. I mean we still have what an hour left of Time we can sort of Have fun sake of having fun There's still building up this range still building up strange, but ultimately this is probably what you want Say price will come down again Ideally you want to hold somewhere here to match with spider is There's don't want to go too low below source where spiders that just because when we start to go too low That that option that's expiring tomorrow starts to be That's you laugh at the bank. So we don't want him to be laughing at the bank Instead we want to sort of make him feel some sort of hurt. So again, we want to Maximum just try and grab stops That's all the clothes that came in here. So maybe sort of Come around here Rory T. I said I'd say Then from there we want to see it price try climb higher at least sort of Factors this off or even be above the large a higher sort of Option that's expiring tomorrow And we didn't really have too much in the way of Calls so it's got puts puts few calls but not too much or 10 And negligible You gain the money made from going higher make is a lot more than the money made to go lower. So It makes more sense to go higher than it is to go low again all these guys So we're already right here. We want to be We want to be higher than this area at all times tomorrow At least on the open again, it's just on the open So that just sort of spells maybe we get that no peak formation let them expire Let them come into the market and Let's continue back into the direction and they were essentially right, but they were wrong in terms of Executing and then you just slowly start to think to yourself are this whole game's rigged and like yep is Like ah, they get to see they get to see where I'm at Ah, but those of you are wondering if trading is a fair game In fact, it's not It's a highly stacked, you know pretty mellow other than that. I mean we can go with some all kinds follow We wait for prices or Uh dictate other things. I mean All coins will give us a good indication if the market is sort of acting Uh together or acting independent. Uh theorem had very much fun day yesterday Uh, you can see sort of I mean morning rather Uh, you can see that I I mean much higher than everything else. So A nice little rise to old points But now it sort of puts itself in risk that okay everyone's sort of funneling their money into theorems like oh, yeah Oh, yeah, baby but again All of them are following at this high here. So We're expecting price to try and Take those guys out if we're coming out retests. We're just looking to probably hold around here Uh, which comes in at what 1925 Around 1925 then if we try to retest it for a high open tomorrow, maybe we move to 2060 but Generally, uh theorem looking a lot healthier now. Uh, that was sort of above this monthly Uh, I mean finally sort of stayed Couple of days above this. I mean since sort of March and it's been above this for a really long time Uh, oh, it's nice. I mean just struggle with it for a long long time Uh prior So it being above this is very nice Just means But we're still still have ranges If these guys are still in and doubt they are And I'll get here those breakouts here. So you have liquidity available to you. Uh, if you hit higher But generally it's it's higher higher as it is. So if you look at something like atoms, see Now that's trending. I mean again, it is also going higher. So the sort of bitcoin, uh Indifference Uh, sort of letting all coins rise a lot higher. Uh, so Nice, but again, we're going higher before market opens before anyone's really awake in the US markets If people like to trade so it just sort of spells that You see price going higher People wake up. They're like, oh Panic panic. I'm missing a trade. They buy and break out or whatever they buy they buy it Uh, then once they buy it they get they get buyers remorse as price goes down Uh, but generally it's just all of this is looking like big fat up hunt Breakout cat capture before we sort of come back down and if you want to go long you want to go long Our retest so all of these coins and all of these uh assets are looking Looks like they do have sort of retest areas pretty close to where they uh Where they are currently which is a good sign for us because that means That spy is more likely to have a retest area as well which sort of justifies our sort of scenario where People are buying this breakout here get punished. We come in Have a retest somewhere along these lines here Uh, just as long as above this area here for tomorrow Uh tomorrow's open we can create that little information for ourselves and it ultimately works up as a double top Uh, so it's a nice little double top as well so that's scenario would most likely be working towards Things or flash or some of these see if there anything that's different again same thing here and uh, yes similar thing here Got these points here, but it's just the magnitude that which they move slightly different but Just the same thing just got a lot of people long very high Which is always a good sign of things to come Um, this one's a lot A lot more neckluster. You've got a lot of a lot more flat Flat trend here, but generally more of the same thing. You've got a lower level to come back into Uh and other than that And you see just slight variations, but generally Similar similar beats again, this looks like a mirror copy of what we saw on the other ones just Breaking a bit higher has a lower level for it to trade to Other than that nothing too Again, you can see all these uh all these sort of trapped liquidity here This is where we'd want to be highly above ideally above Uh come tomorrow Pass that pass the expires. We don't really care what that If we're above these or not, I mean because these guys probably look dated for the most part Uh being well and honest, but I mean you got that nice little retest here Generally speaking A build up of longs warrants probably a a sort of retracement at least on the short short time frame Considering because yeah, we're going to open somewhere higher here. So we're going to open with a decent little gap and too much of a gap because things are in the way, but Generally sort of There's our close we're already at what five eight this five nine seven So we're already uh five nine seven. So we're already around here So we're going to sort of open this bar somewhere here already from looks at things out of the storage line uh nice also It's trading inside opens inside. So we'll see what it does Generally, uh off a red bar. I mean even if we end this day green, which ideologically ideologically that we do but If we do Not really to uh rise some skin just looking to be above this Uh for the most part. So if we come in we open here if people decide to go long for the day You see it being green. They're like, oh, okay. Let's continue back up. They'll play stops around here. So that's probably where you look I'll get for a potential buy-in Then you sort of look at that as a short term trade Other than that, there's not really too much to say. I mean I've run around things Whilst now No, there we go. You got news see what the news is saying The US numbers price signed 0.5% in march to mark the biggest decline in Almost three years potentially a sign of further easing Interflation in the month ahead Economics poll by the wall street journalists forecasted no change in the producer price index And the fine last month was the biggest since the start of the pandemic early 2020 All sailors costs often Herald future inflation trends the increase in wholesale price over the past 12 12 months also slowed again to 2.7% from 4.9% in the prior month. That's the lowest reading since jan 21 A separate measure of wholesale prices that strips fossil food and energy costs As well as trade margins rose a sank 0.1% last month the government said that this was below the wall street forecast Increasing the so-called core prices over the past Year is 2.3.6% from 4.5 API reports captured that companies pay for suppliers such fuel, metal packing and So forth these costs are often passed on to the consumer at the retail level and And the idea of whether inflation is rising or falling How quickly inflation slows will determine whether the federal reverse halts the interest rates increases soon Central bank has jacked up Key short-term rates since last year to slow economy The economy and tempo inflation They're expecting what one more rise and Just looking at how things are fading. I mean everyone's sort of Very bearish come partial expires for I might need to ask two months by There we go And you can see that Look at that. Just like you got you got millions here got millions out on the water as long as we go higher come that So a lot of people are expecting that A couple may and usually mays usually I don't know why they expect it because mays usually what season they're really good February or fall I mean, yeah, you got some ad points, but Usually really good Junim Juni is really good. So it could it could start trending against against them, which is Probably like the most like hood As maybe we get one more hike and after the hike we start. Okay, cool. We start easing into uh greener pastures Because what everyone's calling for a week a dollar And what's really important? What does jame jim crane must say? Because whatever he says is uh, is what we do the opposite of Buying now. This is a really good sign of central downwards On the come today. We're just building up longs, which is always nice Is what we're looking for and again, we're just buying time, which is another nice sign So we're not going too fast Too high too fast. So slowly making our way slowly, but surely Making our way but keeping sort of everything Keeping everything sort of very controlled. So that's very nice Now they come slightly off But very sort of controlled and whether keeping the volume which is again nice sign that we may sort of pop ourselves down Give us a nice little double bottom But that's pretty much all I got for this day Uh, thank you guys so much for coming Uh, because I mean I could carry on but I don't think anything will change My out view on today looks pretty decently set in stone More interested to see what uh What it does for tomorrow, uh, because again all those expires so hopefully We get a nice little retest Come today then we may lead We may get that nice higher burst tomorrow and from there You can go down hopefully And start our descent hopefully start us small little descent to catch Catch lower price again all this inefficiency. It just needs to sort of Start correcting gains again. We're looking for 24 500 As a sort of nice happiness Uh other than that and nothing more So I will catch you guys tomorrow. Hopefully you guys have a lovely day. Make sure to trade safe and peace