okay this is the first video in the introduction to symbolic or propositional logic Series so the first question you might have is why bother studying this why B studying all these p's and q's and uh here's some reasons first it'll help you better understand how computers work at a deep level it'll lay the foundations for that it'll help you better understand and do math it'll create more logical circuitry in your brain so to speak neural networks it'll help you learn habits of thinking that are just more logical and they become ingrained um it'll help you symbolize arguments and thereby quickly and more accurately identify their validity or invalidity so a lot of arguments you don't even have to think about the content you can just put them in symbols in your mind and quickly see they're valid or invalid and finally I think it's fun and challenging okay so hopefully that'll keep you motivated through all this um the categorical vers modern so we're leaving the categorical Aristotelian traditional logic and moving to the modern and we're going to start with this propositional Logic the difference is that categorical logic dealt with uh actual words and categories like if I wanted to say all dinosaurs are funny creatures I would symbolize that with Aristotle's method as all SRP but now with modern symbolic Logic the letters will represent entire statements so um all dinosaurs are funny would just be represented as D it's a whole proposition that's making a truth claim that's either true or false right so we're dealing with propositions not classes and we can do much more with modern logic so with that in mind let's look at an example here s let's say that represents the proposition that I'm wearing a blue shirt so if s is true then it's true that I'm wearing a blue shirt and if s is false then it's false that I'm wearing a blue shirt right now if not s is true then it's true that it is not the case that I'm wearing a blue shirt right and you want to instead of just saying I'm not wearing a blue shirt it's helpful and logic to say it's not the case that I'm wearing a blue shirt for reasons we'll see later okay so um also when you write statements you don't want to have s represent I'm not wearing a blue shirt rather you want to make it make a positive assertion and then put not s if if you're not wearing a blue shirt we'll see why later don't worry about it right now okay um so the letters represent true or false statements but there are some sentences out there that don't make truth claims like um who are you or close that door and uh we won't be able to translate those sentences into our modern logic so keep that in mind it doesn't translate everything well here's a compound statement because it's making two different truth claims so I'm wearing a blue shirt and I'm wearing clogging shoes right um or I'm wearing a blue shirt and clogging shoes that would be represented as S&P or B and C it doesn't matter what letters you use okay but that's a um compound statement so here's some practice see if you can um just put these into symbolic form using some letter so for example for number one you might write J or just s you know for number two you might write S and P right number three might be um I'm either clogging or singing might be C or S okay I'm going to show you the answer slide in just a minute and there you go notice number four I put dogs like cats and then I put a knot in front of that it's not the case that dogs like cats right okay now instead of using these little words here like uh and and or and not we want to symbolize those too we want to put everything into symbols okay so we introduce these five operators you can see the first operator here right here is called the TIY it's a nice name and it represents negation so instead of saying not or it's not the case that I'm going to use a little TIY and then the dot will represent and also and it's what we call a conjunction it conjoins two propositions the the V will represent the wedge it's a disjunction usually or captures that um by the way the and is sometimes represented with an upside down V in some books okay the Horseshoe represents if then uh sentences and that represents implication we'll go over that later and in some books it'll be an arrow pointing to as a right instead of a horseshoe but we'll use the Horseshoe and then the last one is a triple bar which represents equivalence and that's um if and only if statements and some books that's represented with arrows going both ways okay so um here's those same sentences again uh statements I should say and they're represented now with letters and operators so look at number two John Denver is a great singer and pilot J.P right number three I'm either clogging or singing c i capitalized the V I meant to low our case but C wedge s right dogs don't like cats not D and so on okay all right so let's talk about each one of these a little in a little more detail and do some um the truth tables for them okay and uh if you look at the TIY which represents negation um this is uh very let me make sure I'm going in the right yes okay so the ti represents negation the ti is the only operator that can occur right after another operator so if I say I'm eating green beans or I'm not healthy that would be G I'm eating green beans right G wedge not H H represents I'm healthy not H I'm not healthy okay so it can occur right after the wedge operator it's kind of neat anyway P let's say p represents I'm wearing a blue shirt that's either true or false a statement can be true or false based on that we can figure out it's not the case that P so if it's true that I'm wearing a blue shirt then it's not the case that P it's not the case I'm wearing a blue shirt must be false if it's false that I'm wearing a blue shirt then it then it's not the case um that I'm wearing a blue shirt must be true okay so that's what this truth table means okay um you just reverse the values so see if you can translate these real quick monkeys don't fly and it's not the case that if I turn in my homework I'll acis course and so on and here's the ansers not M notice number two it's not the case that the whole statement if H then a because they're not saying um if I don't turn in my homework on Asis course that would be not H then a with no parentheses whether they're saying it's not the case that if I turn in my homework I'll Ace this course look at the last one number three neither Clemson nor Virginia will win the championship so it's not the case that Clemson will win or Virginia will win okay now for number three you might have represented it like this down here in the bottom it's not the case that Clemson will win it's not the case that Virginia will win and these two are equivalent same thing so you would be correct if you did it that way and later we'll use this logical equivalence and proof and so on all right okay the main operator is very important um to we want to understand this concept because we'll later use the the main operator to determine whether the entire statement is true or false okay um it's the one operator that covers the entire statement so for example my socks are not a main operator metaphorically because they only cover my feet my shirt only covers my uh waist and chest and arms so it's it's not a main operator but if I got you know into a well I guess well let's say I'm inside a tent the tent would be a main operator because it covers All of Me Okay so let's look at the main operators here this will help later if I say SNP the main operator is the only operator and right if I say let's say number six Not A or B the main operator is or because it applies to the whole sentence or most of it A or B the tilty is not the main operator because it only applies to a not to B look at number two if s SMP then Q or R the little and Dot here only applies to S&P not to q and R therefore it's not the main operator the little wedge Q or R only applies between q and R only applies to q and R not to S&P so it's not the main operator here is the main operator the little horseshoe because it connects the hole so once I know the value of the Horseshoe whether true or false then I know the value of this whole statement once I know the value of this and in number one this dot I know the value of this whole state statement okay so on this slide I'll show you the answers and there they are here's the main operators and you can kind of tell the main operator will always be outside of the parentheses if you have parentheses more than one parenthesis and um this one was complex right all these little brackets and stuff but if you think it through I think you'll see that uh the tilty is what applies to the whole statement none of these apply to the whole statement the N because the til here only applies to PR Q not to um mrb so the and is the main operator okay the next one is a conjunction and this is um the conjunction is p and Q so I'm wearing a blue shirt and I'm clogging P represents blue shirt Q is I'm I'm clogging so if both are true I'm wearing a blue shirt I'm clogging then the conjunctive sentence the main operator of p and Q is true but if any one of those is false either P or Q is false then it's false so if I'm not wearing a um if I'm not clogging but I'm wearing a blue shirt then this p and Q is false right so they both have to um be uh true in order for the conjunct the dot to be true and this is something you got to memorize you'll you'll um be using it over and over again and it's helpful to talk to through it like with my example with blue shirts and clogging okay let's do the wedge now the wedge over here the main operator for P or Q um this is true unless both disjuncts are false and the disjunct one disjunct is p and one is Q okay now before I even get started I'm going to explain truth tables in a in the next lesson but notice when you have two letters you're going to have four rows because you're giving all possible combinations it's like flipping uh two coins twice right you're going to have four possible combinations heads heads Heads Tails um Tails heads and tails Tails so um that's what we're capturing here all possible combinations we'll get to that later but P I'm wearing a blue shirt or I'm clogging if they're both true it's true it's what we call an inclusive or um and then um as long as one is true either I'm wearing a blue shirt or I'm clogging then this or the wedge is true but if I'm not wearing a blue shirt and I'm not clogging then the uh wedge of course combining them um is false okay now um sometimes when when you have an or sentence in English like I'm in Austin or Orlando Florida I'm in Austin Texas Orlando Florida you can't represent that with um a or o because you can't be in both right so this first line wouldn't apply I can't be in both right now so we'll learn later that you can still represent that sentence by saying a or o but it's not the case the a and o and we'll get to that later okay the next one is a conditional all right so this is called material implication and when you see if then sentences they'll probably be um expressed as if P then q and the best way to remember this is that it's always true unless you have a false an um a true antecedent and a false consequent and I did a little um video on conditionals that you can check out um later but um so um if it's raining then the roads are wet okay so P represents raining Q the roads are wet both are true it's true but if it's raining and the roads aren't wet then my statement must be false now if it's not raining and the roads are wet for that's true right if the antecedent is false then the uh conditional is going to be true it's kind of counterintuitive so Hurley suggests you use the example of um if I make an A and the final then I'll Ace the course okay now if it's false that you make an A in the final but still true that you a to the course well the statement could the teacher didn't lie to you so it could very well be true or if it's false that you made an A in the final and false that you made an A in the course then again the teacher didn't lie to you so if P then Q is true now again there's some English statements that involve causation with if then P then q that we just need to capture in a different way so you have to be careful but for now just memorize the truth table and we'll get to those more complicated ones later finally there's a by conditional and this is p and then you see three lines and Q okay so this is p if and only if q and when I see the symbol I think of if P horseshoe q and if Q horseshoe P that's basically what it means but the bottom line is that this is true whenever p and Q have the same truth value so I'll jump over a cliff if and only if you do or off a cliff right so if you jump over then I will right and vice versa so all now if you don't jump over then I won't so it's true if they're both false P or Q are both false and it's true if p and Q are both true but if you jump over and I don't then it's false or if I jump over and you don't then it's false so that's a pretty easy one to remember right um okay so in the next video I'll go over well formed formulas and give some more practice on um determining um the truth of compound statements thanks