 So what have we looked at so far? We've tried to understand the two basic parts or two main divisions in an argument and that's the premises and the conclusion. We've taken a look at terms. Terms are not what is true or false. Terms are either defined or undefined and that happens by either following the rules or not. I mean he looked at several different kinds of definitions. We've looked at propositions, atomic propositions and propositions are what is true or false. Propositions are what is true or false. Whether an atomic proposition is true or false, it's going to be determined by how the terms relate to each other within the atomic proposition. Atomic propositions are composed of terms. We didn't dive too much into that but we're taking kind of a pretty basic approach. I have a term tree and I have a term tall and therefore the tree is tall. That proposition is true. The tree is fluffy. That proposition is false. So that's our real naive version of the truth or error of atomic propositions. We've also looked at truth relations and these are the relationships, the truth relationships between atomic propositions. How the truth or error of one proposition can affect another. These truth relationships in turn are expressed by logical connectives and complex propositions. Complex propositions express these truth relationships between propositions. In the parts of a complex proposition can either be atomic propositions or even other complex propositions. So complex proposition gets even bigger with more complex propositions composing it. Last time we looked at the connectives that we used to understand or to express these truth relations and these propositions. With this chapter we're going to learn how to construct truth tables. Now truth tables will give us every possible truth value of complex propositions and it does this by first starting with every possible combination of truth values for the atomic propositions. This is going to be very important later on in the next chapter when we start evaluating arguments. We start looking at every possible truth value of the premises compared to every possible truth value of the conclusion. Now before you get too intimidated by this phrase every possible truth, it's really not that complicated. We're going to have a set of rules that's going to give us a very fine set of finite steps to construct our truth tables and while the truth table can get bigger each step is not insurmountable. We just need to follow the steps. Now like I said these truth tables are going to give us every possible combination of truth values and every possible truth value for the complex proposition. So given the truth relationship is expressed between the component propositions don't underestimate the value of this tool. We can literally begin to understand at least one way to understand at least one kind of infinite using just these very simple steps just these very simple steps and all in all it's going to tell us it's going to give us a series of steps which will in turn help us to evaluate every possible deductive argument every possible deductive argument that's huge. Before we dive into the truth tables very much I thought it might be helpful to go over a few practice exercises involving which you know kind of proposition we're dealing with just looking at just using the parentheses and the logical connectives. So here's an example of an exercise we're going to have here looking for what kind of complex proposition is this. So notice we've got parentheses and we've got logical connectives. Well the way to do this is to look for what's you know outside the parentheses right so inside the parentheses that's a conditional but the overall proposition is not a conditional when you look outside the parentheses the overall proposition overall complex proposition we have here is a conjunction. Now it's a conjunction where one of the conjuncts is a conditional but overall this is a conjunction this is a conjunction. So looking at this one uh notice we've got lots of connectives in here we got conditionals and negations and we've got parentheses so take a look at what's outside the parentheses all right look at what's outside the parentheses well that that's a conditional there now the consequent of this conditional is also a conditional and it's a conditional negations that's in the consequent and the antecedent of this conditional there's a lot to say here isn't it the antecedent of this conditional is also a conditional just pq but overall all right this is this is a conditional and we find that by looking for what's outside the parentheses let's take a look at this one now we have a conditional and negations well again i'm telling you look for what's outside the parentheses outside the parentheses and we have a conditional of negations within the parentheses but outside we just have a negation now this is an instance i don't think you've seen it before but this is an instance where the entire complex proposition is just negated and the entire complex proposition is just negated so this is a negation and this one's probably wow look at that got a lot of symbols happening got parentheses and there's junction negations but again look for what's outside the parentheses and what's outside the parentheses it's also a negation this is a negation all right so uh just keep that in mind for when i'm talking about uh complex propositions here i'm not in the truth conditions for these complex propositions well if we're going to understand the infinite we got to start small and we start small with atomic propositions so if you remember atomic propositions the truth value top atomic proposition is just determined by the uh terms that are used to you know combined into the atomic proposition so the tree is tall it's true the tree is fluffy is not true complex propositions are different complex propositions are not determined sorry the truth value complex propositions is not determined solely by the meanings of the terms rather the largest chunk of this work is done with the truth values of the atomic propositions okay so uh when we're constructing our truth tables we have to have enough rows for all the possible combinations of the truth values of the component atomic propositions so this sounds really complicated it's not there's a very simple formula which determines the number of rows for our truth table yeah the top row is always going to contain the values i'm sorry the uh the letters representing the atomic propositions and it's going to have the formula for the propositions you know for the and then later on when we get to the arguments it'll have the uh formula for the uh premises and the conclusion but after that's just the number of rows for every possible combination of atomic propositions and that is determined by the number of variables right raised uh sorry then the number two raised to the power of the number of atomic propositions all right so uh this is rule five right for our truth tables uh the number of rows rule five total in this whole logic course but it applies specifically to propositions the number of rows in a truth table is equal to two times the power of the number of variables or the number of letters used in that uh particular argument so uh if we have an argument with uh p and q that's two that's two variables so that's two to the power of two which you know you don't have to have a calculator after this one that's uh simply four right four uh if we have three variables p q and r that is two that you know so that's three variables so that's two to the power of three which is eight rows if we have uh p q r and s that's four variables so now that's two to the power of four uh which is now 16 so like i said it's not complicated or it's not complicated and uh um i've provided a little chart for you a little table in in the text which gives you the number of you know variables and consequently the number of rows uh in the uh in the uh uh that's available and uh you know it adds up pretty quick now i really seriously doubt we're gonna have i think one of them is over a thousand rows are probably not gonna have a truth table like that uh but it you know it's there nevertheless um now you know when you're doing the homework so you're constructing these truth tables for your for your own work i strongly suggest using a spreadsheet and so something like numbers or excel or google docs google docs has a spreadsheet you know you can go to google docs and start constructing spreadsheets there and then just make your you know make a a table right uh with p and q and have your four rows and then make another one with p q and r have your eight rows and p q r s have your 16 rows and p q r s t keep going and going to go um at least up to you know four or five variables or something like that so you just so you have enough uh that are there and then when it comes time to construct a new truth table like you copy and paste that um but you know you could do it however you like just as long as you make sure you follow uh this fifth rule this fifth rule so it's rule five the number of rows in the truth table is equal to two times the number of variables used in the book okay so the number of rows is one thing and it's not hard to calculate the number of rows well the next step is to assign every possible combination of truth values for all of the atomic propositions now don't don't worry you don't actually have to get creative here and you know kind of make a checklist no no it's actually pretty straightforward um however explaining it verbally is a lot more difficult than say just showing it to you so uh take a look at this here's how we assign every possible combination of truth assignments for the variables and and again don't freak out this is not nearly as complicated as it first seems so uh the first thing you do right following rule five we've got two variables so we'll try this example of p and q we've got our two variables that means we've got our four rows well um we place you know p and q we have our we have our four rows we place p and q and the columns the next step is we take half of the rows of p and assign them as true the top half and then the bottom half receive the value f for false so that you know there we go there's our possible combinations for p q it works kind of the same way except we don't start with the total number of rows per q instead we just start with the rows that are labeled as t for p right the true for p and we take that you know that that's our first you know set that we're dealing with and take the top half of that for q and assign them as t and the bottom half as as false and then we just copy and paste for the rest of the rows below that now that may not have been too clear with just p and q right when we deal with uh when we're dealing with these assignments so let's let's try with you know p q and r so following rule five we have our three variables p q and r that's two raised to the power of three that's eight rows so we have p q and r in our columns and we put at the top of our columns and we have rows one through eight so first things first we take the top half of p and we assign them all as true so that's rows one through four and we take the bottom half of p that's five through five through eight we assign them uh false next step we go move over to q but we're not going to assign the first four rows of q is true right instead we're only going to deal with you know just at the time right deal only with the true rows you know the rows labeled true for p all right the top half of those right so that's rows one through four the top half of those receive true for q and the bottom half receive false for p uh false for q and then we just copy and paste all the way down and you know by the time we the last column should always be true false true false it doesn't matter how many variables you have the last column should always be true false true false so uh you know that's we have our you know true false and then we just copy and paste all the way down if we had another row say we had p q r s right r would not be true false true false instead r would have the we look actually a lot like q does in this one so that's how you uh you you rule six you assign right p you know the top half of p receives true the bottom half receives f and you move to q you deal only with that you know those rows labeled true for p the top half of those receive true for q the bottom half receive f then you copy and paste all the way down you move to r you deal only with those rows labeled true for q the top half of those get t you get true the bottom half get false and you just copy and paste all the way down and so on and so forth for if you have s or t or u or or whatever so if you look in the text you notice that i have given you uh examples of this i think i did it through t uh might have just been through s but i did it through t so you know you could just take those and put them in a spreadsheet for safekeeping so you can see i'm just going to show you here real quick you can see uh what's happening so if i have p q r and s the top half of p that's rows one through eight they receive the value of true the bottom half that's nine through sixteen they receive the value of false so i move on to q and q we're only going to deal with those rows labeled as true for p first okay so you can see that that's rows one through eight well the top half of those that's rows one through four is given true for q and the bottom half that's rows five through eight is given false for q and then we just copy and paste for the rest of that column while the way down and look at r and r we're only going to deal with those rows labeled as true for q first that's rows one through four the top half of those right that's rows one and two receive true for r and then uh the bottom half that's three rows three and four for r receive false you copy paste all the way down and then the last column just gets uh true false and that's true false all the way down okay that's rule six assign every possible combination of truth values for the atomic propositions all right uh so now oh if you notice when we need to go in the text uh you can see that i've provided the truth tables for something like four or five values or forget what off the top of my head uh you know you might just do yourself a favor again use a spreadsheet and uh construct a truth table or you know construct a you know put a table in there and you know leave the rest blank of course but put a table in there or at the very least just something where you can uh you know put all those truth assignments in there following the rules five uh six uh and just copy and paste later on right that'll save yourself a lot of time if you just go ahead and do that and again uh you can see you can see that i've done a lot of that work for you you just kind of have to copy it out of the text keep in mind that the numbers are the rows are numbered and they're numbered for a reason right a lot of the homework will rely upon you finding which row has some particular truth assignment so uh make sure you uh when you construct your truth tables you're going to have to use the numbers as well not just the uh the propositions and the uh truth values but also number the rows um otherwise you're not going to be able to do a fair amount of the homework at least not very well um so we've got rule five produce the number you have a number rows equal to two times raised to the power of the number of variables row rule six assigned every possible combination of truth values and we got that the next step is is rule seven so we've got the combinations of the uh possible truth values well we have to keep that same combination consistent to the rest of the table so with rule seven we know when you have your formula written out on top and you'll see an example of that in just a second when you have your your formula up there uh you know you copy again put this in a spreadsheet to make your life easier copy the column right with this uh truth assignments for p and then everywhere else you see p in that truth table paste p copy q the column for q and everywhere else you see q in that uh truth table paste q right and just keep going on down that way right constructed that way uh and uh that'll save yourself a lot of time a lot of time so we got five rule five number rows equal to two raised to the power of variables rule six assign um you know assigned every possible combination of truth values i'm sure you had to do that and rule seven assigned though it assigned the same uh truth values for each variable as given in that uh as later on in the argument as given in that assignment the next thing we need to look at are the truth conditions for uh all for the various complex propositions uh that we've seen uh and you know express that in the truth table so i think i told you think i mentioned the truth value of a complex proposition which we're evaluating with these truth tables the truth value of a complex proposition is determined by the truth value of its component propositions so for the moment let's just stick with atomic propositions although keeping in mind we could be doing complex propositions and let's start with negations so uh remember that a negation is the claim that some proposition is false right so here's a proposition atomic proposition uh that tree is fluffy that tree is fluffy well obviously that proposition is false so the atomic proposition that atomic proposition is false if i were to construct a negation out of that i would say it is false that that tree is fluffy it is false that that tree is fluffy all right now that proposition that negation that the complex proposition composed of the atomic proposition and and you know the minus sign that negation is true the negation is false that the tree is fluffy is true because the proposition the tree is fluffy is false so this tells us the truth conditions for a negation a negation is true just in case a proposition the proposition is false the negated proposition is false and it is false otherwise so here's another proposition uh the tree is wooden right the tree is wooden that's an atomic proposition and if i were to turn that into a negation it is false that the tree is wooden that negation would be false okay so these are the truth conditions for a negation a negation is true just in case the component proposition is false and negation is false just in case the component proposition is true now this brings us to rule eight in our truth tables we will enclose the truth value of the complex proposition and parentheses this is for the sake of being able to spot it quickly and trust me it's going to make your life easier later on okay let's apply rules five six seven and eight for negations and to look at the truth conditions for negation so just for this illustration we're just going to use the atomic propositions keeping in mind that complex propositions can also be negated but you know one step at a time right so rule five uh one of the few cases we actually only have one variable so the number of rows is two raised to the power of one well that that's just two so we have our variable p we have our two rows one and two and we have our negation listed there in the columns as well notice that the negation symbol gets its own truth it gets its own column for the truth assignment that's because we are looking at the truth value of the negation as opposed to just the truth value of p right so rule six we assign every possible combination for our variables well in this case it's just t and f so we put t and f in our column there for our truth assignment for p then we will follow rule seven we copy and paste those truth assignments over to p all right so then got t and f then in our column for p and now this is where we have our new rule eight enclose the truth values for the premise of the conclusion in parentheses in parentheses and uh so this is what this looks like so if we were to have say a series of propositions or more than one uh each proposition uh each complex proposition would have or each proposition would have its truth assignment enclosed in parentheses so in this case you know here we go so notice you know p where its value is true well negation with with that truth value then is now false and wherever p is false then the truth value for the negation is true following our truth conditions for negations so next we have conjunctions now remember that a conjunction is the claim that both component propositions are both conjuncts are true all right so i'll have a proposition say this you know this let's say one of those trees back there right so here's a here's an on top proposition the tree this organism is a tree uh have another proposition uh this organism is green okay so if i can join those this organism is a tree and this organism is a is green right i have a conjunction and each of those propositions is true or both of them both of those propositions are true this uh this organism is a tree is true and this organism is green is true now since i've conjoined those into a complex proposition and each of the conjuncts is true the conjunction itself is true and if one of the conjuncts is false the conjunction is false so if i say this organism is a tree and i say this organism is blue and i put that together at this organism is a tree and this organism is blue that conjunction is false that conjunction is false and it's certainly false when both conjuncts are false so uh this organism is a mammal and this organism is singing opera right uh okay right both that that conjunction is really really false so the conjunction right conjunctions claim that the component both of the component propositions are true and a conjunction is true just in case both components are true and false when at least one of them is false it's false when at least one of them is false all right this is what a truth table using conjunctions looks like so first we follow rule five we've got our two variables p and q so that's two raised to the power of two that's four rows following rule six we give our every possible combination or truth assignments we've seen that now and done several times following rule seven we place the same truth assignments for each variable within the rest of the truth table so p receives in the in the proposition p and q p receives the same truth assignment as we assigned back with rule six same thing with q okay so the next thing following rule eight we enclose the truth value of the complex proposition and parentheses using our truth conditions for a conjunction and remember conjunction is true just in case both conjuncts are true so that's only a row one right only row one is true for the complex proposition of this conjunction with row two q is false of row three p is false from row four both p and q are false right those conjuncts are false or at least one of those conjuncts is false so the entire complex proposition of this conjunction is false so we've got negations and conjunctions those are the easy ones remember the rest of the complex propositions express a truth relation so uh next we have disjunctions and remember that a disjunction uses an either or connective and expresses subcontrariety right and when what that means is uh you know with subcontrariety if one is false then the other is true right or at least one so if one is false then the other's true well that means that at least one is true and by the way it's it's possible that both are true so let's uh let's start off uh with something like this okay so either this here either this is uh either this organism is a tree or this organism is a bush right that's that disjunction uh it is true right uh the component propositions either the component propositions are this organism is a tree the other one is this organism is a vine sorry is a bush well at least one of those is true if it's one's false the other's true so a disjunction is true just the case at least one disjunct is true at least one disjunct is true now it's possible that both can be true right uh either this organism is a tree or this organism is tall right oh okay right both of those component propositions are true and that's fine right because the condition is still mad at least one is true uh a disjunction is false only when or just in case the component disjuncts are both false both false so either this organism is a bush or this organism is a vine right well those are both false so that disjunction uh is false that disjunction is false okay so a you know we had a negation remember negation is true just in case the component proposition is false and false otherwise you have a conjunction a conjunction is true just in case both conjuncts are true and false when at least one conjunct is false and now we have disjunction disjunction is true just in case at least one disjunct is true and false when both disjuncts are false well that just leaves us with conditionals let's apply rules five through eight for our disjunctions so first we have rule five we've got our two variables that's two raised to the power of two four rows we've got our p and q following rule seven we place our truth assignments in our columns for p that's our following rule six we place our truth assignments for p and q in our columns then for rule seven we paste those very copying paste those very same truth assignments to the every instance of the variables within our formula okay now for rule eight following the truth uh truth conditions for disjunction a disjunction is true just in case at least one of the disjuncts is true so looking at row one well both of them are true so at least one of them is true uh so that uh that row it is assigned t and following rule eight it's enclosed in parentheses for rule uh so for row two p is true so it's true for row three q is true so the disjunction is true and it's only when we get to row four we're both of the component disjuncts is false so that's uh when the disjunction is false well conditionals might be the trickiest of the bunch to understand remember that a conditional expresses the truth relation of sufficiency from the antecedent antecedent to the consequent right antecedent to the consequent so uh we use the connective if then uh if we're already got there is if this organism is a tree then this organism is a plant right so that's true right and that's because the truth of that antecedent necessitates the truth of that consequent right this organism is a tree if that's true that means this organism is a plant that follows okay so uh there's only one way that a conditional is false because remember a conditional is supposed to express sufficiency if the first is true then the second is true it's supposed to express sufficiency uh the only way that is false is you know if the antecedent is true and the consequent is false because then that means it's it's not sufficient if the antecedent is true then the consequent is false that's the only way a conditional is false so I have uh this organism is a tree that's true uh this organism is a mammal so here's a proposition if this organism is a tree then this organism is a mammal that's a false conditional and the only way that a conditional could be false I can say this again the only way that conditional could be false is if the antecedent is true and the consequent is false now this has kind of a strange result uh because that means that conditionals with false antecedents always true and first you might balk at this idea like what no that can't be right well we'll think about it right here here's another conditional uh with a true antecedent I'm sorry a false antecedent but a true consequent right if this organism is a vine then this organism is a plant well that that's true right that conditional is true yet the antecedent is false because that organism is a tree here's uh here's another conditional with a false antecedent and a false consequent if this organism is a dog then this organism is a mammal that's also true right that's also true so conditionals with false antecedents are always true are always true and it might seem bizarre but keep in mind the only way that a conditional is false is if it's actually not sufficient and the only way giving you know not sufficient is if the antecedent is true and the consequent is false okay so the true the condition is for a conditional it's false when the antecedent is true and the consequent is false and true otherwise and then otherwise means that whenever an antecedent whenever you have a false antecedent or a true consequent you have a true conditional and I know that sounds bizarre but you get used to it trust me you get used to it so it's like conditional is false when the antecedent is true and the consequent is false but true otherwise and that just means a false antecedent or a true consequent okay that's the truth of conditions for a conditional now that wraps up uh you know when I call just the real you know basic uh uh connectives right we got negation conjunction disjunction and conditional but we still have two more truth relations right so so we've got subcontrarity and sufficiency but we have to cover yet contrariety and necessity let's take a look at conditionals using rules five six seven and eight with our truth table so rule five pretty familiar with it by now we've got two variables so that's two to the power of two that's four rows and then we've got our p and q at the top column those are uh atomic propositions and then our complex proposition listed after that with the conditional so rule six uh we have every possible combination of truth assignments with the our atomic propositions rule seven we copy and paste those into every instance of the variable within the prop the complex proposition listed afterwards now we have our truth conditions for rule eight so row two notice row two is the only row with uh a true antecedent and a false consequent so it is false uh row one we've got uh a true consequent and a true antecedent and it's also true but you know keep in mind that since row two is the only one that's false the other two rows rows three or four they are also listed as true so one fast way you can do these truth tables is to simply just look for those rows with the false antecedent and you could automatically mark those rows as true and then you look for the uh rows with the true consequent and you can automatically mark that row as true and there's uh uh then you just find those the rest of the rows should be true antecedents and false consequence and that'll be you know kind of a shortcut for these truth tables and if you remember what contrary he said is you know the truth of of one proposition means the other is false uh another way of expressing this is at least one of these propositions is false at least one of the propositions is false and the way that we express this is to use a disjunction of negations now i want to give you a little word of warning here right so to do this we have to uh have a disjunction where each disjunct is a negation all right so we have to uh apply the truth conditions one at a time so in your uh truth table you will first you know you'll have the assignments for the atomic propositions then you'll uh have the you know because you found the rules for negation then you have the rules for the negation but then the truth value of that conjunction since it is a i'm sorry that disjunction since it is a disjunction overall not a negation overall a disjunction overall then the truth value for that disjunction is determined by the negation right not just the atomic proposition so like i said you can have complex propositions within complex propositions it can get complicated pretty fast okay so uh that's our that's contrariety or contrary uh a contrary propositions contrary truth value it's a disjunction of negations for necessity remember necessity was that the error of one means another one is false false makes false and to express this we still use a conditional but the but the antecedent the consequence are both negated so same thing when you have it in your truth table you have your assignments right assignments of the truth values for atomic propositions that's by rule six and then you copy and paste those over to wherever the atomic propositions pop up in the argument that's why by rule seven but then you use the negation on it to you know to determine the truth value of that negation and the truth value of the negations then determines the truth value of the conditional so you follow follow just step by step right and for both cases right following rule eight you enclose the truth value you know for the disjunction you close the truth by the disjunction in parentheses and then you enclose the truth value of the conditional in parentheses uh so the you know the truth value for the disjunction doesn't change just because we got a disjunction of negations the truth value for the conditionals doesn't change just because we got a conditional of negations but it is going to but we have to use the negations in order to express these other two truth relations okay let's take a look at what contrariety and subcontrariety look like i'm sorry contrariety and necessity look like using our rules so we uh have rule five we've got two variables four rows we have rule six we assign every possible combination rule seven where we copy and paste those uh truth assignments over to the variables now here here's where things get a little different and you might think that we need to start in closing the truth values for the negations but overall this proposition is not a negation overall this prop complex proposition is a disjunction it's a disjunction of negations but overall it's still a disjunction so while we still assign the truth value of negations according to the truth value of the proposition that's negated right uh overall we have to follow the truth conditions for a disjunction okay so um you know looking at the looking at looking at this junction here uh row one is the only one where both disjuncts are false so you look at the negations right you look at the negations and each one is false so that's the only row where the entire disjunction is false all right rows two right the negation of q is true row three the negation of p is true and row four both negations are true and so in those rows two three and four the entire disjunction is true so are we enclosed we enclose the truth value the disjunction within the parentheses something similar happens with uh necessity where we have a conditional of negations okay so rule five we have our rows rule six we uh get our truth assignments row seven we copy and paste those truth assignments to every instance of the variable right we have the same truth assignments and now again so the next step we don't go straight for the conditional right because the conditional is not a conditional there's atomic propositions there's a conditional of the negations of the atomic propositions so we have to give our truth values for our negations next and there's still not a close of parentheses because overall the proposition the accomplished proposition is a conditional so we look through um you can do this a couple different ways you can either just take it real fast you so you look at the antecedent and so it's the negation of p is the antecedent and you can just tick those off as true because that antecedent is false then you go to the consequent and you look for those rows where the consequent is true and the negation of q is true only in row four only in row four so row three is the only one where the negation where the antecedent is true and the consequent this falls the antecedent of p is true excuse me the antecedent is negation of p that's true and the consequent is the negation of q and that's false so that's the only one that gets assigned false that's the only row the only way for conditional to be false is if the antecedent is true and the consequent is false and I have this in the text but I thought I just kind of highlighted this just nicely sums up the truth conditions for our for our complex propositions