Video Transcript: Using Parentheses

Hi, I'm David Feddes. And this talk is about using parentheses in logic to translate complex  sentences into Symbolic Logic may sound a little boring how to use parentheses. But it is a  very important part of logic to properly be able to group our thoughts and the ideas that we're dealing with. In many sentences, there are multiple operators. In previous talks, we've talked  about complex propositions, and usually those just had two atomic propositions and just one  operator. But now we're going to start talking about having multiple operators in a  proposition. Here's an example. Bob will not go to class, but will play video games. Now when  we break that down, we have two assertions, Bob will not go to class. But we can't have that  as an atomic proposition because it contains the negation, and the negation is a logical  operator. So we can't say that that itself would be an atomic proposition. And then Bob will  play video games. When we break that down into the atomic propositions, we'll have Bob will  go to class symbolized by the constant C, and Bob will play video games symbolized by the  constant G. And then if we want to say Bob will not go to class, but will play video games, we  have tilde C, that means the negation of Bob will go to class. So Bob won't go to class. And  remember but is a conjunction, which can mean and or just joining two different statements.  So tilde C then the conjunction the raised dot, and then G, he will play the video games. Now  that's a sentence with multiple operators. And yet, we can clearly see what it means. But it  does, it doesn't always work out so well. To just string things together and read them and say, now we've got the sentence translated into symbols. Here's an example Bob will not both go  to class and play video games. So what we're trying to say with that sentence, Bob will not go to both go to class and play video games is we're saying Bob might go to class, Bob might  play video games, he might do neither. But he won't do both. But how do you say that in  symbols, because you can see that that's what the sentence is saying he will not both go to  class and play video games. So he's not gonna do both. But how do you put that into  symbols? Well, let's get our atomic propositions Bob will go to class. Remember, you can't  have a not as part of that, because it's an operator. So Bob will go to class is C, and Bob will  play video games is G, then what will we say is the resulting translation? Well, we know he  won't go to class. So we put a tilde in front of the C. And we know that this is a conjunction  because it's talking about two different things. He won't go to class. And so we just say, tilde  C and G he won't go to class in video, he won't go to class and video games. But there's a  problem with that. Because if you if you read that, you'll see well, it might mean Bob doesn't  go to class, and he does play video games. You see what I mean? Tilde C means he won't go  to class. Raised dot means and, and G means he will play video games. But we're trying to  say that he won't go to class. He won't both go to class and play video games, though he  might go to class. And he might play video games, he won't do both. So this one won't convey what we're trying to do. So again, we have our atomic propositions. But if we use  parentheses, we can say it the way we want to. We're trying to say that he won't do both.  Now how do we symbolize the bothness? Well C raised dot G means he goes to class and play video games, they were trying to say he won't do both. So we put parentheses around that  whole thing. And then we put the tilde out front of the whole parenthesis and that says not C  and G so he's not going to do both. Now the main operator of a sentence when you're doing  parentheses, the main operator is the one that ranges over the whole sentence or influences  the whole sentence. So if you have a tilde in front of a set of parentheses with some  propositional information inside, then the main operator is the Tilde because it covers the  whole rest of the sentence. But if you have the parentheses around outside the whole thing,  there's really no operator outside the parenthesis and so the main one inside is the raised dot  the conjunction of the two atomic, the two different propositions there. Let's try to translate  Noelle will either feed the dogs or clean her room, but she will not do the dishes. Well when  we try to translate from an English sentence into the more symbolic logic, we first have to find the atomic propositions, and the constants for those. So the first statement is Noelle will feed  the dogs remember, you can't have a not in an atomic proposition. Noelle will feed the dogs  Noelle will clean her room symbolized by C Noelle will do the dishes symbolized by D. Now,  what's our next step? Well, let's start breaking it down. She will not do the dishes, let's  symbolize that that would be tilde D, right? D means she will do the dishes, but our sentence  says she won't. So let's go with tilde D, there next step, it says she will either feed the dogs or

clean her room. So that's a disjunction. So we take F, she will feed the dog's wedge for or and  then C for clean her room. Next step, there's a but there's a conjunction there. And that  means we use the raised dot so we've got Noelle will either feed the dogs or clean a room  that's F wedge C and then but a dot and then we've got a tilde D. So is that the answer?  We've got F or C and not D. Well, that could mean different things. So we still have to figure  out how to group those symbols once we've got it. Because without parenthesis, F wedge C  raised dot tilde D can mean different things. It could mean F wedge C and not D, or it could  mean F wedge C and not D or to put an English again, it could mean well, let's go back for a  moment, it could mean that F means feed the dogs. C means clean the room. D means do the dishes. So it means she'll feed the dog or, and then C and not D, clean the room do the dishes and not do the dishes. But it could also mean something different. It could mean that she'll  feed and that she will clean her room and not D you see, those are two different meanings.  And so we have to have one meaning we can't have a symbolization that could mean two  different things. So what we want to say is she will either feed the dogs or clean a room, but  there is the conjunction, she will not do the dishes. So you put F or C in parenthesis, then the  conjunction but translated as the raised dot or and and then not do the dishes. And that's how you would symbolize that you have the three atomic propositions, the negation of the last  atomic proposition and you put parentheses around the disjunction F or C. Overall when  you're trying to translate and this will take a little practice and you'll have opportunity for  such practice. I'm sure you're just dying to get at it aren't you, you'll have a chance to  practice in your exercise of how to translate from an English sentence into the more symbolic  logic, the first thing you do is you identify the atomic propositions. And remember, you can't  have a not in those you'll use the not as the negation operator later. But start by identifying  atomic propositions, your next step is pick a unique constant for each of those propositions to  symbolize it. And then you ask some questions, which atomic propositions are grouped  together? And what operator connects each group? Because then you're starting to and the  fourth question is, what's the main operator of the whole sentence? So you have to figure out  your propositions and how they ought to be grouped. And, and part of that will be determined  by figuring out what's the main operator of the whole sentence. And then once you've, once  you've written it out as a logical translation, then read it back, put substitute back in what  those constants mean, you know, you have an A, what did that a mean? That you read the  atomic proposition that A represented and, you read the whole thing, you read it back and  you see if your translation matches the English sentence, and if not, well, go back at it again.  And make sure that you have the right constants, make sure that you have the right  operators, and then make sure you have the parentheses in the correct place to give the  meaning that you want. Those are the basic steps for translating and then you just have to  keep on practicing until you get the hang of it. Let's try Bob cannot fly a plane or pilot a ship,  but he can ride a motorcycle. Okay. The atomic propositions are the first thing to do and then  assign each of those constant atomic propositions Bob can fly a plane represent that by P. Bob can pilot a ship represent that by S Bob can ride a motorcycle, represent that by M So you've  got those propositions and now, you say he can't fly a plane or pilot a ship, so he can't fly a  plane is not P, and he can't pilot a ship. So not P and NOT S are both true. And you put  parentheses around that. And you say, he can ride a motorcycle. So then outside the  parentheses, you put the raised dot and the M that can also be expressed, it's a similar, you  know, it's written differently, but it means exactly the same thing in logic, you could say, he  can't fly a plane or pilot a ship. So if you took the proposition P or S, Bob can fly a plane or  pilot a ship. That's what it means in parentheses, P wedge S is Bob can fly a plane, or Bob can pilot a ship. But we're saying he can't do either one. So we put a tilde in front, which negates  it says he can do neither of the above. And then you can have the raised dot represent the  and or the but and he can ride a motorcycle, you see that both of those expressions written in yellow actually mean the exact same thing you can say that if Bob can't fly a plane or pilot a  ship, you can say not P and NOT S, or you can express it as not P or S, and with parentheses  around that. Well, that's enough for our use of parentheses, I hope you enjoy getting a little  bit of practice in and getting the hang of it. One of the things to do and this is not just to get  frustrated, or bored and say I don't want to bother with all those symbols. A real aim of all of 

this is to make you a more exact thinker. To make you a more logical thinker when you read a  bible passage or a sentence and know how the different parts fit together. What are the truth  values, sometimes that are implied of various statements, and what is being claimed and not  being claimed? Whether you analyze biblical statements, whether you analyze political  

arguments, whether you're just trying to figure out what something implies and what it  doesn't, the use of logic can be helpful to you. And like many things, sometimes you just got  to toil at it. If you're learning a new language, you have to learn some of the vocabulary. If  you're learning mathematics, you have to learn what some of the symbols mean and then  practice operating with them. And here in logic too, that's what we're doing. You're learning a  little different kind of language, and a few symbols that are used in certain ways and we need to master that and when we do, then our minds will be better trained to really think through  matters of truth and falsehood.