 Thought I'd do a little city park for a change of pace my confluence park here in San Antonio Today we're going to talk about rules of implication and Truth tables are great. They're a nice mechanical way of demonstrating validity But truth tables don't necessarily tell you the steps that you take with some of these premises It looks downright mysterious how you get from the premises to the conclusion Uh We'll use the rules of implication To show the steps these steps of inference from these premises to the conclusion and what these rules tell us to tell us is What sort of? Proposition we can infer From other propositions Well, the first rule we're going to deal with is called modus ponens Monus ponensis is about as straightforward as a rule of implication as it gets um It's you know composed of two Propositions or premises one's a conditional And one is the assertion or the second is the assertion of the antecedent So remember a conditional is composed of the antecedent the consequent and the antecedent is sufficient for the consequent Meaning if the antecedent is true the consequent must be true And when we assert the antecedent we can infer the consequent Like I said very straightforward So uh to use an example we've used repeatedly If my pet is a dog then my pet is a mammal. Let's say that's our first premise My second premise or second proposition is my pet is a dog We can infer from those two propositions. My pet is a mammal And we use the rule modus ponens to make that inference Now in this course, we're going to be Uh creating formalized arguments, right? We're going to label the premises And we're going to show what steps we take to get Uh from the premises to the conclusion in this case, we only have one but don't think that every As a matter of fact, very few of the arguments we're going to look at have only one step But let's just deal with this one right now, right so In formalizing our arguments just like we had rules for our truth tables We're going to have rules for our uh for these for for these uh for the rules of application Okay, so rule 10 Continuing our list rule 10 Provide the sequent first So in our formalization, we're going to write the sequent. Let's just deal with the same argument So the you know we're not getting lost anywhere, right? And our sequent has two premises or the conditional p greater than symbol q that represents a p than q That's the first premise. We have our comma then the second premise is we have the assertion that they've seen That's just quite simply p So we put if p then you know p than q comma p and we have our two vertical lines And we have q That's our argument. All right That's rule 10 rule 11 It's a number each line each proposition in our form in our argument Okay, that's rule 10. So just keep that in mind. We're always going to have line numbers for each each of our uh For our argument for each step Rule 12 says to label or cite provide a citation For our premises and the citation we'll use will be the letter a Could have used p for premise, but that would probably have been more confusing All right, since we have p all over the place. So we'll use letter a and let's say stands for assumption Okay, so line one We have p greater than symbol q And we're going to cite a right this we did and what a means is we didn't infer it from anything We're starting with this Okay line two we have p and again we cite it with a So we fulfilled rules 11 and 12 we've numbered our lines And uh 12 we provide the citation namely the letter a okay Next step is we're going to infer q from our first two premises And this brings us to rule 13 Rule 13 says that we cite in our citation We got q right In citation first we provide the line numbers from which we made that inference and in this case, it's 1 comma 2 comma 1 comma 2 comma and we'll put a space And then we provide the rule that we're using and in this case, it's modus ponens and we're abbreviate modus ponens with mp okay Like I said, you probably think well, this is a no-brainer and it is It's it's about as straightforward as it gets as far as the rule of implications concerned But that's also why I start first Helps us get us helps get us started. So that's modus ponens. Let's take a look at the rest of the rules All right, the next rule of implication that we have is modus tolens and modus tolens Again uses a conditional Fp, thank you. But instead of asserting the antecedent, we deny the consequence Instead of asserting the antecedent, we deny the consequence. So, uh, for instance, we can go back to My pet is a dog and my pet is a mammal Well, let's try something sound this time, right? If my pet is an iguana then my pet is a reptile, right? Okay, so that's still a true condition If my pet is an iguana then my pet is a reptile Well, it you know, it's false that my pet is a reptile It's false. My pet is a reptile. So it's false that my pet is an iguana So as before we're going to follow our rules For a formalization We're going to have line one. Uh, it's the additional fp than q Line two We're going to have the denial of the consequence And remember use that little minus sign on our keyboard for the denial of the consequence for for, you know negation excuse me negation And we label we cite both with a Right. Oh, I forgot about the sequencer. I got to make sure you put the sequence in there You know p than q comma negation of q two vertical lines negation of p Thus line one and two with our citation. We're going to use a is our citation Line three Is where we make our inference of negation of p We have our inference at line three, but we're also going to provide our site or our citation, right? We got we made this inference from line one and two We have comma, you know one comma two comma no space right between the numbers in the comma one comma two comma Then there's a space right space between the last comma and the citation rather than the other rule use and we'll use m t to For abbreviation for modus tolens So that's modus ponens and modus tolens Now I want to offer a little word of caution here. We're talking about an invalid inference Back with modus ponens. We assert the antecedent to infer the consequence and that's just fine but You know, don't think that you can assert the consequent and then infer the antecedent, right? That's an invalid inference Uh, so for instance, let's try this right We'll use the conditional we just talked about if my pet is an iguana. Then my pet is a reptile Okay, or let's try this. Let's try this another true proposition if my pet is a cat And my pet's a cat We talked I've talked about my dog before you see my dog my pets a cat Uh, then my pet is a mammal Now that's your true condition An animal being a cat necessitates that it's a mammal. All right Uh, well, let's have the uh assertion of the consequence Yeah, uh, my pet is a mammal By the way, we've got two true premises If my pet's a cat then my pet's a mammal and my pet's a mammal We have two premises and from this we infer that a pet is a cat Well, I think my dog will be very upset about that accusation So no, this this is an invalid inference. We've got two true Propositions We've affirmed, you know by trying to affirm the consequence. We we infer the antecedent We infer the antecedent but then we get something false and then my pet is a cat So this is an invalid inference and it's a classic error in logic called affirming the consequent Now affirming the consequent is not valid affirming the consequent is not valid and then by inferring the antecedent, right? Similarly, right that's affirming the consequent and it kind of looks like modus ponens, but it's a mistake, right? It's a mistake Similarly, we've got denying the antecedent So let's take our true conditional if my pet's a dog I'm sorry, we take that conditional if my pet is a cat then my pet's a mammal. All right That's true Second premise my pet is not a cat This is also true, right? This is also true. My pet's not a cat But then from this we infer that my pet is not a mammal Right, that's another error and reason we've got two true premises and a false conclusion Two true premises and a false conclusion. So that's not going to work Okay, well golly. So we got modus ponens. We got modus tolens These are two good rules of inference And we talked about two errors two formal formal errors and reason affirming the consequent and denying the antecedent Well, let's keep going with our rules of implication We're going to cover two more Okay, so far we've looked at two rules of implication and two errors and reason right two formal fallacies Uh, let's take a look at another rule of implication again using conditional But instead of just one conditional and you know either the assertion of the antecedent or the now the consequent Uh, we're going to have a second conditional Now hypothetical soldierism Allows us to infer a further conditional But you know there's a cat. So we got two conditionals and we infer a third The catch is this the consequent of one conditional has to also be the antecedent of another So Let's stick with my dog My favorite topic for this logic course my favorite topic with logic course my dog Uh, if my pet's a dog then my pet's a mammal That's true. And what's also true is if my pet is a mammal Then my pet is uh endothermic, which means warm-blooded It regulates regulates warm-body temperature Well, so we had those two conditionals and uh, let's by the way, let's formalize these right using our rules that we studied before for formalizing premises We have fp then q is our first premise And then if q then r for our second premise And from this we're going to infer if my pet's a dog then my pet is endothermic all right, so uh We have just setting up our sequent right following our rules to start off our formalization. We have our sequent p, you know greater than symbol q comma Then we have q greater than symbol r We have our two vertical lines and from this we infer p greater than symbol r Well, let's uh, uh, put it into our symbolized argument our formal argument We have fp then we have our line one p greater than symbol q And again following our rules give the citation a Then we have q greater than symbol r And that's line two greater q greater than symbol r again. We have our citation a And then lastly we have our inference once again. We have p greater than symbol r And uh, we have our lines right following rule 13 We get our two lines one comma two comma space and for hypothetical syllogism We cite the rule h a abbreviate the rule h s So as three rules have been for its modus potens modus tolins hypothetical syllogism We got one more to go So the last rule we're going to look at Using a conditional at least for this chapter. We're not more later, but at least for this chapter Is called conditional proof Now conditional proof is a little strange Because you're approving something by assuming something else Now that seems odd I you know, I can't just I can't just go around assuming that You know unicorns exist for the sake of what validating Tokens claims about unicorns if you made any I don't really know if you did That that's a bad idea But uh, well we we can still prove something by making an assumption And you know, believe it or not you already do this, right? So if you've ever had a discussion with somebody And what uh, they were wanting to talk about let's just say bigfoot right say they're talking about bigfoot And you don't think bigfoot exists Um, but you say something like this well assume for the sake of argument that bigfoot exists Well, that you're making an assumption in order to prove something else You say assume for the sake of argument that bigfoot exists Then his feet would be substantially or bigfoot's feet would be substantially bigger than any human beings foot Uh, but it would retain the same bone structure since in all likelihood Uh, this bigfoot is a you know, uh, as a has a common ancestor with uh, homo sapiens Okay Suppose that's what well, you know, then what you're doing there is you're creating a conditional if bigfoot exists then what? If bigfoot exists, then bigfoot has a common ancestor with homo sapiens Uh, as since bigfoot has a common ancestor with homo sapiens, uh Bigfoot's foot bone structure on his feet will be a lot like human human beings. Okay So, uh, you know, you're not saying bigfoot exists, but assume for the sake of argument that bigfoot exists Well, this is this is and you're not saying bigfoot exists, right? You're not saying bigfoot exists All you're doing is saying that certain things follow on the assumption that bigfoot exists certain things are implied On the assumption that bigfoot exists. Okay So we uh, do this in formal reasoning. It's called a conditional proof And then you're you're inferring a conditional right using the rules of implication to infer a conditional We've already seen one rule that does this Well hypothetical syllogism now hypothetical syllogism is great for inferring conditionals But sometimes you don't have the necessary conditionals in the argument In order to make the inference using hypothetical syllogism so, uh You know, maybe you have to create your own And you can create your own conditionals Given what you already have in an argument using conditional proof all right So to do this right, let's try this argument. Let's try this sequent uh fp then q Now From this It's probably pretty clear, but you know, let's say it anyway. This is real simplest simplest thing Let's let's try this you know if my dog if my pet is a dog then my pet's a mammal. We've been using that Well from that conditional another one follows Uh, if my pet is not a mammal then my pet is not a dog Now that that should be pretty straightforward. I mean you're looking over modus tollens and like, yeah, I mean I see that I get that But notice, right? We don't have the denial of the consequence In order to therefore conclude that my pet's not a dog And by the way, that would be an invalid inference, right? If we conclude if my pet's a dog and my pet's a mammal my pet's Not a dog will somehow make that inference. We've done something invalid because like I said my pet's a dog. Okay But we can still conclude that conditional And we do this using conditional proof so Here's what the sequence looks like If p then q It's a one premise argument p greater than simple q double vertical line not q Greater than not p If p then q therefore If not q then not p Now you look at this like whoa, hold on a second Um, that's it. We only get one premise. You only get one premise But we could do this using conditional proof so Um, let's start let's start the argument, right? So we got rule 10 We put our sequence at the top of our formalized argument Rule 11. We're numbering our premises rule 12. We have our single proposition our single piece of evidence of p Then q we place it there and we have the citation a That tells us that it's our premise. Well, what do we do next? Well We want to prove this conditional if not q then not p In order to do this, um, we're gonna have to make an assumption Now you might wonder what assumption can I make can I make just any assumption? Let's assume not r Well, you can't do that. It's supposed to be not going to do much with it right Well, you know, we want to prove the conditional if not q then not p so to do that We assume not q so, you know, here's a little clue if you're going to use conditional proof to prove a conditional Or to infer a conditional you start by assuming the antecedent So we have our line two here We have our antecedent, you know, we It's the antecedent that conditionally you want to prove right not q. That's our assumption Now you don't just assume if not q then not p therefore are proven using conditional proof. No, that's not what's happening You have to make an assumption and infer something from it in order to infer the conditional So we have our assumption not q but we don't cite it as a Right, that would be a mistake. That would list it as one of our premises And we're setting up modus tollans, right? That's not what we're doing We start with uh, we have not q but we cite it as a c p Which is going to be short for assume for the sake of conditional proof Okay So we've assumed it for the sake of conditional proof What now will we draw an inference from it? Well, we're drawing inference from it and Uh, what are we going to infer? We're going to infer not p We're going to infer not p And we'll use lines one and two And our rule would be a modus tollans Okay, so we're using modus tollans with ink conditional proof and in all likelihood you're going to have to I don't know what you could assume or what you could prove just by making an assumption You have to use something else along the way. Okay So, uh, we have we started with an assumption not q We made an inference from it not p All right, so what do we do now? Well, that's our conditional, right? That's going to be our condition was started with the assumption not q So we put our conditional in there not q then we have a greater than symbol Not p All right, and we did this Using our assumption at line two So this is the way the citation for conditional proof works the first number Is the number where you made the assumption? The second number Is the number where you drew your inference? Or the line where you drew your inference? And the rule cited will be c p Now I just a little word of warning It's not going to be always going to be the case that The that what you infer is going to happen immediately after there might be some steps in between Uh, and you might use a variety of different rules, but we're you know so far we're kind of limited. We only have a couple at this point, right? Now you might worry That uh, you know, we could just start concluding You know not q since we made that assumption. Well, you can't right? You can't just do Yeah, if you're going to infer anything From that assumption, it's got to be included You know, it's got to be included in you know in that whole line of reasoning, right? And the last bit of that line of reasoning has always always always got to be the consequent And your inference Your conclusion that it's got to be the uh that conditional where the assumption is the antecedent and that last bit of inference is the consequent Okay, so we've got four Rules of you've got four rules of implication modus ponens modus tollens hypothetical syllogism conditional proof Now you're going to want to use these correctly in order to make valid inferences, right? You have to make valid inferences Um, and we'll go we'll go over some problems in class