PHIL 126 Lecture 2. Basics of L

0:01 All right everybody. today we are going to continue our portion of the class on validity by sort of talking about how we're going to approach validity and then doing some of the basics of of the approach that we're taking. So an argument is valid a deductive argument. Remember, deductive arguments attempt to provide logically conclusive evidence for their conclusion. We say an argument is valid if it succeeds in providing this conclusive evidence. As it turns out, validity has nothing to do with the with what an argument's premises are about.

0:42 It instead has to do with the logical structure of those premises and how they relate to one another within the argument and how they relate to the conclusion. And so if this is right, that logic, that validity has almost nothing to do with what the premises are about and instead has everything to do with the structure, the best way to assess whether an argument is valid is to construct a logical language that lays bare the logical structure of our premises and conclusions. And then we can just look at the structures and see whether they fit together in the right sort of way.

1:23 And so what we're going to do is we're going to build a formal language called L for deductive logic. We're going to see how it works. We're going to introduce some logical relations and then we're going to manipulate the symbols. Feels weirder than it is. So bear with me. You got this. I promise you. I'm going to be following fairly closely the le the chapter that I wrote. So this is going to look very similar to what's going on in the chapter. So what that means is if I were you, I would go read the chapter first, try to gain an un read the relevant portion of the chapter first, try to gain a relevant un a relative understanding of what's going on and then come back and watch this video in full where I can walk you through things in more detail, explain it using English words rather than just whatever's on the page. You know what I mean? Okay, so let's begin. So our logical language L this is what's going to happen. you don't need to know this right now.

2:31 Just to refresh what validity is. Deductive arguments premises are offered as logically conclusive support for some conclusion. A valid argument is one where the logical structure rules out the following situation. All the premises true and the conclusions false. We'll come back to that later at the near the tail end of our portion on deductive logic. So the basic unit of L or logical language is a declarative sentence. Something that can be true or false.

3:11 We will translate declarative sentences using judiciously chosen capital letters. So the sentence one that says water is wet becomes W. Use a sentence letter that looks nice to you. In this case W for water is wet and that's what represents the entire sentence water is wet. So sentence letter does not stand for a word. It does not stand for a concept. It stands for an entire declarative sentence.

3:42 Now, there are logical languages, especially if you take Phil 241 later, that have subsential logical phrases that don't just assign a sentence letter to a complete declarative sentence, but instead gets into the messy structure of a sentence. But that's not going to happen in this class. We're doing a much simpler version here. So, we translate ordinary sentences, declarative sentences using capital sentence letters. Then, we can mess with them. We can do things to those sentence letters. We can connect them using what are called logical connectives. So, we can manipulate these symbols in five ways that we're going to do in this class that can iterate on top of one another.

4:37 So we will translate the logical connectives not and or if then and then if and only if sometimes paraphrase as iff. So a logical connective is a way of logically manipulating these sentences. we will go through each one of these shortly in detail. You can write these down now but we'll go down we'll go through them again. Okay.

5:08 So the first one is negation. What negation does is it denies the truth of the sentence. So take sentence two here. It is not true that water is wet. We know that W stands for water is wet. So in order to say in our logical language, it is not true that water is wet. What we do is we put the tilda the negation sign in front of the statement water is wet so that we get 2 L here.

5:42 Tilda W not W. So this translation here till the W is the proper translation of the English sentence it is not true that water is wet where W means water is wet. Okay, so that's the first one. Negation takes as its object only one sentence. In this case, that one sentence is W. Later on, we'll see that that sentence can be big long complex sentences.

6:19 Okay. The next two are called conjunction and disjunction. So we can combine multiple basic declarative sentences together to say more complicated things. So for example, we can conjoin W with another sentence here. Three, water is wet and grass is green. So we can replace water is wet with W and we'll introduce a new sentence letter for grass is green, G.

6:52 And to translate the logical connective and we're going to use the symbol like little carrot. It's called a carrot. It's a it's a little peak. That is how we translate and so water is wet and grass is green is going to be w carrot g. So the next one in addition to conjunction is called disjunction. So disjunction is A or B which will be translated using what's called the wedge. That's the V symbol. It's like an extended V.

7:33 So take the sentence either salamanders eat insects or turtles eat algae. Let S stand for salam salamanders eat insects. Let T stand for turtles eat algae. The translation of six would then be s wedge t. There are two more that we got to go over. First one is what are called conditional claims. These are sort of getting at the logic of if a then b. We're going to translate these using the arrow symbol.

8:11 I'll say a little bit more real quick. Do I have an example here? Let's look at one example real quick. Ignore the rest. Look at seven. Just look at seven. So seven says if umbrellas are open, then visitors stay dry. So umbrellas are open. We're going to use you to stand for that basic declarative sentence. Visitors stay dry.

8:43 We're going to use V to stand for that sentence. And so the way we translate this, as you see right there at 7 L, is U arrow V. So that's what a conditional statement looks like. Let me say one more thing about conditional statements. actually let me let me introduce by conditionals first and then I'll say this one thing. So the by conditional we're going to use the double arrow that it's the arrow that points left and right. It's also called the left right arrow. Double arrow is fine. This means a if and only if b.

9:27 Now, this is a weird turn of phrase that you don't see that often except in legal contexts and other similar sort of places. we're going to use it. We're going to show you the way that people use the logic of statements like this. So, a if and only if b we're going to translate as a double arrow b. So, let me identify something for you real quick. I don't have all of them up here. Okay, I have all of them up here.

9:54 So we have four connectives that take two objects. A and B, C or D. If E, then F, G if and only if H. Right? of these four the bottom four almost all of them are commutative meaning there a and b means the same thing logically speaking as b and a they can be written forwards and backwards in all cases except one the one case in which the left and right directions don't don't always work is the condition the conditional statement the single arrow.

10:48 So the single arrow the because it doesn't it's not commutative because a arrow b means something different than b arrow a we have special names for the thing that goes to the left and for the thing that goes to the right of these of the arrow. So the thing to the left of the arrow is called the antecedent. The thing to the right of the arrow is called the consequent. Anticedent comes first. It's antecedent.

11:18 Consequent comes second. It's the con. It's consequently there, right? You don't have to fully memorize anticedent and consequent. At this point, you will need to when we get to one of the later lectures here, you will need to know what the difference between an anticedant and a consequent is. But for right now just know that there is this distinction kind of sort of get yourself starting to think in terms of it.

11:49 So if we look through all these four these five sentences take four parrots do not understand English. If we take the sentence P to mean parrots understand English then our translation into logic will be tilda p five. Quitting saves effort and reasoning takes time. We're gonna choose letters to stand for the declarative sentences. And then we see that this is an and statement which we will translate using the the carrot the and symbol. Q carrot R. Number six, either salamanders eat insects or turtles eat algae. Again, once we've chosen our sentence letters, we see that this is an or statement, a disjunction. And so what this means is we translate this sentence as t wedge s.

12:42 We already went over seven. If umbrellas are open then visitors stay dry. We have an if then sentence. This uses the arrow again. Anticed and consequent. Let me tell you. So umbrellas are open is the antecedent of this conditional. It comes to the left of the arrow. And visitors stay dry is the consequent. It goes to the right of the arrow. Finally, number eight, water freezes at 0 degrees C at sea level if and only if you use the Celsius scale.

13:22 So this is going to be W double arrow Y. Okay. Now, often we're going to want to translate more complicated English sentences than what you see here at 4 through 8. We're instead going to want to translate sentences like this. If both Alice and Bob are invited, then Carol will attend. Right? This is the sort of thing we say all the time. This sentence has two logical connectives in it. If both Alice and Bob are invited, then Carol will attend. In order to properly translate sentences like these, we need simple rules of grammar that allow us to parse sentences with multiple logical connectives.

14:16 In this case, we need the translated sentence to show that Carol is going to go only on the condition that both Alice is invited and that Bob is invited. So what we'll do is we'll put parentheses around both Alice and Bob are invited in order to indicate that this is a a you a package deal. So the way you should think of this sentence is if both Alice and Bob are invited then Carol will attend or A and B goes in the parentheses A wedge sorry A carrot B arrow C.

15:07 So what we see here look at 9L right there. So 9 L the antecedent of that conditional right it's a conditional statement because it has an arrow the consequent is C the antecedent of that is the whole chunk A and B. So when you have grammatical sentences like A and B that is going to behave like a single unit. So a and b in parentheses that whole thing is the antecedent.

15:46 We can treat it as if it were just any old sentence letter. So we're left with 9 L here. This allows us to build more and more complicated translations by working piece by piece using parentheses as a way of capturing the logical form of English sentences. Additionally, not every string of symbols in L counts as grammatically correct. We're going to have two rules of grammar.

16:24 Here is the first rule. The tilda can only precede a sentence letter or a parenthetical sentence including already negated sentences. Okay, so let me say that again. The tilda can only precede can only come in front of a sentence letter or another parenthetical sentence.

16:55 Our second rule is on either side of the wedge, carrot, arrow, and double arrow, there must be a sentence letter or a parenthetical sentence, which could mean a negated parenthetical sentence. In other words, we need a complete unit either after a negated sentence or on both and we need a complete unit on both sides of the carrot wedge arrow or double arrow. So here's a here's an example of one that is not grammatical.

17:43 A carrot B arrow C. This is unrammatical. Why? Because when you look at, for example, the carrot, to the left of the carrot is what? Is a a sentence letter. Cool. That's fine. What's to the right of the carrot? It's B arrow C. That's not a parenthetical sentence. And so, as we can show later on, it's really not clear what this sentence means.

18:16 There are multiple ways of parsing out what this sentence means that mean different things. So we can go through some examples of these grammatical or ungrammatical sentences. Let's just go through the grammatical ones first. Number 10. A that's grammatical because it doesn't violate one of the two rules. Sentence letter is grammatical. 11 is also grammatical. Why? Because it doesn't violate any of the rules. The only rule that applies is rule one, which says a tilda can only go in front of either a sentence letter or a parenthetical sentence. Cool. It does.

19:04 It goes in front of a sentence letter. 12. We already saw this one, but this one is grammatical as well. Why? because it doesn't fail to meet rule two. Because it doesn't fail rule two. So rule two is why can't I speak? Rule two says that to the left of the arrow there needs to be a complete unit. And to the right of the arrow there needs to be a complete unit. To the right of the arrow is C. That's totally fine. We know that's okay.

19:38 On either side of the arrow, there must be a sentence letter or a parenthetical sentence. Okay, cool. To the right of the arrow on 12, there's a there's a sentence letter. To the left of the arrow on 12, there's a parenthetical. A and B. Cool. So, the arrow is totally fine within the sentence. What about the the carrot? Is that a grammatical phrase also? Well, to the left of the carrot is a to the right of the carrot is b. Cool.

20:05 That works also. So number 12 also is grammatical because it doesn't violate the rule anywhere. 13 gets a little crazy now. Okay. So 13. How do we do this? How do we figure out whether this is grammatical or not? Well, here's what I suggest. You work from the outside in. Actually the better way is look at every look at every logical connective and ask whether it violates the the rule or not. So first let's consider the first connective we see that tilda that's out in front. So it says tilda double open parentheses a lot of stuff double closed parenthesis. That's what 12 says. Now, our rule for tilda, let me refresh, says tilda can only precede a sentence letter or a parenthetical sentence. Okay. Does tilda precede a parenthetical sentence?

21:14 Well, what tilda's in front of is a or is parenthesis a or b double arrow parentheses if c then not d. Okay, that is a parenthetical sentence. We can verify that by going through each one. So the double arrow, what's to the left of double arrow? Well, that's A or B. We already know that's appropriate as a sentence. Cool. What's to the right of the double arrow? Well, that's C arrow tilda D. Is that okay? Well, let's look.

22:04 Let's look at the rule. on either side of it, there must be a sentence letter or a parenthetical sentence or a negation of one of those two things. So C arrow till the D. To the left of the arrow is a C. To the right of the arrow is a negated sentence letter. Tilda D. That is okay. according to rule two.

22:36 Additionally, tilda d is okay according to rule one. So this counts as fully grammatical. There's no problems with it. Let's do the ungrammatical ones because that's how you can learn a little bit more. So first off, 14 tilda arrow a. This fails both rules. Remember, rule one says tilda can only precede a sentence letter or a parenthetical sentence or a negation of either of those.

23:06 What does tilda precede? Well, it precedes arrow a that is not either a sentence letter or a parenthetical sentence or a negation of those. So, so 14 fails on rule one. 14 also fails on rule two. Why? Because rule two says on the left of the arrow, there must be either a sentence letter, a parenthetical statement, a negated sentence letter, or a negated parenthetical statement. A negation by itself does not count as one of those.

23:44 So 14 fails both rule one and rule two. What about 15? 15 says, we already went over this one. We don't need to do it again. What about 16? 16's got a parenthesis. Isn't that okay? Actually, no. Here's why. 16 says a wedge B carrot C. Now, look at the two connectives, the wedge. What is to the left of the wedge?

24:16 Well, there's an A. Cool. That's fine. What's to the right of the wedge? Well, it's B carrot C, not in parenthesis. Not a parenthetical statement, not a negation of a parenthetical statement. Same thing with the carrot. To the left of the carrot is a whole sentence, a whole string of symbols. That's not a parenthetical sentence. And so 16 counts un grammatical on this sentence. It fails rule two twice over.

24:48 Number 17. Now let's do so number 17. This one looks fairly promising complicated. So let's go through each of the connectives. The tilda. Well, let's start with the innermost ones, the ones that are like small small ps. So the arrow a arrow b is that grammatically correct? Yeah, a arrow b is grammatically correct. Cool. Now, what about parentheses a arrow bend parenthesis double arrow C carrot D? Is that grammatically correct?

25:30 Well, no, that's not. Why? Because the double arrow here does not have does not satisfy rule two. Rule two says to the right of the double arrow needs to go a parenthetical or a single sentence letter or a negation of one of those things. To the right of the double arrow is not that. It is instead C carrot D. So 17 fails rule two.

26:03 Okay. A concept that is related to this grammatical ungrammatical stuff is the concept of scope. So scope is kind of weird. This so it's logical connectives that have scope. It's a property of a logical connective within a string of symbols in our logical language. So the scope of a logical connective is what it ranges over in some sense. So for negation the scope of the negation is whatever is being denied, whatever is being negated.

26:47 So in for or in until the p the scope of the tilda is just P. It's the thing that it's in front of. In this slightly more complicated sentence, tilda parentheses P wedge Q. The scope of the tilda is the thing it ranges over, which is that whole parenthetical statement P wedge Q. So the scope of tilda is P wedge Q in that one.

27:19 So for these, okay, so that's for negation. Negation is whatever is being negated is the scope. For the compound connectives, these binary ones that take two things, one on each side, the scope will be whatever's to the left of it and whatever's to the right of it. So in Q carrot R the scope of the carrot is Q and R in parentheses A carrot B and parenthesis arrow C. The scope of the arrow is the stuff to the left of it which is the parenthetical statement A carrot B and the stuff to the right of it which is C.

28:06 Okay, so that's what scope is. The concept of scope allows us to define a really important concept for translating sentences, which is what we're walking towards right now. And that is the main logical connective. So the main logical connective of a sentence is the connective whose scope contains the rest of the sentence. So consider the following three sentences A, B, and C. All three of these are grammatical. You can verify this on your own. They don't violate either rule.

28:48 A tilda P wedge Q. What is the main logical connective of this sentence? Well, what we need to do is we need to find the connective that has the rest of the sentence within its scope. for this that is the wedge. The tilda only applies to P. So the scope of tilda is just P. There's other stuff left over. The wedge has to the left within its scope to the left of it till the P to the right of it Q. That's everything. So the main logical connective in A is the wedge.

29:27 What about in B? tilda parentheses P wedge Q the main logical connective here is the tilda right right it's scope the scope of the tilda what the tilda ranges over what's being negated is the whole rest of the sentence parenthesis Q wedge P what about C is tilda big parenthesis little parenthesis P wedge Q and little parenthesis arrow R and big parenthesis.

30:10 What's the main logical connective here? Well, if you look at the arrow, which you might initially be inclined to think is the main logical connective, to the right of the arrow is the R. Cool. To the left of the arrow is the P wedge Q. Cool. That's the whole sentence, right? No, you're missing the negation out front. So, the arrow does not contain within its scope the rest of the sentence. The only connective that contains within its scope the rest of the sentence is the tilda.

30:46 And you can verify this if you'd like, but here's a fun little tidbit. Our rules of grammar, rules one and two, guarantee that every grammatical sentence will have precisely one main logical connective. That's why we have these rules of grammar in fact is so that there can be one main logical connective that tells us what the sentence is really saying.

31:17 Okay, let's now go through a couple of examples. Sorry, before the examples, one last thing on scope and and stuff. So we'll notice that the way we can capture the sense in which these two sentences mean the same thing is one way at least is by thinking about scope. So till the P wedge Q means either not P is true or Q is true.

31:59 Now what this tells us the main logical connective of this sentence is the wedge right and the way that gets represented in the sentence is it's either not P or Q in this second sentence till the parenthesis P wedge Q. This means it is not the case that either P or Q. So we got the same symbols going on except for the parenthesis, but the parenthesis give us different indications of the scope. And so it shows us that there's different claims being made here.

32:42 Okay, let us translate these. So here's what I want you to do. I want you to translate sentences one through five using these sentence letters and I am not going to do that. So what I want you to do in a couple seconds is to pause the video, go through it, make sure you understand these and then hit resume. Okay, pause in three, two, one.

33:17 And we're back. Okay, I am not going to give you the answers to these. They are fairly straightforward. You can get it. If you really need help, type it into chat GPT. Here's some more. These ones are slightly more complicated, but not that much more complicated. Here they are. So, what I want you to do is I want you to use these sentence letters. They're nice and easy to translate these sentences.

33:53 So, again, I'm not going to give you the answers to these. I want you to work through them yourself. and then yeah, so pause if you need to and then I'm going to continue speaking for the rest of the time. there's nothing else going on. I just want to wrap up. Okay, so that I'm assuming you've unpaused by now or you're watching until the end and then you're going to go back. That's also a possibility. so that is the basics of our logical language.

34:33 This is weird stuff. takes a couple tries to get it to crack into the head in the way that makes intuitive sense. again, if this didn't make sense, I would go back and read the chapter, listen to this again, and if none of that works, come talk to me. Happy to help you. Happy to walk through this stuff. It's my favorite thing to do is logic. So, use me as a resource.

35:04 Okay, catch you next time.