PHIL 126 Lecture 3 Advanced Translations

0:00 All right, everybody. Welcome to the lecture. it says lecture two, but that's incorrect. This is lecture three. we are now going to move on to the next portion of the logic chapter which is going to be on advanced translations in our little logical language. Let me turn the light on. Is that even better? Who knows? I guess it's a little better. Okay. so let's move on to advanced translations. So here is our game plan for the day. I want to say what I mean by advanced translation a little bit first and then I'm going to give you a stepbystep method for how to do translations.

0:49 Now, this isn't like the only way to do translations, but in my professional experience, having taught this for many, many years, this will reliably get you the right answer in a way that other methods might not. It's very easy to make small mistakes when doing logic. This method will help try to reduce the amount of mistakes that you make. can't eliminate them, but it can help reduce it. Then I want to go over common English forms. So we'll see and probably intuitively you can figure out that there are many different ways of saying something in English that gets translated the same within our logical language. And so I want to give you a spread of terms that we're all going to translate using the same connectives for the for each connective.

1:55 Then I want to go over three common tricky things that happen in logic. These are the terms if, only if, and unless. Those are very frequently hard to remember. I have a couple tricks for remembering them. so hopefully that will help and then we'll do a bunch of examples and show why these get especially tricky. Okay, let us talk about advanced translations. So what do I mean by advanced translation? An advanced translation is not like a new symbol or anything like that, but it's new logical structures, right? Our rules for the grammar of the logical language L permit us to create arbitrarily long sentences within our logical language that are grammatical.

2:45 Accordingly, many English sentences that are extremely long will have unique translations within our logical language. They're not actually unique, but that's fine. they're logically equivalent to other sentences. There's an infinite amount of sentences that every sentence in our language is logically equivalent to. Doesn't matter. That's not our issue here. But here is how things can be advanced.

3:16 You can have multiple connectives in one sentence. You can have connectives inside of connectives in appropriate ways. You can have tricky particular phrases, the sort of only if unless things. but the main strategy to do all of this is to find the main logical connective for each sentence or subsent you're concerned with. I'll show you what I mean by that. So, let's first go over our step-by-step guide. Oh, sorry.

3:52 Before we do that, I want to give you two reminders. The first reminder is the concept of scope. So the scope of a logical connective is what that connective ranges over within the sentence. So in this sentence, tilda P or parenthesis P or Q. The scope of the tilda is the whole thing parenthesis P or Q in the sentence parenthesis P or Q. arrow R. The scope of the arrow is what's to the left of it, the P or Q, and what's to the right of it, the R.

4:31 So that's scope. Just a reminder, go back to the last video if you need a more precise understanding. The main logical connective then given this concept of scope is the connective whose scope contains the rest of the sentence. So getting the right translation usually means you're getting scope things right. Not always, but that's usually the way that people go wrong is by messing up the scope a little bit. So here's how to translate.

5:01 There's actually a step zero to all this. So it's a five point five-step strategy. There's a step zero. Step zero is read the damn sentence. Try to understand the English sentence before you begin trying to symbolize it. Figure out what is being said by the sentence using you're not going to believe this your brain. Use your brain.

5:31 Figure out what the sentence is saying and then try to translate it. So that's step zero. Step zero is read it. Maybe read it multiple times. That's what I often do. Then step one is I honestly I recommend that you write these down. Step one is to replace the simple sentences with sentence letters just to make your life easier to stop thinking in terms of English and start thinking in terms of the logic. Make sure with step one that if there is a negation you are keeping the negation clear. Right? So don't lose the negation with step one.

6:15 We'll see this in in the first example we do. Step two is to identify the main logical connective in the sentence. The one the connective the English word connective that has the rest of the sentence within its scope. And then just to for your mind you can put parentheses or brackets around its scope just so you know what the main logical connective is so you don't lose it.

6:45 Step three is to identify the remaining connectives. Add parenthesis if we need to. Step four is then to replace the English logical connectives, the words or, if, and with the symbols of L. Repeat step three and four as needed. And then step five, once you have a completed sentence, read the sentence, the logical sentence. You know how to read it. read it using English words and see if that matches the original sentence intuitively.

7:28 Let's see this with an example. So, here's an example. One of my favorite sentences. If I study too hard, if I study too hard, then I will not enjoy college. Comma. And if I flunk out, then I will not enjoy college either. Okay, read the sentence a couple times. All right, if I study too hard, I won't enjoy college. If I flunk out, I won't enjoy college either. Let's first create a translation scheme.

7:59 So S will stand for I study too hard. E will stand for I enjoy college. and F will stand for I will flunk out. Those are our three sentences. We will identify a capital letter for each one of those three basic sentences. So the first step after step zero which is to read the sentence and understand what's being said. The first step is just to replace the English sentence with the sentence letter that we have identified or picked. So if I study too hard then I will not enjoy college becomes if S then not E.

8:38 Right? This is crucial. So S stands for I study too hard. So the 19 the sentence says if I study too hard. So if S then and then it says I will not enjoy college. My translation says I will enjoy college. So I have to say if s then not e comma. So keep the comma in step one. And if I flunk out then I will not enjoy college either. If f then not e again because it says I will not enjoy college. We have to make sure to keep the word not in there. So step one should yield if S then not E, and if F then not E. Period.

9:31 All we've done is replace the basic unit with the letter, but we've kept the other everything else remaining the same. So what we should have left is English connectives if then not and if then not and grammar meaning or punctuation. hear the comma. Okay. Then what we're going to do is we're going to identify the main connective which in this case follow the punctuation. The comma tells us that that is the main logical connective. So in this case it is the and right the comma and is our main logical connective. And so step two, recall, is to identify the main logical connective and put parentheses around its scope. So we'll put parentheses around the thing to the left of and which is if S then not E and the thing to the right of the and which is if F then not E. Okay, so that's step two. We've identified the main logical connective and we've put parentheses around it. Moving on to step three, which is to then identify remaining connectives and add parentheses as needed. We'll do this a couple in a couple iterations. So the first one is we'll do the easy one first. The word not I know is a tilda.

10:59 So let me replace the not with a tilda. If s then till the e comm, and parenthesis if f then tilda e period. Cool. Then what else do I know? I know I've got if then sentences here. Those are pretty easy to translate. It's just an arrow between the thing to the left of the then and the thing to the right of the then. So in this case, we'll put the arrow and we'll put the carrot because I know that and is translated with the carrot. So I get parentheses S arrow till the E end parenthesis carrot parenthesis F arrow till the E and that's our translation.

11:46 Okay, cool. Now let's go back and see if this makes sense. It says something like both. If I study too hard, then I then it's not the case that I will enjoy college. And if I flunk out, then it's not the case that I will enjoy college. Cool. Yeah, that's right. That's exactly what this original sentence said. Sorry, that should have been step five.

12:18 Okay, so that was step five. Okay. So that is how you do these translations. The advanced translations is you follow this fivestep procedure. Okay. Now let's move on to common translations. So here's the basic idea. English has many different what are called surface forms for the same logical structure of a sentence or a phrase.

12:58 What this means practically for our purposes is that many different English phrases will translate to the same symbol within our language. So here are some common ones. These are not the only ones, but these sort of give you a vibe or a flavor of the connective so that you can find them on your own. So, here they are. I'm just going to read through them because I think it's really helpful. You have this table. Screenshot it. Write it down. Writing it down is better so that you don't miss them in your first pass through.

13:31 So the connective negation which is the tilda will be tra will have will be the translation for all of the following English phrases. Not it is not the case that it is false that it is not true that no neither cannot does not will not all of these things will translate to negation.

14:03 There are also some tricky cases but you don't have to worry about those for this class. Second one conjunction the carrot. Some of these might feel weird. Let me read them. And, but, although, even though, while, as well as, both and. Some of those feel pretty different than the others. Consider but, although, even though, and while.

14:40 Usually when we say but what we mean is that there's some sort of tension between the two things you know I'm sorry but you were being a dick at the time right we want to what we're expressing is that there's a tension between the two things that we said that is true and philosophers of language and logicians recognize this as well they know that this is the case but if you're thinking about the logical structure of the statement but and and contribute the exact same thing to logical structure. They say both of these things go together.

15:22 You might additionally want to say on top of that that those two things are intention, but at the very least they go together. And so when we translate and and but both using the conjunction, we lose some stuff. We lose flavor of language. We lose a lot of communicative abil ability. But what we are trying to do is we're trying to isolate the logical structure of the sentence. And and and but contribute the exact same thing to the logical structure. So we will translate them the same.

15:57 Next is disjunction. So disjunction or the wedge will be translated as or will will be the translation for or either or unless otherwise. We'll talk about unless shortly because that one seems really weird but we need writing on the slide in order to accurately talk about it. I'll flag this now. I'll talk about it again when I talk about disjunction in this class.

16:35 Disjunction is supposed to be what's called the inclusive variety. So disjunction means either or or both. That's what disjunction is in this in this logical language. Talk about this again when it becomes important. But just so you fl you if you're worried about that right now, it really doesn't matter. It's just a choice that we make within our logical system. This is the standard choice that logicians make. So I will continue doing that.

17:05 The next one, the hardest one to translate usually is the conditional or the arrow. The conditional, as you all know, is the translation for if then, but here are a ton of other ones. provided that. So you might see a sentence that says provided that blah blah blah, blah, blah, blah. That's going to be translated as a arrow where the thing to the left of the arrow is whatever immediately follows the phrase provided that.

17:40 On the condition that blah blah blah, blah blah blah. Same thing. Only if I will do this. Sorry. Only if I go to the store will I buy milk. Something like that. Another one that is translated with the conditional is if. But things get tricky here with if and only if. We will talk about them shortly. I have a dedicated slide to them. Unless. Again, we'll talk about shortly. Whenever whenever X happens, Y happens. That is going to be a conditional where the X goes to the left and the Y goes to the right.

18:27 Implies is another conditional phrase and sufficient form. We'll see some of these with examples. Finally, we have the by conditional. The number one mistake I see students make with the byc conditional is they think it happens way more frequently than it actually does. The by conditional will wear its logical structure on its sleeve. It will say if and only if or something along those lines. If you just see if by itself or if you just see only if by itself is not a by conditional. A by conditional is a conditional both ways.

19:13 And so you need if and only if for it to work. Sometimes we shorthand this as iff. This happens occasionally in math and in philosophy. Another phrase that gets translated as the by conditional is just in case. Now this doesn't mean like as a precaution, you know, I'll bring my umbrella just in case it rains. Although there is something related with that. Instead, you should think of just in case when you're doing translations as meaning something like just in the cases where or precisely in the cases where P occurs precisely in the cases where Q occurs. That's going to be a by conditional.

20:04 Another phrase that means something exactly something similar is exactly if this will happen. Exactly if this will happen or when and only when precisely when all of these things mean the by conditional. Another is necessary and sufficient. Talk about this slightly. Okay, now let's talk about the weird cases, the ones that I promised individual slides on. So, these are the ones that reliably produce the most mistakes for students.

20:48 Number one is the phrase P if Q. I promise you just write this down and we will figure and you can convince yourself that it's correct later. By later I mean like in a minute when I explain it but write it down. Write down the rule so that you have the rule. So P if Q will translate as Q arrow P.

21:22 P if Q translates as Q arrow P. Okay, using our technical terminology P if Q the Q is the antecedent the P is the con is the consequence. Okay, so that's the first one. Moving on to the second one. P only if Q. P only if Q translates as P arrow Q. So notice that those go the opposite way.

22:00 So P if Q is Q arrow P. P only if Q is P arrow Q. Again just write this down. I will convince you that it is correct. The third one is unless. This is a very strange phrase. Strictly speaking, you can translate unless by just throwing a wedge in between it. P wedge Q is a totally appropriate translation of P unless Q.

22:32 However, that doesn't make sense in my brain. In my brain, if you say unless, I translate it differently. So if you say so if you say something like we will go to the beach unless it's cloudy. What I hear in my head is if it's not cloudy we'll go to the beach. So what that means is my rule for translating this isn't like a this isn't a rule rule but my trick for translating unless is P unless Q is going to be translated as if not Q then P.

23:19 We will go to the beach unless it's cloudy. If it's not cloudy then we'll go to the beach. Okay. So let's talk about each one of these in more detail. So P if Q what this means is if Q then P. So here's an example and I will give you a rule for this. Bob will go if Alice goes. What this is saying is Bob will go on the condition that Alice goes. So that means if Alice goes then Bob will go. So this should be translated as a arrow b if Alice goes meaning if Alice goes then Bob will go.

24:10 Here's another example. The alarm sounds if the door opens. What this means is if you open the door then the alarm will sound. If so, it's D arrow A. Now, here is the rule. The rule is thus. I don't have it written down. So, look at me. Eyes up here. If you see a bare if, meaning if without the word only in front of it. So if you just see if as in P if Q or the alarm sounds if the door opens, what follows the bare if is the antecedent of the conditional, meaning it goes to the left of the arrow. So what follows the if is the antecedent of the conditional. The alarm sounds if the door opens. What follows the if is the door opens. So that means the door opens is the antecedent of that conditional.

25:17 Okay. What follows if a bear if is the antecedent. Why do I say bear? Well, it's because we also have the phrase only if. So let's move on to that one. So now let's consider constructions of the form P only if Q. What this says is that Q is necessary for P. The way we translate this is if P is true then Q is true.

25:49 So P only if Q means if P then Q. Let me try to explain this a little bit. So Bob will go only if Alice goes. That means what this means is if Bob goes then that's because Alice also goes. So Bob will go only if Alice goes means if Bob is going to go then you know that Alice is going also. So that's going to be translated as B arrow A. Here's another one. You pass my class. This isn't true but you pass my class only if you submit the homework. meaning if you're passing my class then here's one thing I know you submitted the homework.

26:42 So if P then H okay the way we're going to trans the so that's that's the rule P only if Q is going to be translated as P arrow Q. Here's the rule that corresponds to that. If you see an only if what follows that is the consequent of the conditional. So what follows a bare if is the antecedent. What follows an only if is the consequent meaning goes to the right of the arrow. So you pass only if you submit the homework. You submit the homework comes after only if. So that means that is the consequent means it goes to the right of the arrow.

27:27 So that's your rule. What follows a bare if is the anticedent. What follows only if is the consequent. Finally, let's talk about unless. So again the way I tr so unless is tricky because there is a in my opinion unintuitive but totally correct way to translate it which is just to throw the wedge. Just put a wedge in between P and Q. That's appropriate to translate P unless Q. You can do it anytime. I won't mark you off. You're totally fine to do that. However, I need these to make sense for myself. So the way I translate these sentences is P unless Q is going to mean if not Q then P.

28:23 So for our example, we will hike unless it rains. What this is going to mean is if it doesn't rain then we'll hike. So if not R then H. Boom. That's the translation. This also works if you put unless at the beginning of a sentence. So unless the server restarts, the update will fail or fails. This is going to be if not R because that's the unless portion. Then the update fails then F.

29:02 Okay, those are our three common mistakes. two, one, two additional things to keep in mind. These are sort of just things that somewhat frequently come up, but neither nor and not both are sort of strange constructions. So, let's look at 23. Neither the alarm is on nor the door is locked.

29:38 When somebody says this, what are they saying? They say neither the alarm's on nor is the door locked. What they're saying is it's not the case that the alarm is on and it's not the case that the door is locked. So neither nor can be appropriately translated as not a and not d. If I'm neither hungry nor tired, what do you know about me? You know I'm not hungry and you know I'm not tired. So that's the way you're going to translate neither nor. Both and sorry not both is literally just going to be not both of those things. So, it's not the case that both the alarm is on and the door is locked. What does that mean? Does that mean the same thing? No.

30:30 What that means is, let me tell you what I haven't done. I haven't both set the alarm and locked the door. One of them maybe, but not both of them. So, this is going to be tilda both A and D. So these are different logical structures. So let's do a couple examples and then I will give you your own to work on.

31:05 So either Alice is invited or Bob is invited and Carol will attend. Okay. Going through the steps. So let's first find our translations. ABC. Nice and easy. you follow the so we want to follow the punctuation here which will be our guide. So that tells us since there's only a comma that that marks the main logical connective meaning whatever goes to the left of it is one chunk and whatever goes to the right of it is another chunk. So what this means is the wedge or is our main logical connective where to the left of the wedge is going to be Alice is invited to the right of the wedge is going to be Bob is invited and Carol will attend. So we go a wedge parentheses b and c.

32:13 Going back this means look either Alice is going to be invited or both Bob is invited and Carol attends. Cool. That's what the sentence says. Now contrast this with the following sentence. parenthesis A or B and C. Now, what this means is either Alice or Bob is invited and Carol's going to attend.

32:45 Notice how those two things mean quite different things. We'll be able to prove shortly that these things mean different things, but just using your intuitive senses, you can see that they mean different things. Okay, let's move on to the next one. If the key is turned, then neither the alarm sounds nor the camera records. Okay, we have three sentences here. Key is turned, alarm sounds, camera records. We need to come up with K, A, and C for these again. So, step one, find the main logical connective.

33:31 Use punctuation. We have a comma here. To the left of the comma is this if phrase. To the right of comma is this then phrase. Put your parentheses around the then phrase. And put your parenthesis if you want them around the if phrase. So, that means that the conditional is our main logical connective. To the so to the left of it is the key is turned. That's just going to be a K. To the right of it is this complicated thing. Neither the alarm sounds nor the camera records. Take this slowly. We've got this neither nor statement. How do I translate neither nor statements again? Oh, I don't remember. Go back to your notes right here. We just talked about neither nor.

34:16 Neither nor. I'm neither hungry nor tired means I'm both not hungry and I'm not tired. So that means you do a negation in front of each thing and you do a wedge between them. So neither the camera sounds nor sorry neither the alarm sounds nor the camera records. That's going to be not alarm sounds and not camera records. Remember that was all within the parenthesis. So this is going to give us k arrow parentheses not a and not c.

34:57 So you take it chunk by chunk. So I know to the right of the arrow is this whole chunk. Let me think carefully about how to how to symbolize this exact chunk. Cool. Great. Awesome. That's how we do it. Moving on to the next one. either the train is late or if it's on time then we'll arrive by noon. Okay, again look at the punctuation. We've got a comma. It's an or statement. So that means the wedge is going to be our main logical connective. To the left of the wedge is this thing, the train is late.

35:37 I'm just going to translate that as L. To the right is if the train is on time, then we will arrive by noon. I know how to translate that. That's a straightforward arrow. So, O arrow n. Cool. Easy. Finally, here's a weird one. It's not true that Bob is invited, then either Carol attends, then Dana in attends.

36:09 So, what is the main logical connective here? The main logical connective is given by the colon sort of telling us here's what the scope of the negation is. It is not true that then it gives you all this stuff. So that means that the parentheses are going to go around all that stuff. So you know this sentence is going to be a negation and then a big parenthesis. So what goes within that parenthesis? If Bob is invited, then either Carol attends or Dana attends. I know how to translate this. This is going to be B for Bob attends or Bob is invited. Then an arrow and then I want either Carol attends or Dana attends C or D. So I get this large translated sentence tilda big parenthesis if B then small parenthesis C or D.

37:11 So when you're going back and thinking about the sentence, step five, trying to reconstruct it. If you ask what is the sentence doing, that will help you identify the main logical connective. What is the sentence doing here? Denying something. Denying that if Bob is invited then either Carol attends or Dana attends. Sentences might also introduce two options either this or this or say that something happens only if something else happens.

37:48 All of these are options. So asking yourself big picture, what is the sentence trying to do? And then the parenthesis should help you out with that when you go back and look again. Finally, I have three sets of pro practice problems. I'm not going to give you the answers. If you would like them, let me know. We can figure it out. But I'm going to give you these practice problems. And I want you to translate these.

38:22 So, these are sort of easy ones that give you some basics of translating. I'm going to give you a moment to write them down or pause. do that right now. Okay. So, practice A, fairly easy translations. Let's move on to practice B. So, practice B, slightly more complicated. Here is your translation scheme.

38:54 I'm going to give you a second to pause or to write these down. Okay. And let's move on to the last practice set which is C. These are again about the same difficulty level as B but just slightly different examples. and that is all I've got for you today. So these are three practice sets. we will be doing more and more difficult ones as we go along. so each step of logic iterates on the next one. So it gets more and more difficult as we go along.

39:39 So that's all I got for you. We will see you next time. Godspeed.