 Hi there and welcome back to deductive logic fill 320. Today we're going to begin working with our first formal system what we'll call sentential logic or SL for short. This is a system sometimes referred to as propositional logic in other classes and we're going to explore some of the basics of it today. Before we get into it though I want to start us off with a little logic puzzle of the day. So let's have a quick look at this. In a certain flower garden each flower was either all red, yellow, or blue and all three colors were represented. A statistician visited the garden and observed that whatever three flowers you picked at least one of them would be red. A second statistician observed that whatever three you picked at least one was bound to be yellow. Two logic students heard about this and got into an argument. The first said it therefore follows that whatever three flowers you pick at least one is bound to be blue, doesn't it? The second student said of course not. Which student was right and why? So think about that puzzle as we go through our lecture today and I'll come back to it at the end. So I want to start by looking at a little brief argument, right? It goes like this. The dog is on the rug. If the dog is on the rug you should let the sleeping dog lie. Therefore you should let the sleeping dog lie. Now NSL will represent basic sentences with capital letters. We'll call them sentence letters. Let's try it out like so. We'll replace the first sentence with A, the second sentence with B and the third sentence with C. Now that doesn't help us too much so let's look again at what we might do to represent this using sentence letters. The first premise the dog is on the rug is contained in the second premise here in the first part of it and the conclusion is also contained in the second premise and the two are connected by an if. So how about instead of replacing it with three separate sentence letters we include A and C in the middle sentence and say if A then C as a representation of that. So this allows us to think about a distinction that's important to SL which is a distinction between atomic sentences, sentences like A and C where the structure has been completely abstracted away and complex sentences that have atomic sentences as their parts as well as logical connectives that connect them together. Here I'm using still the English if then but we'll look in a moment about how we represent that with a formal symbol. So this distinction between atomic sentences and complex sentences is crucial to SL and the reason we call it sentential logic is because the basic units themselves are sentences or propositions we talk about it that way. So I mentioned that if then is a logical connective here all the logical connectives we're going to use in SL. I'll give you the symbol the name that we refer to the symbol by and what it means in English. So the first is negation which we represent with the hook and it means it is not the case that. Conjunction which we represent with an ampersand which means and or both and disjunction which we represent with this kind of V shape which means or. The conditional are if then which we'll represent with an arrow and finally the biconditional which means if and only if which we represent with a double arrow. So for each of these connectives I am going to tell you something about common translations how we what kinds of English sentences we translate using these logical connectives. I'll also try to give you a better idea of what each connective means by showing you three different representations a Venn diagram that visually represents when the logical connective is true and when it's false. A truth table which provides more or less the same information as the Venn diagram but in a tabular rather than visual form. Both of these the Venn diagram and the truth table emphasize that the connectives of SL are truth functional and idea will discuss further in unit three. And the third representation I'll give you some of the characteristic inferences that we use to infer to or from that connective. You'll see what I mean in a moment. Let's start with negation it's not the case that. Here are some examples of sentences we will use the negation to translate. Rolex does not make corn flakes so if our atomic sentences Rolex makes corn flakes the negation of that sentence would would be Rolex does not make corn flakes. Here are some other examples it is not the case that Rolex makes corn flakes it is false that Rolex makes corn flakes. Rolex makes anything but corn flakes. The last thing Rolex makes is corn flakes that's a little that's a little hairy but I think negation would be the best way to translate that and then one that I find one of my favorites yeah yeah Rolex makes corn flakes sure they do right sarcasm might best be captured with a negation in this case. So look at this Venn diagram here the red space represents when the connective is true and the white space is when it's false so if the inside of the circle is our atomic sentence a outside of the sentence is the negation that's when it's true right or here's our characteristic truth table for negation you can see the same thing whenever a is true not a is false whenever a is false the negation of a is true you can see that not a is truth functional because it takes the truth of its part a and spits out a new truth value for every truth value that's input gives you an output and that's that's the sense of it being a truth function a truth functional connective we can also understand the meaning of any connective in terms of the kinds of inferences it tends to figure in negation is perhaps not the easiest place to start on this but the main characteristic inference for negation looks something like this if you have some premise P and you can show that P implies something like Q and not Q that's a contradiction right what that little sideways T represents we call it a single turn style what it represents is that there is a proof that starts with P and ends with Q and not Q right so if you have such a proof then you can infer that not P because a contradiction is impossible right the characteristic inferences for most of the connectives are a little simpler so you'll see how this works a little bit more clearly for our next connective which is conjunction so here are some examples of the kinds of things we translate using conjunction Ford builds trucks and Chrysler builds mini vans Ford builds trucks but Chrysler builds mini vans you might think those two are different but they both say that both things are happening and that's what we're trying to capture Ford builds trucks however Chrysler builds many vans although Ford builds trucks Chrysler builds many vans both Ford and Chrysler build many vans Ford builds many vans but Chrysler builds them better in the last one of course it's a different different atomic sentence involving Chrysler but there's still the but captures conjunction what's interesting I think about some of these is that whether it's and or but or however it's captured by conjunction whether or not you mean a kind of contrastive thing when you're making a statement like this or not that's not captured by SL that's some structure that SL leaves out if you look at the Venn diagram for conjunction it looks something like this so if our circle on the left means a and the circle on the right means b the conjunction of a and b is true when both are true right so a and b is true there and we can see this again in our characteristic truth table only when both a and b are true is the conjunction true every other situation it's false what are some of the characteristic inferences we can make with conjunction well if we have both a and b we can infer that a and b is true on the other hand if we if a and b if we have a and b we can infer a or we can infer b right so those are some of the characteristic ways we use conjunction in our proofs in our inferences next let's look at the connection we call disjunction which is an or it's translate it translates sentences such as rabbits are lagomorphs or pigs can fly either rag rabbits are lagomorphs or pigs can fly rabbits are lagomorphs unless pigs can fly unless rabbits are lagomorphs pigs can fly rabbits or pigs are lagomorphs those are some of the uh representative kinds of things we would translate with an or or a disjunction now the venn diagram for disjunction looks like this the the disjunction is true whenever either a or b or both are true right and you can see this also in the characteristic truth table when a is true or b is true or both are true the disjunction is true otherwise it's false now that might seem wrong to you right that first line might seem wrong because uh it's saying it's saying either or um but on the top line it's both right so that's what we would call an exclusive or and it's a it's it's a convention of s l that our disjunction is an inclusive or so both can be true right we could have uh used an exclusive or we can represent an exclusive or in a complicated way but it's just a choice it's a choice we made when we designed s l right here are some of the characteristic inferences for disjunction from a we can infer that a or b is true right you might say well we don't know about b how can you say that well as long as we know a as long as we start with the premise of a we can we can infer the disjunction we can also infer from a or b um and not a that b is true and of course we can infer from b that a or b is true right so those are some of the characteristic kinds of inferences that we make based on disjunction next let's look at the conditional the if then so here are some of the typical translations of course if you let the wine breathe then it will taste better that's a that's an if then statement that's a conditional statement um also the wine will taste better if you let it breathe the wine will taste better only if you let it breathe the wine will taste better provided you let it breathe breathe. The wine tastes better, which implies that you let it breathe. These are all different aspects that we can translate with a conditional. And it's important to note that where the if falls is going to tell you which direction your arrow goes, right? I mean, the arrow always goes from left to right, but I mean which part of the sentence is going to be on the left or the right. So in the first one, the plain old if then, it's, if you let the wine breathe, then conditional, it will taste better. Using B to represent, you let the wine breathe and T, it will taste better. The second sentence actually means the same, right? The if is in front of you, you let it breathe still. Although that's in the second part of the sentence, it still means the same thing, right? So the order is reversed in English, but because the if is next to you let it breathe, you let it breathe is still the first part of the conditional. We call that first part the antecedent of the conditional. We call the second part the consequent of the conditional. You might think, okay, well, the third sentence is probably the same, but no. Only if swaps the order, the wine will taste better only if you let it breathe. The only if comes before the consequent, not the antecedent. This fourth sentence, the wine will taste better provided you let it breathe, like the second sentence is translated this way. The wine tastes better, which implies that you let it breathe is represented this way. So the implies is kind of like the only if. Also here are two more sentences that we can translate using the conditional. Letting the wine breathe is a sufficient condition for it tasting better. You might not see this sentence outside of a philosophy class, but we can translate it using a conditional. Also letting the wine breathe is a necessary condition for it tasting better, can be translated with the conditional. In the first case, sufficiency means the arrow goes like this. The letting the wine breathe is sufficient for it tasting better means that if you do the one, you'll get the other. Letting the wine breathe is a necessary condition for it tasting better is an only if kind of thing. It reverses the order. It tells you that if it tastes better, you must have let it breathe. So the necessary condition goes in that direction. It's a little complicated, but you'll get the hang of it. The Venn diagram for conditional looks like this, right? It is only false where you have A but not B. It's true everywhere else. It's perhaps a little clearer if we look at the characteristic truth table for the conditional. Here you see if A is true and B is true, the conditional is true. And if A is true but B is false, the conditional is false. In the case that A is false, the conditional is true. These first two lines are pretty clear, right? If A is true, then B needs to be true. That's what the conditional says. If that doesn't pan out, then the conditional must be false. But what does it mean? What is A if A then B mean when A is false? It's not so clear, right? So what should we say here? By convention, we say the conditional statement is true. We call this the material conditional, okay? And it makes a certain amount of sense. When I say if I wear a coat, then I will be warm. It doesn't mean necessarily that I'll be cold. If I don't wear a coat, maybe the weather is warm or I have a personal heater or some other thing, right? Makes me warm. So if I don't wear a coat, I might or not, might not be cold. I might or might not be warm. It doesn't matter to the truth of the statement. So we'll say that the statement is true whenever the antecedent is false. Now, this conditional, the material conditional doesn't capture everything we say in English with if then talk, right? It doesn't capture a kind of counterfactual what would be the case if A were true. It doesn't say anything about a causal or stronger connection between A and B. And so it doesn't necessarily capture everything we mean by if then. That's a feature of formal languages. It can't capture everything. And the material conditional captures a lot of what we mean with if then statements. Here's some of the characteristic inferences that we use conditional for. If we have if A then B and A, we can infer B, right? This is an inference form sometimes referred to as modus ponens, right? It's fancy Latin. You don't have to worry about it. Also, if we have if A then B and not B, we can infer not A. And again, these make sense based on the characteristic truth table for the conditional. Also, we can infer from if A then B that either not A or B, right? That is an equivalent statement. If we look at the Venn diagram for not A or B, it looks the same as the Venn diagram for if A then B, right? So they're equivalent statements and we'll make use of that later on. Finally, we have the biconditional if and only if, right? Now, the biconditional could be just represented with a conditional in a conjunction. But for convenience, we have a separate connective to represent it. We translate the biconditional with sentences like this. Bill is going to pass the class if and only if Ted is or we say Bill passing the class is a necessary and sufficient condition for Ted passing. The Venn diagram for the biconditional looks like this, right? So it's true where they intersect, where they're both true and it's true where both are false outside, right? I know that the biconditional is not the same as logical equivalence. The biconditional is a single sentence, right? It doesn't mean that A and B are logically equivalent sentences. It rather has the following kind of truth table structure, right? If A and B are both true, it's true. If A and B are both false, it's true. If A is true, but B is false or vice versa, the biconditional is false, right? So it has a similarity to logical equivalence, but it's not the same. It's a logical, it's a truth functional logical connective, right? Not a property of sets of sentences. That's an important distinction. As I said before, the biconditional A if and only if B is equivalent to the conjunction of if A then B and if B then A, right? And if you remember how we translated if then or if versus only if, you can see why we say it's if and only if, right? Here's some of the characteristic inference forms, right? From A if and only if B and A, you can infer B. From not B, you can infer not A. From B, you can infer A. From not A, you can infer B, right? So it's the same kinds of inferences you can make from the conditional, but they go in both directions. Those are the main logical connectives that we use in SL. That's the system we're going to use. We don't need the biconditional, but it's nice to have it. Actually, we don't need all these logical connectives. One could actually build a complete formal language equivalent to SL with all the same expressive powers of SL with only one connective. There are two versions of this. One is based on neither nor, what we call nor. The other is based on non-and, right? Or what we call NAND, right? So logical nor or non-disjunction, right? What's sometimes called Perse's arrow from the philosopher Charles Perse or Ampec, which is what Perse called it, or Quine's dagger from the philosopher Quine or the web operator from the scientist web are all different ways of representing this not or, right? Non-disjunction. You can see that it's just the negation of disjunction, right? Interesting fact, the computer used in the spacecraft that first carried humans to the moon, the Apollo guidance computer, was constructed using entirely this logical operation, the non-disjunction operation, and you could build a whole logical system using this and a whole computer. NAND, what we might call non-conjunction, what's sometimes called alternative denial, what's called the Schaeffer stroke after the philosopher Henry Schaeffer, which we represent with this upward-facing arrow, is also sufficient to define any other connective we might want to use in SL. This kind of operation is used as the basic the basic logical operation in most modern computer processor design, right? You look at the characteristic truth tables for NOR and NAND, right? They look like this and you can see why we call them non-conjunction and non-disjunction because they are the opposite of disjunction and conjunction, sorry, conjunction and disjunction, I mean. If you replace the T's and F's with ones and zeros and you think about these as operations in a computer, you can start to see how you can build computers out of these, right? Because what are compute, all the numbers in computers are bits, ones and zeros, and the functions that you need to transform them work like this. I'm not going to right now prove to you that you can derive the rest of the connectives from them, but I think it's an interesting thing to think about. So those are the logical connectives that make up SL, negation, conjunction, disjunction, conditional and biconditional. In the next lecture, we'll practice translating between English sentences and arguments and sentences and arguments in SL and look at some more complicated examples than the ones we've looked at today. Before we wrap up, I want to come back to our logic puzzle. Did you come up with an answer? If you did, share it on Discord, but don't forget to use the spoiler tag for the benefit of other students. All right, I look forward to seeing you in our next lecture where we'll talk about symbolization and translation in SL. Bye.