YOU NEED MATHEMATICAL LOGIC!

good evening everyone I wanted to make this video because I mentioned in my previous video something that was bothering me for a very long time and that is the the lack of proofs that so many students today are you're just not exposed to having a good mathematical Foundation to create proofs you're a lot of you are you're you're Con Academy your way through life through math and way maybe not through life but through math and it that bothers me because to truly grasp mathematics and its arguments you have to first conquer logic and logic is

a skill that's now seemingly as rare in our schools in America as cursive handwriting and there are many people out there they fancy themselves as logical thinkers but the the I'm going to say it right now the Casual mental gymnastics of Tik Tock debates so-called they're not even debates but the mental gymnastics of Tik Tok and everyday living fall woefully short of the Precision demanded in mathematics and by the way I am glad that the the the United States Supreme Court upheld the ban on Tik Tok praise God and I hope that uh president-elect Trump

uh although I won't hold my breath because I don't uh I'm very disappointed in American leadership for the past several decades so uh I I would be very surprised if Trump did the right thing or any American president in the past several years and upheld the ban Tik Tok is owned by an enemy a foreign enemy government the Chinese government is a foreign enemy government and no decent American no patriotic American should ever support a uh an enemy nation and I know there's going to be a lot of people in the comments perhaps just like

in other comments there's always some push back there's always some loser who trashes the work of others there's always people who they try to um they they they they have no qualifications at all but they think but you are what we used to call a knit all you have no qualifications you have no training in education you've never even taught a class but you come to this Channel and you go to other channels as well and you try to trash the work of other people but I'm not interested in those kind of people I I

laugh some of the times I read some of the comments and most of the comments are positive by the way they're they're they're good comments but there's always that one or two individual they're angry at life and they they lash out at me uh which is which is hilarious I had a uh someone yesterday they they made a comment it was hilarious they said have you ever actually taught a college course well what do you think I you know why do you think I started this channel in the first place I mean you you some

of these people are just IR they are irredeemable people they they are just beyond the scope of any hope but I'm not worried about them I'm making these videos because I know there's there's there's more people out there who understand what I'm talking about but we have a perhaps if logic were taught as rigorously as scrolling we'd have a generation that is more Adept at proving theorems than proving their point in some in some kind of common section the language of mathematics is basically a collection of declarative sentences you can think of the declarative sentences

as an AC quied well- behaved cousins of the drama fil post you see online if a sentence can be neatly tagged as true or false we call it a statement it's truth or it's falsity that is that's its truth value it's a term that might sound fancy but really we what we're really referring to is just is it real is a statement real or is it nonsense and interestingly for a sentence to qualify as a statement we don't even need to know if it's true or false actually just that with a little effort it could

be one or the other as we're going to discover in this video so it's a level of clarity mathematics Demands a level of clarity that seems to have escaped the logic lessons of most gen Z math classes but here we are and if you're if you're not this is why some of you if you are easily offended if you're the type of person who is easily offended it's because you don't you don't accept truths and Mathematics is about finding the truth and that's why a lot of people fail out of bath because they cannot they

cannot handle axiomatic truths and that's why you get people who come into the common section they have no idea what they're talking about but they lash out and they they they are angry because they know that I'm pointing out something in their own life and it doesn't take me to point it out they already know it intuitively it's just that the difference is that I actually am the person to point it out they're used to not being told that they're wrong they're used to being told uh as I mentioned in my previous video there's a

religion of self-esteem in America that's it's been happening since the 1990s it's the religion of self-esteem and it's the religion of don't hurt anybody's feelings right well guess what I don't I don't mean to sound like I'm using a cliche but the cliche in this case is true facts don't care about your feelings so in this video I'm going to introduce you to logic and I'm going to create a series you know I wasn't going to do it today because I've got a wonderful dinner awaiting me I've got a really good Southern barbecue it's uh

ribs barbecued ribs with homemade vegetables we don't I don't have processed food in this house heaven forbid we don't do processed foods here I want everything natural including my the milk by the way I want everything from I I do everything from scratch and that's why I mentioned in my previous video a lot of you would not be able to survive on that regime you'd not be able to survive under a lot of men have surrendered in America discipline and when you surrender discipline you no longer have a foundation in which to build your character

which to build your identity you know you don't find you you hear a lot of American men which is pathetic you hear them say that they're looking for happiness I'm looking to be happy well you don't find happiness by looking for it you stumble over it on the road to duty and when you create a foundation for yourself when you have a moral Foundation when you've got character that everything else falls into place you know the measure there's an old saying that says the measure of your character is what it's going to take to stop

you how easily do you give up how easily do you stop when you feel threatened about something and I one of the things I've noticed more and more as I I work with a lot of young people and they don't it's I'm not blaming the young people by the way it's not their fault they're just not trained to have strong criteria about things what I used to be the type of person I I would look at some students who were of a certain age and I fell into the Trap of saying oh it's just the

age or their gen Z and falling into the labels but have you looked at the parents lately have you looked at some of the the the young people's parents lately it's just it's appalling it's not the young people's fault is that they have no Foundation at home and at City tutoring we're going to talk about that next week but we some one of our principles is you are not going to be allowed here to use your identity to couch you to to Shield you from accountability there's a lot of students today if you're a teacher

watching this channel you know exactly what I'm talking about there's a lot of students today that they under the excuse of some invented identity they Shield they try to Shields from any accountability oh I'm this and I'm that you created something new and as a result because I consider myself X or Y you have to accept that and you have to bend your will to my needs that's not going to happen here at City tutoring if you are the type of person was always coddled if you are the type of person who was told you're

wonderful uh you have no religious Foundation you have no Christian Foundation you don't have a foundation of Truth and you're not going to do well in this channel you are not going to do well in this channel you're going to lash out you're going to be one of those people the angry people who lash out but you're never going to fix your life because you always be people who lash out at others and try to trash the work of others you will always be the failure that you are you were made to be some people

were made to be a failure predestined uh but you have the opportunity now to do better you always have the opportunity we are we in in America of all places you H you do have you still have even in 2025 you still have the opportunity to do improve your life so you need to stop blaming other people for your own misfortunes you need to stop blaming other people for your own lack of adequacy you are inadequate because of your behavior because of the way uh the values that you've been taught so in this video we're

going to start the logic uh we're going to start talking about logic but this is going to be an entire Series so I decided to do that because we really really need to improve the logical The Logical foundations in this country so the first thing to note is we're going to talk about the types of sentences that exist so I'd like you to consider the first these uh well five sentences actually five sentences to consider the first one says we're going to dissect these with the Precision of a mathematician and the flare of a commentator

who hasn't had their critical thinking like so many people today outsourced to Twitter or X nowadays it's called x to the mobs right we have seen a shift in that as well but anyway I'm not going to dwell on that but let's look at sentence uh a here sentence a says 5 + 5 = 10 and this is really the a quintessential statement of arithmetic truth is it's really unassignable it's a beacon of clarity in a world that increasingly questions even the basics so that is a true statement however sentence B is a falsehood any

serious calculus one student knows that and it's a delightful reminder that when you have sometimes when you have sweeping generalizations they don't always hold true uh and then we have sentence C that's really a Fickle creature it's true for some values of X but it's false for others and if we were feeling generous we'd call it situationally dependent like certain ideologies that flourish only in Echo Chambers right we've seen that these days but with a if if we have a specific context statement C be becomes a proper statement context does wonders for clarity but there

are still people in our society that you prefer slogans over substance uh then we have sentence d now sentence well sentence d and e they go into deeper waters because even if you hav a clue about their truth or falsehood rest assured there're still there're still statements because yes even complex ideas have definitive truth values it's a concept many modern so-called intellectuals I don't call them intellectuals but some Modern intellectuals they might find a ra they might find Radical the idea that yes that we can find truth and for the record by the way if

you look at statement D that could be demonstrably false and for E well its truth value sort of remains a mystery it's much like certain political platforms its appeal lies not in its proof but perhaps in its promise there's a lot of people today they uh they allow themselves to be lied by uh easy lies you know one of the one of the biggest lies we have today is the uh when people promise you your life is going to improve because the minimum wage goes up all the minimum wage increase does is increase inflation everyone

any any serious Economist knows that but anyway uh sentences D and E are are re really rather more difficult because you may or may not know whether they are true or false but it's certain that each sentence must be it has to be one or the other so D and E are both statements and as we said you know you can show statement D to be false and the truth value you know we have not you're not concerned with the truth value of any particular simple statement to be a statement it must be either true

or false it cannot be both again let me say that again to be a statement it must be either true or false and not both but it's really not important it's immaterial uh which condition applies what will be important is how the truth value of a compound statement is determined by the truth values of its simpler parts that's what we're going to be looking at and you know you have in everyday English you have a lot of statements when you have a a typical conversation there's a variety of ways to change or combine statements for

example what what would be a simple statement like it is hot and you can negate that statement it is not hot right most people understand that now it may be questioned whether or not if you say it is hot is a statement because the term hot here can be vague we haven't defined what we mean by hot but if we assume that hot is given a if we give it a precise definition then a particular place uh in a within a particular context then it becomes easier to understand but it's not always as precise as

we would like and what about if we said something like it it is um a compound statement what if we said it is hot and I don't know the waves are high something like that well here you have two parts the first part is it is hot and the second part is the waves are high and these two parts you can be you can also combine them in other ways you could say that it is hot or the waves are high you can also say conditional if it is hot then the waves are hot then

the waves are high you can also say it is hot if and only if the waves are high table if you take any logic class you know what I'm talking about you have p and not P so if p is true then not p is false however if p is false then not p is true and of course uh you know you can let usually if you're taking a logic class right like let's say I said p q and R and we said that P is today's Monday uh let actually let me write it out

because sometimes it's easier if you see it you can visualize it so let p q and R and we can say be given as the following statements we can say p is today is Monday q five is an even number and let's say R the set of integers is countable so they're negate if you negate this you could write them with that symbol that I told you if I write the not p then you would say today is not Monday and if you say uh the not Q then you would say five is not an

even numbers or the set of integers is not countable or the set of integers is uncountable if you prefer so you have you have here the connective end and it's used in in logic in the same way as it is in ordinary language so p and Q are statements and the statements p and Q in that case are called we talked about that in my video on inequalities we we said that that was a conjunction a conjunction of p and Q and how do you label that with logical symbols the first symbol we're going to

learn today is it looks like a carrot it's the carrot symbol actually and it's written like this p and Q so this is true only when both p and Q are true and it is false otherwise it is false otherwise so we also have we talked about in our previous video we also have the connective or and if you have the connective or then that is called a disjunction so in common English in spoken English the word the English word or it can have two meanings if you have a sentence like we you know let's

say you said we are going to New York or Paris so the intended meaning here is New York or Paris but not both and this is known as the well we have an exclusive meaning here of all but what if you have another statement like let's say you said somebody asked you a question do you want to eat uh let's say ribs or let's say soup for lunch so the intendent meaning here would mean the uh it includes the possibility of having both and this inclusive meaning is the only way that the word or is

used in logic so if we use the symbols the disjunction P or Q by P uh and then it's if you say or it's an upside down carrot symbol so you would have the following truth table if you have p and Q then the the disjunction if p is true and Q is true then you have the a true a truth value here but if one of them is false you could still have truth if Q is false you still have truth or vice versa right false and true is still true if they're both false

however then of course you have a false disjunction so the next thing to know is what we call a conditional statement if P then Q is called a conditional the if makes it conditional so the if statement here you have something called the in the implication this is called the antecedent and the then statement the Q part is called the consequent so you have an antecedent and you have a consequent so if you're going to decide on an appropriate truth table for implication you have to consider a a sentence like a typical sentence would be

something like typical sentence used if if if it stops raining by Saturday then I will go to the I don't know the football game I always used to skip those back in school I never cared for football but some people do uh so but what I'm trying to say here is uh if you if one of your friends a lot of you are high school students so you you're probably into sport some of you if one of your friends made a statement like this the question is under what circumstances could you call him a liar

and certainly you know if the rain stops and he doesn't go then your friend did not tell you the truth your friend is a liar and if you have friends who are liars I would reconsider those friendships by the way but what if the if what happens if the rain doesn't stop well he hasn't said what he will do then but so whether he goes or not either is all right in that case right because he hasn't made a statement so it could be argued that other interpretations make equally good sense but mathematicians have agreed

when you have an implication an implication will be called False only when the antecedent is true and the consequent is false so if you are going to denote the implication if P then Q by by saying that P implies Q then you have a new table and you can use the following table to come to that conclusion you have p and Q once again and P implies Q if both of them are true then the of course the implications are true if one of them is false however then the entire thing is false if one

of them and and again but however if the antecedent is true notice what happens though if it's false and true then the whole thing becomes true and of course if they're both false then the entire thing becomes true so if really it's important to recognize when you're when you're dealing with mathematical writing the conditional statement can be disguised in sometimes you have several equivalent forms so there are many Expressions that mean that all mean the same thing like for example if I told you and they don't teach grammar in schools anymore I know for a

fact they don't teach grammar I dare you to tell me in the comment section that your school has a gra a specific grammar class you you'd probably be lying but correct me if I'm wrong I I and if you and if you do go to a school that has a a separate grammar class I want you to name that school so for example if I said if P then Q you could say it as Q provided that P if I say p implies Q I could say Q whenever P there's just different ways of writing

if I say p only if Q I can also write that as p is a sufficient condition for Q if I say Q if P I can also say that Q is a necessary condition for p so we have antecedent and the consequent in a lot of statements and of course it it does have an application with math right if I told you for example if n if n is an integer then 2 * n is an even inte integer and we would have to find out the truth statement of that so we create truth

tables based on that we have antecedent and we have consequent I will leave the practice for you and hopefully in my in my next video we're going to talk about quantifiers quantifiers we're going to go back to the statement with the quadratic equation that I gave you in the beginning of video and we're going to see how we can uh make that a uh how it relates to quantifying s in other words and then we're going to talk about other symbols you you need to start really becoming comfortable with symbols in logic a lot of

you don't even know how to manipulate a logical symbol one of the biggest problems in math is that students are used to uh you're used to Con Academy your way through you Con Academy your way through everything and you think you know it all you don't Con Academy is a joke that's that's not math that's not real math they're just spoon feeding you but when you get to my class I guarantee you KH Academy is is not going to save you if your life depended on it we take math very seriously at City tutoring so

that's why we teach The Logical foundations here if every high school student had a specific logic Class A lot of the problems that we see at the at the uh University at the college level would be avoided so that's why we do this we emphasize this any good school will emphasize logic you will have a separate logic course so this is the first I hope this video was helpful because it's going to be a series so there's a lot more to talk about we have to talk about the The Upside Down symbol that looks like

an A and I will do that next week we need to talk about the element symbol etc etc if this video was helpful I ask you to please continue to subscribe please continue to support this Channel and what we do we are here to work for you we take work very seriously at least I do I take this work very seriously so thank you all thank you for all of you who have left positive comments and to the haters you know you you you need to you need to straighten out your relationship with you know

with others I'm not g I'm not here to be your therapist uh you know and I'm not interested in your in your mental health I'm interested in whether you are right or wrong I'm interested in whether you are you are you can be successful in math I'm here to help you in math I'm not here to be to be your therapist I'm not your therapist I'm not qualified to be a therapist I don't care for therapy I don't care for psychology I care for truth so thank you all who leave the positive comments and I

I appreciate every single one of you we'll be in touch next week and uh let's be vigilant of our freedoms always and let us be as uh let us be always defending our constitutional rights and never ever accept tyranny thank you all