4.4 Conversion, Obversion, and Contraposition

hi this is an introduction to logic I'm mark Thoris B and this course overviews the basics of categorical proposition or predicate logic today we're taking a look at categorical logic in particular we're looking at the logical operations of conversion aversion and Contra position ok welcome back everyone so today we're going to be taking a look at or logical operators quickly let me do a sort of quick review here just to remind you what we've looked at before well we discovered last time is in categorical logic there are four standard types of propositions there's the a proposition which is basically all S or P right there's the e proposition which is that no SRP there's the AI proposition sub X or P and then finally there's the o proposition which is that some s are not P okay now by this point you should be fairly familiar with this last time we talked about is well the square ball position right which essentially looks like this it would least we looked at the modern square of opposition in which the I the e they and what these are contradictions of each other okay now one of the things I'll remind you there's going to be important here to remember is what each of these things are remember that this thing right here is known as the quantifier and it's a quantum as a quantifier it tells you how many write how many of the class were actually talking about this of course is the subject in the predicate I won't write that since this is P and of course but this last thing here is R this is note against the copula okay and that's going to be important today especially as I diagram these and go through the now because we're going to see here that what the logical operators do is they switch around and mix these either they mix around the subjects and the predicate and so on and so forth so as to come up with statements that have logical equivalence so let me sort of mention what is logical equivalence I'm sorry my handwriting is not that good logical equivalence logical equivalence means that if you have two statements it also lies that by s 1 and s 2 that if s1 is true then s2 will be true conversely if s1 is false s2 will be false so logical equivalence means that they're the same we actually don't use the equal symbol we actually use a double colon here to symbolize logical equivalence so if s1 is a true statement or a true proposition then s2 will be a true proposition if they have logical equivalence so that just means they're equal in terms of their truth value ok so maybe we can write that they a statement is logical clothes is a statement that has equal two statements who have equal truth values necessarily oops equal truth-values okay and what that's going to be an important feature because we're going to see that we do logical operas essentially we're going to do is we're going to do something to this statement to cope with a second statement that's going to have logical equivalence in most cases okay and it's interesting here because logical equivalence in many ways has to do with distribution let me give you an example here let's take a statement let's give you an example here else but you X here for the example what if I said that no Democrats are Republicans okay no Democrats or Republicans how do we symbolize this statement well let's do a quick Venn diagram right we'll put D here for Democrats and R for Republicans whereas some of us an e statement it looks like this you fill in this little football shape here in the middle which me and remember when you fill that means there's nothing there which means that these two things are totally separate now remind ourselves how does a distribution on an e stated work remember we said that a term is distributed if in the proposition the entire class it you learn something about the entire class and an e statement since there's a separation of both classes both terms are distributed both terms get destroyed and one of the interesting properties here is that since both terms are distributed we can switch around Republicans and Democrats right I can write no Republicans are Democrats and that actually means the same thing because what would the the Venn diagram look like for this statement whether it would look like this right if we add D and our blows keep to the same here right if no Republicans or Democrats the Venn diagram is identical okay and so since the Venn diagrams identical will we what we see here is that no Democrats or Republicans and no Republicans or Democrats both of these statements are logically equivalent right so if we call this change the color we call this state at 1 and we call this statement 2 what we see here is that statement 1 is logically equivalent to statement 2 okay so this is just to give you an example here one of the things you see here is that each of these operations actually that we're going to go over today actually involves distribution distribution is really interesting because it's actually pivotal to the whole function of these things okay so let me so let's just erase this make it a little longer for myself okay so that's going to be pretty important here so let's go over these logical operators and there are actually different types of logical operators but in categorical logic at least for this section we were just going to talk about three logical operations three logical operations now to say that it's a logical operation or what is it operation it's just an operation here it just means you can think of it almost as a medical operation it just means we're going to do something we're going to do something to a statement in order to derive a second statement so that's all we mean biological operation and to say that we have three logical operations maybe we're going to have sort of three ways of operating on a specific sort of statement and the flips talk about the first logical operation I guess number one here the first logical operation we're looking at is what's known as conversion and conversions actually quite simple conversion consists of switching predicates and the subject so to convert a statement is to switch the subject and I don't why this computer does this switch the subject and the predicate terms okay so if I was going to draw a diagram of it right remember we always have a quantifier for some reason this computer cannot recognize that right you're quantify your subject your copula and finally you have your predicate that's essentially what every categorical proposition looks like and so to convert a statement all we do is just switch these two guys around okay and in fact if you'll notice in our earlier thing of an e statement this is what checked out here's the thing when you convert a statement it only works for E and I statements alone okay so remember it so that means you can go from no SRP is logically equivalent to no PRS okay and conversely remind yourselves what's a nice statement and I say most that some SRP this is logically equivalent to some p RS okay but here's the trick is the logical equivalency only works for these two and I'm going to write in red here what about the the a and the other statements these are not logically equivalent there's no logical equivalence at least there's no necessary logical equivalence there's no logical equivalence here for the a in those toons it has to do with distribution right and we could draw Venn diagrams I'm not going to do that because if you look in your textbook he does that and I'll show you that here just a second for those of you that don't have the text on hand but why does it matter it goes to distribution remember in an e statement both terms are distributed in an i statement neither term is distributed so as a result if you've switched them it doesn't affect the truth value so these statements are the same so no Republicans or Democrats no Democrats Republicans that's means the same thing some let's say let's say some dogs are nice and some nice things are dogs right those are logically equivalent students but it does not work for the a in those statements and if you convert an Ana no statement this commits a fallacy and the fallacy that we're going to call it here that your book calls it is it's called an illicit conversion and illicit conversion so that's a no-no you don't want to commit an illicit conversion and you're going to see here that one of the exercises you have to do in this in the text here is if you discover an illicit conversion you have to cite it okay so that's the first logical operation let's go take a look at the second logical operation here and that's called ab version of version an up version has two steps okay the first step for nerves of version is to change the quality change the quality I wish I knew why this thing doesn't work a virgin first you have to change the plight and second you have to replace the predicate term with its class complement okay so let me write this down replace the predicate term with the class complement now at this point in the course we haven't actually talked about what a class complement is and so I'm gonna have to sort of give you the background there okay well what exactly is a class complement so let's sort of do this right here what is a class color well let's remind ourselves what is a class and we'll use a Venn diagram to symbolize let's say we're talking about Republicans okay and in this circle here it's everything in this circle every Republican would exist within a circle or something like that okay so what is the class color well this is the class what complements it is what's around it right so all of this let's sort of highlight it all of this space right here this is what complements the class right this is what's known as the class complement but how do we define that exactly so if R is Republican what's the class complement well the class comment would be non Republicans or non are okay because anything around here is not a Republican so if we say non Republican right and that includes a lot of things in fact includes everything in the universe that's not a Republican so the class complement is really simple it's basically whatever it is with the non in front of it so it's the non Republicans okay so the class complement is that that's how you understand the class compliment so let's give an example here let's take let's take an a statement which looks like this let's say that all dogs are animals okay so how does this work remember let's go back here first you have to change the quality now remember this right here is the quantifier what is the quality the quality is either going to be affirmative or negative so that's the thing to remember there you're always put it's either affirmative or it's negative okay so you're not changing how many you're changing what since what's the what you're trying to say so if you're changing the quality of an a state all dogs are animals you're going to change it to no dogs are and remember but you have to so you have to change this because this is where the quality is contained but you have to change animals here because this is the predicate term so it's a no dogs are non animals now that's sort of an odd way of talking because we don't usually talk about not animals but strictly speaking this is true right because that means that no dogs are not animals will that make sense right what you think about like that okay now here's the thing is that a version works for all statements okay so this works for the a the e the eye and the odo statement okay so it works for all the statements so that means that there's no fallacy that we would call the Aleut there's no illicit aversion because it could work for all of them now we did list let's do something let's do a negative particular because this is a universal affirmative statement let's do another example here what if I said that some cats are not brown some cats are not brown animals let's just say that that's our statement okay some cats are not brought animals so how would I avert this statement now that was in a statement this is a no statement so how would I do this okay well remember first time to change the quality not the quantity right notice that before I chained in the this statement up here I had changed the quantifier in order to change the quality but here I'm not going to do that remember this goes back and if you need a reminder refresher take a look at one of our earlier videos where we actually discussed why there's no qualifier but there is a quality at each proposition so remember the quality is either affirmative or negative so that means we got to get rid of this nut right so we don't to change how many so we leave some so that means we want to get rid of this thing okay and the predicate term is this brown animals okay so it's gonna so if we're gonna avert this statement it's going to look like this change it some cats are non brown animals and you can see here it almost it almost seems like we just chained we just moved the negation into the term itself and that's actually exactly what we've done is we've taken the negative out of the statement and really sort of negated the term by using the class compliment some cats are non brown animals and this statement is logically equivalent with this statement and if we did Venn diagrams here and I'll show you this in a minute we would see that the Venn diagrams are identical okay so that's the second operation that's a version okay that's a version so let's take let's move down here to the third logical operator that we're talking about today and the third logical operator is what's known as contra position or I don't know I think I was using a member of sorry three contra position you contre position and there's two steps to do it a culture position right the first thing to do is to switch the subject to the predicate so first thing you do is something you do sort of a conversion so you're going to switch the subject and the predicate I'm just going to write it like that first thing you do is you switch the subject to the predicate and the second thing that you do is you replace both subject and predicate with their class or term compliment okay replace both subject and predicate with their respective with their with their respective class compliments okay okay now here's the thing this works right notice we said that the conversion works with the e and the I and we say that a version works with the all of them a that I though the aversion only works with the and the L statements it doesn't work with the e in the I okay the e in the I if you if you Contra pose an e statement or an i statement you will not end up with the statement fit is logically equivalent right so that means that this will create a fallacy which we're going to call the no surprise here the illicit contra position Microsoft Word is giving me trouble today sorry about this guys okay the illicit culture position okay so let's do let's do an example here of a contra position we already switch the subject right and you replace each of those with their term or class compliments so here's our example okay let's take a hos statement well let's take it yeah let's just take in a statement here which is something like all citizens our voters okay all citizens or voters again so how do we do a culture position here well the first thing we got to do is we gotta switch the subject to the predicate terms and then we're going to replace them with the term complements suppose like this goes right here replace with complement okay so it's going to look like this right it's going to be all non voters are non-citizens I'm sorry guys this is really annoying me all voters are non-citizens okay now this is logically cool too and actually if you think about if I say all citizens are voters and then I said well all the non voters are non-citizens that actually makes sense and if we did a Venn diagram it would work okay so real quick let me show you here what the Venn diagrams actually do look like for each of these logical operations and here I've just we're just taking a look here at the textbook okay so you can see here here's the here's an for if we look at the conversion of an a statement you can see here right that these Venn diagrams don't look the same thus they're not logically equivalent see how these Venn diagrams do the same they're logically equivalent same they look the same logic lumen but the X's on two different sides they're not logically equivalent okay so there you can see the conversion if I did a Venn diagram for for this notice that each of the Venn diagrams looks identical for each of the statements dust for a version logical logical equivalence works for any of the statements okay then let's scan down here to contra position take a look at the vent Grimm's notice for the a statement they look the same so that means you can do a contra position both take a look here if no A or B if I do a corner positioner to no nod be or no nay what about trying to vit if I was trying to do a Venn diagram I'd actually have to fill out all this stuff those don't look the same so they cannot be logically equivalent right same thing these look different they're not this logically equivalent down here they do look the same so they are logically equipment is true this statement will also be true if this statement is true because they're not logical hood the logical value of this the chunks are the truth value of a contra posed eye statement is unknown it's unknown the logical values unknown so that means it's a fallacy and licit contra position okay and that in essence is what the three logical operators were looking at our contra position aversion and of course conversion now one of the other thing I should mention here is that one of the things you what's this one of the things that your book is going to do is it's going to be you can use logical operations to test immediate inferences what is it immediate inference and immediate inference is basically and it's not really an argument that's sort of like an aura it's if you have a premise and then what can you conclude immediately from that so there's only one premise of one conclusion right so for instance using these logical operations right well actually here let me go back here let's do an example here let's say I want to avert an i statement right now remember you can avert I statements the because a version works for all of them so let's say if I said it's some people or mean are mean individuals or something like that I'll put our mean I'll put I here for mean individuals some people are meaning individuals if I wanted to avert that how would I do it now remember a version was right to change the quality so I have to put this into a negative and they have to replace the predicate term so it'd be that it would change to some people are not non mean individuals now you can notice here that I obviously don't talk like that but this is what a diversion would look like here okay so this you can but you can see here since these are logically equivalents if you made an immediate inference here if someone actually said in ordinary argument which would be sort of a lot but imagine if someone said to you some people are mean individuals therefore some people are not non mean individuals right if someone said that to you you could test the immediate inference and you could realize that all they've done is a further statement and since the truth value of this one after it's averted is logically equivalent that that would be a valid and media inference whereas for instance if I try to try to convert an O statement right if I don't know statement and then I converted it to something that and I tested me the inference I would say no that's not logically equivalent and in fact that actually commits the illicit conversion fallacy okay one of the things you'll find if you have the textbook that is quite helpful is there's this sort of diagram here right where right here's the here's the e the I statement here's what their converse like and these have the same truth dive they're logically equivalent whereas these have undetermined truth lies and therefore commits the fallacy of the illicit conversion and it's a a version they're all the same but here's what they would look like and then for contra position these have data on the same truth five but these are undetermined the E and I okay and so what your homework is going to ask you to do if you're doing the homework here is they're going to give you statements and then they're going to say do a conversion what was the new statement look like what's its truth value okay and as you go through this you even get some that say here's what's david here's another what operation was done and what's the new truth value okay and so this is so it's actually pretty fun to do and it's not difficult but we're going to see later on that these three operations actually are very helpful to do when you're actually assessing arguments but right now all I want you to get is to get how this works now you can see here that those are what they're going to have you do the Homer is they're actually give you statements and then when you're they're going to ask you here perform a version conversion contribution as indicated so they want you to convert the statement all of these statements let's take this one right some murals by Diego Rivera our works that celebrate the revolutionary spirit but conversions you switch subject to the predicate so if you converted this it would be some mere maybe some works that celebrate the revolutionary spirit our murals by Diego Rivera then you got to figure out whether or not that statement commits a fallacy or not okay so that's what the homework is and that's essentially how you do the logical operations of conversion of version and contra position okay well thank you guys for watching the video I hope this was helpful in our next video we're going to be looking at the traditional square of opposition and you're going to see that things get a little more complicated but actually much more interesting and it we're obviously we're going to also go back to the importance of talking about the existential fallacy okay thank you very much for watching I'll see you guys online

Channel: Mark Thorsby

Views: 35,950

Keywords: Screencast-O-Matic.com, logic, Catgeorical Logic, Conversion, Obversion, Contraposition, Class Compliment, Term Compliment

Id: wGwGEOIAGEI

Length: 29min 56sec (1796 seconds)

Published: Fri Aug 10 2012