 Hello, and welcome back to deductive logic fill 320. I'm Matthew Brown, and today we're going to talk about basic concepts in logic Before we get into the content of today's lecture. I want to share another logic puzzle with you so Have a look at this According to this puzzle the figure shows a small island on which is a tree in the middle of a large and deep lake Which is 300 yards across on the shore is another tree How might someone who is unable to swim with only a length of rope significantly longer than 300 yards long? Get from the shore to the island So that's your logic puzzle today Keep that in mind as we go through the lecture and we'll talk about it again at the end So we're going to proceed today by giving some basic concepts and definitions And applying them to some examples. We're not going to get into any formalized logic today Just going to get into core concepts Okay, we're going to start with I think the most central concept in logic, which is the concept of an argument now when we say Argument We're going to define that to mean a series of sentences any series of sentences will do Although we'll talk a little bit later about what exactly we mean by sentences. It's important But let's just start with any any series of sentences The sentences are going to be typically split up into premises and a conclusion so in the canonical form the the first set of Sentences in an argument are the premises and the final is the conclusion Although we'll see that it doesn't always work that way in ordinary English language arguments Now a good argument gives you reasons to accept the conclusion Any kind of argument that's a good argument gives you some Increased reason to accept the conclusion might not be a knockdown argument But as long as the premises are true that give you a reason to to believe in the conclusion The premises in an argument are the sentences at the beginning of the argument as I said now That's in a kind of canonical form Which we're going to write down when we analyze an argument in ordinary language arguments The premises and conclusions might be in a different order often you'll see a conclusion at the beginning You know kind of like the topic sentence and the can premises come after but We'll see a little bit about why we might or reorganize that when we're doing the analysis of the argument Again in a good argument the premises provide reasons to accept the conclusion Sometimes there are some indicator words or terms that Tell us that a sentence is a premise Here's a long list right if you see words like sense in that Seeing that as indicated by Because for as given that and some of these other words and phrases Those are signals that that sentence might be a premise Now not all premises will have these indicator words But they can be signs that we're looking at premises Conclusion of an argument is the final sentence in an argument again in that canonical form right in Ordinary language arguments the conclusion might be in a different place maybe at the beginning Sometimes although more rarely it might be somewhere in the middle If an argument is a good one we sometimes say that the conclusion follows from the premises And if it's bad we say it does not follow right That is to say that In a good argument the premises should sort of push you to accepting the conclusion Conclusions can also sometimes be Have some indicator words that tell us that we're looking at a conclusion Therefore is our is a sort of our favorite and we're doing when we're talking about logic class, but also thus so hence whence where for consequently when we say Something implies that something else that that something else might be the conclusion We may infer it follows that there are many different words and phrases that might indicate a conclusion and again You might not necessarily see these indicator words, but they can be a clue when you're trying to figure out In an ordinary language paragraph, which is the conclusion Let's look at some examples So in this sentence it says corporate raiders leave their target corporation with a heavy debt burden and no increase in productive capacity Consequently corporate raiders are bad for the business community That word consequently is an indicator word that typically indicates a conclusion Which means that the rest of this argument is a premise or premises Depending on whether we think this sentence is really one or multiple statements So I've got premise here in blue conclusion in yellow. All right. Let's look at another Expectant mothers should never use recreational drugs since the use of these drugs can jeopardize the development of the fetus again Since here is an indicator word, right? Now if we go back and look at the previous the previous Argument right we have two separate sentences grammatically speaking one is a premise one is a conclusion in This argument is just one sentence, right? Can one sentence be an argument? Well Grammatically it might be one sentence, but we really have got two different thoughts Forming a full argument the since indicator word tells us that this second half is actually our premise and the first part is the Conclusion, right? So what we want to convince you of is that expected mothers should never use recreational drugs and The premise that comes after the sense about the development of the fetus is supposed to give us a reason to believe the conclusion So we see already here that sort of grammatical sentences and logical sentences don't always line up exactly the same Let's look at a longer example The space program deserves increased expenditures in the years ahead Not only does the national defense depend upon it, but the program will more than pay for itself In terms of technological spin-offs Furthermore at current funding levels the program cannot fulfill its anticipated potential. Okay Take a minute pause the video if you like and try to identify which sentences are the premises and which sentences the conclusion Okay Let me start by using some colors to separate out different sentences here Now you notice for example the first grammatical sentence, that's one sentence I've used the green and orange here to indicate that there really two separate considerations Being suggested here, which we might treat as different sentences and the purple is yet another one, right? We don't have a lot of indicator words here to help us parse this So we have to kind of think about what what we're trying to argue in favor of and Here, I think it's pretty clear that this first the first part in yellow is really the conclusion That the argument is trying to convince us of so we can rearrange this in a more canonical form We have three main premises and one conclusion Okay So that's kind of how we start to break down English language arguments analyze them To understand their logical structure Now remember I said an argument is a series of sentences, but Grammatical sentences don't always line up with logical sentences For our purposes in this class, we're gonna use a pretty restricted definition of sentence A sentence is the kind of thing for our purposes that can be true or false that can be asserted or or denied Accepted or rejected When the book and when I in lecture say sentence typically we mean a declarative sentence right one that makes a Statement and indeed some logic textbooks use the word statement instead of sentence for this Other kinds of sentences like questions or interrogative sentences imperatives or commands Exclamations, right? Those are not sentences in our sense in the logical sense Let's look at some examples. So here's three sentences Istanbul was Constantinople that that could be true. That is true, right? That's a sentence Now it's Istanbul not Constantinople It's very similar to the first sentence, but it's definitely a sentence whereas Take me back to Constantinople is Is is not something that can be true or false It's a it's a command, right? Take me there, right? That's not a sentence not the way we mean in this class, right? Look at these other two Sentences or these are the two lines four and five and Think about whether they Counted sentences in positive video if you need to Do you have it so number four a sentence you cannot go back to Constantinople is Is a sentence Why did Constantinople get the works as a question? It's an interrogative sentence in grammatical terms But it doesn't state something. It doesn't make a claim. It doesn't assert or It doesn't it doesn't say something that could be true or false, right? So that's how we're gonna use sentences in this class Next I want to talk about the term validity, which is our kind of key way of Analyzing arguments, right? We say that and when we talk about what's valid we mean deductively valid and sometimes we'll use that phrase Deductively valid sometimes we'll just say valid An argument is valid if the conclusion must be true If the premises are to so to say an argument is deductively valid is to say that the conclusion must be true if the premises are An argument can be valid even if the premises are false, right? what we're interested is in What would happen if the premises were true would the conclusion have to be true and By contrast we say that an argument is invalid if it is possible for the premises to be true and the conclusion to be false It now it's possible that you have an argument all the premises are true and the conclusion is true Happens to be true but the argument is still invalid because it's Possible that it could be otherwise that it could be that all the premises are true and the conclusion is false It's the form of the argument that tells us whether it's valid or invalid Not whether the premises happen to be true or the conclusion happens to be true Now in contrast to deductively valid arguments we Often make inductive arguments an inductive argument is one that doesn't guarantee the truth of the conclusion But that makes the truth of the conclusion more probable, right? often inductive arguments generalize from the premises to a more General and stronger conclusion. These arguments are not deductively valid But they can still be good arguments in many contexts many scientific arguments are inductive in nature Here's an example of a kind of interesting inductive argument It goes like this the meerkat is closely related to the suricat the suricat thrives on beetle larvae therefore Probably the meerkat thrives on beetle beetle larvae, right? This Argument is not deductively valid. It is a kind of argument by analogy That makes the conclusion more probable, but not it doesn't guarantee that that's that's what the meerkat eats, right? So it's a kind of inductive argument This is often a reliable form of reasoning and it's often used in science and other places But it is valuable and it needs to be checked, right? You don't you can't be sure, right? The second kind of argument let's look at this the meerkat is a member of the mongoose family All members of the mongoose family are carnivores therefore the meerkat is a carnivore This argument is deductively valid If both the premises are true, then we know the conclusion is true, right? So we say that's a deductively valid argument Here's another example, I want you to think about whether this example and then in the next couple of examples I give Are deductively valid? arguments inductive arguments Invalid arguments that aren't inductive or not even an argument, right? So this one says Bergen is either in Norway or Sweden if Bergen is in Norway Then Bergen is in Scandinavia. If Bergen is in Sweden, then Bergen is in Scandinavia therefore Bergen is in Scandinavia Take a second and think about it This is also a deductively valid argument If all of three premises are true the conclusion has to be true. There's no other possibility, right? Let's look at the next argument It is snowed in Massachusetts every December in recorded history Therefore it will snow in Massachusetts this coming December Pause and think about it if you if you need to Figure out which kind of argument this is It's an inductive argument. We generalize from some specific cases all of the December's in history To another case, right? This coming December It doesn't guarantee that it's true, right? It might be freakishly warm this December In Massachusetts and no snow, right? But it does make it pretty probable if it's always happened before it's likely to happen again, right? Here's another one all odd numbers are integers all even numbers are integers therefore all odd numbers are even numbers kind of argument is this This is an invalid argument. It's not an inductive argument. There's no generalization It's not even a good argument. It's a very bad argument, right? So we just say it's an invalid argument not an inductive not even an inductive one Here's our last here's our last argument Place your cursor in the file at the spot where you want to insert the symbol on the insert tab click symbol If you see the symbol you want listed in that gallery just click it to insert Otherwise click more symbols to open the symbol dialog box Scroll up or down to find the symbol you want to insert when you find the symbol you want double click it What kind of argument is that? Pause if you need to think about it. This is not an argument at all Why because it is not made up of sentences not made up of sentences in our sense of Statements or declarative sentences. These are all imperatives. These are all commands. They don't have truth Values, right? They can't be true or false So I just use this term truth value. It's a really important one for understanding sentences And the properties of sentences, right? so we say that the the truth value is that is The property a sentence has of being true or false if a sentence is true We say that its truth value is true What's the use of that? Well, it's a way of talking about a statement's truth or falsity Without knowing which it is now. It's worth taking a second to to talk about the word truth and how we use it in this course Truth is sometimes a kind of intimidating term In its sort of deep sense truth with a capital T We say sometimes the kind of truth that philosophers and theologians worry about It generally doesn't have a ton to do with what we're talking about in this class, right? We could we don't even have to talk about truth We could use ones and zeros in place of truth and false falsity to talk about the truth values of statements Typically what we mean by by truth is just you know, you know, is it so is it a fact? Do we believe it? Do we accept it? You know, we Nothing too deep About the concept of truth as we use it here, right? So we talk about truth values of different kinds of sentences We have contingent sentences. They could have either truth value. They could be true or false. It depends on Some facts outside of the sentence itself, right outside of the form of the sentence itself Maybe it's empirical facts. Maybe it's some other other kinds of information We need we don't can't just look at the sentence and know that it's true or false You might say well is that's all sentences, right? No, there are some for example What we call tautologies that must be true in virtue of how they are written, right without any reference to any other underlying facts or external information We sometimes call these logical truths because they necessarily have the truth value of true And sentences that Necessarily have the truth value of false we call contradictions a contradiction must be false just in virtue of how it is written Without reference to any kind of facts. So we sometimes call these logical falsehoods Let's look at some examples. So We say if it is raining or it isn't right That's a tautology, right? That is we don't have to go outside and look to see Whether it's raining or not We don't have to check the weather report to know whether sentence one is is true It has to be true, right? Either it is raining or it is not, right? There's no middle possibility You might say well, what if it's kind of like drizzling or misting? Either drizzling or misting count is raining or they don't, right? So we might we might Argue about how the term applies, but either the term applies or it doesn't, right? It's a tautology So this number two if it's raining than a disnowing You might say well, I mean is that a contradiction when it's not it's it's contingent It it might be false. In fact, I think it is false, right? It's not always the case that it's raining and it's snowing If maybe possible in some cases Wintery mix is that what they call it in the weather? but it's raining Doesn't always go along with it's snowing. So it's a contingent statement and maybe it's false But that depends on like meteorology or other kinds of information about weather phenomena Here are some more examples, why don't you pause and do these on your own think about them for a moment on your own? Okay, do you think you have it? Let's look at these other examples number three says If it is raining, I'll wear a raincoat, right? That's contingent. Maybe I'll wear a raincoat. Maybe I won't It it's you can't tell by looking at the sentence whether it's true or false Number four it is raining and it is not the case that it's raining Both of those things both of those parts of the sentence can't be true at the same time So the same reason that number one is a tautology number four is a contradiction, right? There's a there's a this doesn't matter it could be it could say anything it It is it is snowing and it is not the case that is snowing We could substitute any term for raining and it would still be a contradiction, right? It's by the form of the sentence that we know it's false Number five is a tautology if it is raining then it is raining, right? It's always the case that if the first part of that sentence is true The second part is true and we know therefore that it is a tautology It's not possible for this sentence to be false Lastly If it is raining then it is not raining, right? That might seem like a contradiction But actually it's contingent, right? This is a little bit of a trick question It's it's based on how we kind of interpret if then sentences and if then statements If it is raining then it is not raining So if it happens to be the case that it's raining right now, right? Then we know the sentence is false, right? Because it can't be raining and not raining at the same time but If it is not raining right now It's not clear how to evaluate the sentence and for reasons that we'll get into in the next unit The way we interpret the sentence is that if it is not raining the whole sentence is true, right? So it's a contingent sentence If that doesn't make a lot of sense to you, don't worry. We'll get there, okay? Logical equivalence is Not a property of a sentence but a property of two or more sentences not an argument per se just a collection of sentences, right? Two or more sentences are logically equivalent if they must have the same truth values in all situations That's how we define logical equivalence It's helpful to think about the notion of truth value in General because it we can talk about logical equivalence without knowing whether the sentences are true or false in particular cases now All tautologies are logically equivalent to each other because they're always true all Contradictions are logically equivalent to each other because they're always false But there are actually some contingent sentences that are always that are that are logically equivalent that are The sentences could be true or false, but they're always true or false in the same situations, right? Another property that two or more sentences could have is that they could be consistent or they could be inconsistent Consistent sentences could possibly all be true at the same time. There may be other situations in which Some are true and some are false, but there is some Possible situation where they're all true at the same time that it's possible, right? We don't know if it's true or not A set of sentences is inconsistent if they could not all possibly be true at the same time Maybe they're all false maybe some are true and some are false, but they can't all be true at the same time, right? Let's look at some examples So number one is two sentences and it's a it is raining since be it is not raining Both of these are contingent sentences and They are inconsistent if a is true then B is false if B is true then a is false. There's no There's no way that they can both be true at the same time Example number two A says it is raining B says it is sunny. Are those inconsistent? consistent or logically equivalent They're consistent, right? You might have been outside on a sunny day Where it's also raining. Maybe the clouds only cover part of the sky and the sun is shining while the rain is falling It's unusual, but it is consistent Even if it never happened in the real world, it would still be consistent because you'd need to again to know some facts about the weather To know that they that they weren't both true at the same time It's nothing about the logical form that guarantees that they're inconsistent Why don't you pause the video and think about three four and five? Let's see number three says A says if it rains, I will wear a raincoat B says if I do not wear a raincoat, then it is not raining these two sentences are Equivalent they're logically equivalent ever in every case where if it rains, I will wear a raincoat is true If I do not wear a raincoat that it is not raining is also true and every every case in which a is false B is also false, right? What about number four number four is Kind of depends on the meaning of the words comprise and compose That is if a comprises B then B composes a Those are complementary terms You might say well, isn't that some outside factual information? I need to know but we typically you know We acknowledge that that kind of that kind of word the is relevant to the logical meaning, right? So we would say these are equivalent, right? To say a house comprises several rooms is to say a house is made up of several rooms to say several rooms Compose a house is to say that several rooms make up a house, right? And so made up by and make up they go together, right? What about number five a this three sentences here? They say it is warm. It is cool. It is tepid, right? Are those consistent inconsistent or logically equivalent? You might think it is consistent, right? If you don't know anything about temperatures You don't know whether it's it's true. Maybe, you know, it feels warm to me It feels cool to you feels tepid to Alice You know, they're consistent You might think that Warm cool and tepid are kind of mutually exclusive terms just like raining and not raining And that might make you think these are inconsistent, right? Not on the basis of Facts about thermodynamics or whatever but just in terms of that the terms are kind of Contraries or compliments to each other. They're mutually exclusive It's a bit of a trick. I mean it depends on how we interpret these terms I think it's probably best to see these as consistent, but it would be possible to represent them In a in a logical form that that makes them mutually exclusive in which case we would have to say they're inconsistent I just use a concept of logical form So let's talk about that logical form is the part of a sentence or an argument that has to do just with its logical structure Not its content, right? just just the the part of the The argument or the sentence that has to do with concepts like truth value logical consistency, etc That don't have that we don't have to look outside of the sentence to understand that that Really gets at this sort of formal structural information, right? and Then a formal language is a language Typically an artificial human created language that tries to capture the logical forms of everyday language While extracting Everything else away everything has to do just with the content is abstracted away Formal languages are usually made up of things like atoms connectives inference rules There are many different formal languages out there There's Categorical or syllogistic logic sometimes called Aristotelian logic is a very simple formal language and very old The two formal languages we will use in this class are Called in the book sentential logic or SL and quantified logic or QL all in other in other In other books or by other people they might be called propositional logic and predicate logic or first-order logic There's just a different names for the same type of formal language. There are other formal languages modal logics deontic logics Mathematics Is typically a formal language formal mathematics made up of formal languages No formal language captures all of the interesting formal logical formal structure In ordinary language or or other kinds of discourse The reason there are many different kinds of formal languages is that each captures different kinds of formal structures The formal lot languages are kind of models of language or models of discourse. They're abstractions That are useful for different purposes But there is no kind of one best formal language that serves all purposes equally well An important property of formal language That some formal languages have the formal languages we look at in this course have is by valence We say a formal language is by valent if No, if every sentence has a single truth value That is either true or false, right? In a by valent language, no statement is neither true nor false No sentence is both true and false. It has to be one or the other Not both not neither. Those are all the options. Those are both those two are the only options right Not all formal languages are by valent They're trivalent. They're they're they're para consistent They're all kinds of other formal languages that have different properties not by valence By valence is a kind of you know simplifying assumption But it's a powerful one. It's it's an important one And it's we'll see ways in which it's useful. It helps us Understand a lot about the kind of arguments we typically make in a lot of ordinary situations Contingent sentences in a formal language might take on different truth values Depending on other facts, right in different situations, maybe but they can only have one true value at a time That's the assumption of by valence So those are the key concepts that were we I wanted to cover today that are help set the stage for the Investigation of formal languages that will do in subsequent units in this class If you have any questions about what we talked about today, please feel free to reach out to me come by my office hours Send me an email or a message Or you could pose the question to on discord If you don't if you is a concept in this lecture that you're struggling with you could discuss it on discord With other students and I will weigh in there as well might help sort of Other students might have a way of putting it that could help you if you pose the question the right way Before we end today, I want to come back to our our logic puzzle, right? So let's have another look at this Did you figure it out? Once you have a guess, I'm not gonna again, I'm not gonna tell you today, which it is But once you have a guess, I encourage you to Go to the logic puzzles channel on our class discord server and share your solution But make sure you use the spoiler tag on discord to to Sort of mask your solution so that other people have a chance to guess before they see Your answer to the logic puzzle, okay So that's our discussion for today, I hope that that was helpful for you and I wish you good luck with With the quiz with the practice problems and the and the exam for for this unit And I look forward to To seeing you in the lecture for the next unit All right. Bye