 Hi, welcome back to fill 320 deductive logic. I'm professor Matthew j. Brown and today we're starting unit for on proofs in SL and we're going to be talking specifically today about the basics of proofs and The rules of what we call direct proof, right? So Let's get into it First let's start off with the question of what is a proof? What does that term mean to you in everyday language? Think about it for a second. Where have you encountered it? You might have encountered it say when someone asks you to provide proof of something that means to provide some evidence, right? Or And that's certainly how we mean it in a court case when we say we're going to prove something beyond a reasonable doubt We're going to provide a certain amount of evidence, right? But what does it mean informal logic? That's what I really want to get down to formally a proof like an argument is just a sequence of sentences, right? The first sentences of the sequence are the premises and these are given at the beginning every sentence later in the Sequence after those premises is derived from earlier sentences by one of the rules of The proof system that we use, right? The final sentence of the sequence should be the conclusion of the argument proof is crucial concept for deductive logic It's what we mean when we say deduction, right? It's a certain kind of process of derivation Proofs are another way of showing the validity of an argument Now we can already do that with truth tables, right? So if we have a simple argument like this if a then be a therefore be, you know We can give the truth table here, right? And we can show everywhere in the truth table that the premises are true The conclusion is true that tells us this argument is valid But there's something pretty intuitive about the argument that the truth table doesn't really help us understand It doesn't help us understand why this argument is valid and others aren't it's kind of a brute-force way of showing it without really explaining it Also, remember, this is a simple truth table, but truth tables get exponentially Larger and more complicated as the number of sentence letters we're dealing with increases if we have two sentence letters We only have four lines in the truth table if our if our argument has three Sentence letters, then we have eight lines if it has four sentence letters We have six lines and if it has only ten sentence letters We're already up to one thousand and twenty four lines, right proofs by contrast can get at the underlying reasoning involved in an argument and help us understand why the argument is valid and Sometimes with less Complication or less less you have to lay out unless you can possibly get wrong In a large truth table in this class. We use an approach to proofs called a natural deduction system, right? There are different kinds of deductive proof systems for Logics like sentential logic or SL But in a natural deduction system Every connective has two rules associated with it. It has an introduction rule that allows us to prove A sentence with that as the main connective So for example the conditional introduction rule allows us to prove a sentence with a conditional as the main Connective as the conclusion of our argument every connective also has an elimination rule It has allows us to prove something given a sentence with it as the main logical operator So every connective has an introduction rule or an elimination rule that helps us derive or Derived from that rule The book introduces all of the rules together But today we're just going to talk about the simplest rules rules for what we call direct proof And we'll learn what that means a little bit later first. Let's look at how our proof notation works We use something called a Fitch proof Notation and the one we use has this sort of guidelines on it, right a sort of vertical line on the left side and a horizontal line Above the horizontal line are the premises of our argument To the left of the vertical line. We number all of the sentences in our proof, right? Everything under the line is a kind of intermediate step and the last line of the proof is our conclusion To the right of every line in our proof With the exception of the premises We write the rules that are allowing us to write down that line So let's start this one simple rule. The only rule that's not connected to any connective is the rule of reiteration, right So here is an example of applying the rule of reiteration. We have on line one. We have a simple atomic sentence a On line two we can reiterate a on another line and our rule is is abbreviated are With one as the line number that We are applying the rule to right Reiteration is a very simple rule almost seems too simple But we will see it's useful under certain conditions when combined with other rules Now this is just an instance of the rule of reiteration when we when we write the rule itself We have to use some meta variables. You'll remember meta variables We used in previous units, right? So I use here the script sort of scripty a To stand not for a sentence letter a but any sentence any arbitrary sentence of SL the M here is a line number could be any line number and You see here that the reiteration rule allows us to write the exact same sentence, whatever it is As long as we write the rule are and the previous line that where that sentence was on M Let's look at some other of our of our rules of Direct proof you see how this plays out, right? So the next one I want to mention is conjunction introduction, right? This is the rule that allows us to derive a conjunction, right? and if on some arbitrary line earlier in the proof we have a Sentence a and On the line in we have a sentence B. We can derive a and B Using the conjunction introduction rule Referring to lines M and N Now M and N don't have to be right next to each other. They don't have to be right above the line Where I am writing down the conjunction. They just be anywhere up in the proof earlier, right? This will make more sense. Perhaps when we look at some examples, but I want to introduce you to the rules first We also have an elimination rule for conjunction that tells us if we have a conjunction a and B on a previous line We can derive either a Or B right both of those are valid applications of the conjunction elimination rule now Again, I want to remind you script a and script B are meta variables that stand for could be very complex sentences All that's all that matters here is that the conjunction the and is the main connective for this sentence, right? We have a disjunction introduction rule, right? That tells us it whenever we have a sentence We can in we can introduce a disjunction Doesn't matter which order We put the a on it could be the first or the second disjunct and B can be any arbitrary Sentence, right? You might think why why can I introduce anything that seems that seems crazy But it's because you already have a right because a or B is true Whenever either a or B or both are true if you already have a You know the conjunction is going to be true I hope that makes a little bit of sense disjunction elimination is a little bit of a more complicated rule, right? It requires not only that we have the disjunction But also that we have the negation of one of the disjuncts, right? If online M, we have a or B and online in we have not be we can derive a and likewise if we have a or B Online M and we have not a online P We can derive B. We also have a conditional elimination rule We I mentioned earlier how intuitive this is whenever you have if a then B and a you can derive B That makes a certain amount of sense given that the in the truth table, right for conditional Whenever a is a one B also has to have a one whenever a is true B also has to be true I'm not going to tell you the conditional introduction rule yet because again, that's not a direct proof rule That's what we call an indirect proof rule, which we'll talk about in the next lecture Biconditional elimination is also a direct proof rule. It works like this whenever you have the biconditional and One side of the biconditional so if you have a if and only if B and a You can derive B and vice versa. So let's look at some examples. Here. I have Simple argument relatively simple argument and I want to show you how we're gonna prove it and to do that I'm going to take us over to Carnap and show you how we write down proofs in Carnap. So what I'm gonna do to demonstrate how you do proofs in Carnap is I'm gonna head over to what I call the proof Playground and this I've assigned this to you in Carnap. So if you log in at Carnap.io You should be able to find this and what this has in it among other things is a set of Reminders for how we generate connectives using the keyboard what kind of sentence letters we can have And also how we write the rules that justify our proofs But what I want you to focus on right now is the actual proof playground itself This is gonna allow us to write down proofs and check them for Validity, okay So let's go back to our To our problem and see we've got Again, if a then B or C a and not B as our premises, right? So I'm gonna put those in if a then B or C and To mark a premises a premise we use this colon PR says that's one of our premises, okay? A was also one of our premises and so was not B And that's all the premises and you can see over on the right how Carnap has Nicely sort of formatted it for us based on what I put in with a keyboard and what I'm trying to get is C Okay, and I look at what I've got And I see that C is embedded there in in line one I also see that I've got everything I need for a conditional elimination. So I'm gonna start there, right? I can derive B or C based on the conditional Elimination rule using lines one. That's the conditional and line two that has the antecedent in it Okay So that's that's a good start now I've got a disjunction and it looks like I need to use the disjunction elimination rule to get that C out and it looks like I have everything I need to do that as well so I can derive C using the disjunction elimination rule on Line four, which is our disjunction and line three, which is the negation of one of the disjuncts Okay, and that is an example of how we do a proof in an SL Here is another example of an argument. I want us to try to prove that we can prove with direct proof What I'm gonna do is I'm gonna go back to Carnap. I'm gonna reset the proof playground so I have a fresh screen to move from and all I had for my premises were A and B and what I wanted to get Oops, I misspelled premise. All I want to get what I want to get is this longer sentence Which I will write down here as a or a or C and B or D, right? Obviously Perhaps not so obviously I'm not gonna be able to get this directly, right? So Let's see how I can use disjunction introduction to get the first part of it here, right? Let's Think about how disjunction introduction works, right? All I need is to name the line that a appears on and there I have The start of my disjunction Introduction, okay. I got it. Similarly. I can get B or D through disjunction introduction on line two and Now that I have both of the disjunctions I can get a or C and B or D through conjunction introduction Conjunction introduction takes two lines three and four in this case and now you see I've proved a and B Validly imply this longer expression Okay, let's look at yet another example a if and only if B a Therefore a and B Let's see if we can do that in Carnot. I'm gonna reset the proof playground again by reloading the page and I've got the Biconditional a if and only if B That's a premise and I have a that's another premise And then I want to get The a and B, right? Well, I can get B through Biconditional elimination on Line one and two if I have either a or B I can get the other right and now that I have a and I have B I can get I can use the conjunction introduction rule on lines two and three to get what I wanted to prove right so there is your valid proof of This argument before we wrap up. I want to talk a little bit about this language I've been using of direct and indirect proofs, right? Direct proofs, which is what we've seen so far Derive their conclusions from the starting premises using rules of introduction and elimination The only things in a direct proof are the starting premises the intermediate steps derived from those premises and From prior intermediate steps and the conclusions similar similarly so derived So every line in the proof after the premises is derived via a specific rule Indirect proofs, which is what we'll talk about in the next lecture in one way or another Introduce additional assumptions in the course of the proof They're indirect in the sense that there are some lines of the proof that we don't directly derive from a prior line or or set of lines We'll talk more about what that looks like in our next lecture until then I suggest you have a look over the first part of chapter 6 in for all x and Let me know if you have any questions Bye