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0:05
Samantha Emswiler: future logicians, back to understanding a little bit more about logic. Here is on how to categorize deductive arguments and how to provide Samantha Emswiler: Rebuttals that are quick and efficient.
0:23
Samantha Emswiler: So how to categorize and rebut deductions is the topic.
0:28
Samantha Emswiler: We're going to just start with reminding you what the goal of logic is. The goal of logic is to preserve truth and inference. Ideally, if the premises are true, they offer complete or partial support for the truth of the conclusion. Samantha Emswiler: Whether they're complete, offer complete, or partial depends on what type of argument it is. So we're going to start with talking about deductive arguments. Samantha Emswiler: Here's an example of a deductive argument.
0:56
Samantha Emswiler: No plants or tacos. Samantha Emswiler: Some green things are plants, Some green things are not tacos. Samantha Emswiler: You'll be learning much later in the semester how to use VIN diagrams like this in order to express Samantha Emswiler: simple con- simple claims, like, in this case, no plants are tacos. In other words, nothing can exist between this overlap. Samantha Emswiler: Between, plants and tacos. Samantha Emswiler: And there is something…
1:30
Samantha Emswiler: That is green and is a plant, so something exists, between plants and green things. Samantha Emswiler: So when we consider what the relation between green things and tacos is, we find that there is something Samantha Emswiler: That is a green thing! Samantha Emswiler: But not a… Not a taco. Outside of the realm of tacos. Outside of the circle.
1:54
Samantha Emswiler: Okay, so this is just giving you a visual diagram to consider the claims. You'll also be learning other ways to be able to deduce, whether an argument is good or bad.
2:08
Samantha Emswiler: Given the standards of logic.
2:14
Samantha Emswiler: Boop! Went too far. There we go. So there are two main types of arguments.
2:19
Samantha Emswiler: differ on the nature of the purported inferential link, okay? So deduction, the purported inferential link. Samantha Emswiler: is necessariness, okay? So if supremacies are true, it is necessary that the conclusion is true, is the inherent claim of the inferential link of necessary or deductive reasoning. Samantha Emswiler: Okay. Samantha Emswiler: So the conclusion must be true if the premises are true, is the idea.
2:49
Samantha Emswiler: Induction offers what we call probabilistic reasoning. Samantha Emswiler: This is wherein, if the premises are true, it offers partial support, not absolute or complete support. Samantha Emswiler: Of the truth of… of the likelihood of the truth of the conclusion. Samantha Emswiler: A more precise way of thinking about the difference between deduction and induction is to think about the nature of the conclusions. Samantha Emswiler: So, deduction is non-ampliative.
3:23
Samantha Emswiler: Whereas induction is amputative. So, ampliative means Samantha Emswiler: that the conclusion contains more information than the premises. So in induction, you learn more about the world if it is truth-preserving in inference, whereas with deduction, if you know what the value of the premises are, okay. Samantha Emswiler: Then you can deduce the conclusion, necessarily, from that.
3:54
Samantha Emswiler: So the deduction doesn't contain any more information in the conclusion than was present, given just the premises.
4:07
Samantha Emswiler: Deductions are evaluated as either invalid or valid. I'm always going to start with talking about invalidity, because that's what we test when we test, deduction. We test to see if it's invalid. If it's not invalid, it's valid. Samantha Emswiler: So, invalid means that an argument Samantha Emswiler: Exists where it's possible for the premises to be true while the conclusion is false.
4:36
Samantha Emswiler: So just that it's possible that if supremacies are true, there's still an arrangement or possibility wherein the conclusion is false. Samantha Emswiler: That's an invalid argument. Samantha Emswiler: A valid argument is one wherein that's not possible. It's not possible for the premises to be true while the conclusion is false. Samantha Emswiler: Sound just means a deductive argument that is valid with all true premises.
5:03
Samantha Emswiler: Thus, unsound means either it's invalid, or it has at least one false premise, or both.
5:13
Samantha Emswiler: So, we're gonna go through some very basic types of deductions. Samantha Emswiler: We're gonna start with, argument by definition. Samantha Emswiler: So, for this, structurally, what you'll often see is just one premise and one conclusion for an argument by deduction. Samantha Emswiler: And the entire inferential link just revolves around the, words or phrases being synonymous, okay? Samantha Emswiler: So, it's based on the definitions agreed upon for the word.
5:44
Samantha Emswiler: Look for synonyms in premising and conclusion. Samantha Emswiler: Again, look often for one premise and one conclusion.
5:53
Samantha Emswiler: So, Lily is a philologist, therefore she studies the development and history of texts.
6:01
Samantha Emswiler: Second example, theft, by legal definition, requires intent to deprive permanently Samantha Emswiler: Therefore, intention is necessary for stealing legally.
6:15
Samantha Emswiler: So this is just operating from the definition to a necessary conclusion, given if that definition is true.
6:25
Samantha Emswiler: Now, the next set are logical terms, okay? These operate in an agreed-upon way. I like to think of these as functional words, okay? Samantha Emswiler: So, one of the ways that they function is to categorize. Samantha Emswiler: So, categorical reasoning is based on inclusion or exclusion from groups or categories, okay? Samantha Emswiler: Categories are just simply, things that are similar or have, properties in common, okay?
7:01
Samantha Emswiler: So, indicators of reasoning categorically include words like all, each and every, no. Samantha Emswiler: Not all, some, some are not. Only and accept.
7:17
Samantha Emswiler: So, here's a, argument, a categorical argument. Samantha Emswiler: All humans are mammals. Samantha Emswiler: How mammals are vertebrates, thus all humans are vertebrates. Samantha Emswiler: Here's another example of categorical reasoning. Samantha Emswiler: Notebooks are immense! Samantha Emswiler: Conclusion, all immense things are not books.
7:39
Samantha Emswiler: So, these are the keywords you're going to look for for, indications of categorical reasoning, but also look for indications of categories as well.
7:52
Samantha Emswiler: Now, I want you to conceptualize early on how these words function. Samantha Emswiler: Because you use them intuitively often very well. We have what we call a natural deduction system, so we sort of inherited ways of thinking about this, and some of us are precise in our… in our understanding of it, though we've never been given the terminology to explain it.
8:18
Samantha Emswiler: So, I want you to think for yourself, so this is active reflection. What do these logical terms mean? Samantha Emswiler: What does accept mean?
8:29
Samantha Emswiler: What about unless?
8:32
Samantha Emswiler: And finally, ALL. Samantha Emswiler: How are they related to addition? Samantha Emswiler: Subtraction, multiplication, Division, or other mathematical ideas. Samantha Emswiler: So, think for yourself. I want you to imagine and conceptualize how these logical terms operate, or what they mean. Samantha Emswiler: Does not all mean the same thing as no? Samantha Emswiler: Explain!
9:04
Samantha Emswiler: So, that gives you an idea about, categorical reasoning. Next, we're going to do disjunctive. A disjunction is simply an either-or sentence, okay? Either he or works like a both, both-and, okay? So, it's just an indicator of disjunctive reasoning. Samantha Emswiler: So normally it's just one single Samantha Emswiler: what we call a disjunctive syllogism. So it's got two sentences, okay, that are premises and a conclusion. Samantha Emswiler: it goes like this, either A or B,
9:38
Samantha Emswiler: A, therefore B, or it could go the other way. Samantha Emswiler: either A or B, not B, therefore A. I hope I did the other one that way. So here's my example in… without using letters to substitute for concepts. Samantha Emswiler: Either you eat pie or cake, But you don't eat cake. Samantha Emswiler: Therefore, we can conclude, you eat pie.
10:04
Samantha Emswiler: For this, just look for the word OR. We just don't have a lot of other words for that in the English language, that's a key word. And look for the word not, because the second sentence is a negation.
10:20
Samantha Emswiler: But negation, we just mean, literally uses the word not.
10:25
Samantha Emswiler: Hypothetical reasoning is based on how a conditional, or a hypothetical, an if-then sentence works. Samantha Emswiler: For example, if there are no clouds, it won't rain. Samantha Emswiler: There are no clouds, therefore it won't rain, okay? Samantha Emswiler: This is a hypothetical syllogism, just as the other one was a disjunctive syllogism. It has two premises and one conclusion.
10:54
Samantha Emswiler: Now, I want to get into conditional sentences a little bit before we move on, because these are very important to logic, and we'll spend a lot of time with them later. Samantha Emswiler: So I want to sort of investigate the nature of them a little bit. Samantha Emswiler: Edith and sentence… Like, if it rains tonight, then either the rain will leak. Samantha Emswiler: Oh, sorry, the roof will leak, or the roof will hold. Samantha Emswiler: We have… Types of conditions here.
11:24
Samantha Emswiler: We have both necessary conditions and sufficient conditions. Samantha Emswiler: Okay? So, what are these different types of conditions? A necessary condition is one that must be satisfied for a statement to be true. Samantha Emswiler: For example, if you're gonna get your driver's license, a necessary condition for you to get that is for you to pass a driver's test. Another way to put it is, if you didn't pass the test, you can't get your license, okay?
11:51
Samantha Emswiler: Now, a sufficient condition goes, operates a different way.
11:57
Samantha Emswiler: It's one that, if satisfied, assures the statement's truth. Samantha Emswiler: So, it's enough that if you meet this condition, then you find another statement to be true. So, for example, a sufficient condition for being a dog… animal is being a dog. Samantha Emswiler: Okay? A sufficient condition for being an animal is being a dog. It's enough you're a dog… Samantha Emswiler: To qualify you as being an animal.
12:26
Samantha Emswiler: Now, it wouldn't work the other way, right? If you were a, Samantha Emswiler: Animal, that doesn't mean that's sufficient for you to be a dog.
12:38
Samantha Emswiler: So, I want you to practice this. Samantha Emswiler: And we'll… we'll return to these, conditions later and just, work through them when they pop up logic… in our logical topics more. Samantha Emswiler: So this is your first time working with necessary and sufficient conditions. So fill in the blank with the correct type… correct type of condition. Necessary or sufficient. Samantha Emswiler: Being 18 years old is a blank condition for being older than 16.
13:12
Samantha Emswiler: Being at least 35 years old is a blank condition for being president. Okay, so let's do the first one. Samantha Emswiler: If you said being 18 years old is a sufficient condition, you are correct. Samantha Emswiler: It's enough that you're 18 to be older than 16. Samantha Emswiler: But it's not necessary. You could be 17, you could be 19, okay? Being 18 is a necessary condition for being older, but it is sufficient.
13:41
Samantha Emswiler: Whereas being at least 35 years old is a necessary condition for being president. Samantha Emswiler: Now, try on your own. Being an opossum is a blank condition for being a marsupial. Samantha Emswiler: And finally, being a mythical creature is a blank condition for being a unicorn. So, which way does it work? Okay? Samantha Emswiler: Is the concept of marsupial inclusive of a possum? Samantha Emswiler: Okay, or is the idea of a possum… Inclusive of marsupial.
14:19
Samantha Emswiler: Okay.
14:26
Samantha Emswiler: Finally, we have mathematical reasoning. This is based on mathematical relations. When we are talking, and logic really comes at the intersection of this, we can use language to represent reality. Samantha Emswiler: We can use other things, we can use music, we can use art, we can use filmmaking. Samantha Emswiler: But we can also use math to describe and represent reality, okay?
14:52
Samantha Emswiler: But often these two are related, but not quite clear what the relation between mathematical reasoning is and reasoning and language, okay? But where that overlap occurs, often this is where logical terms pop up, because they operate. Samantha Emswiler: In a similar way, analogically, to, to mathematical relations.
15:20
Samantha Emswiler: So, if you see anything that's algebra, arithmetic, or geometry. Samantha Emswiler: Then it's mathematical reasoning, most likely. Samantha Emswiler: So, for example, if a classroom must be redivided from two classrooms to three. Samantha Emswiler: And given an equal desire… a desire for equal distribution of 24 students, one must conclude that each classroom Samantha Emswiler: Should have 8 students.
15:45
Samantha Emswiler: Here, if you're not sure if it's mathematical, check to see if you're doing mathematical, the inferential link in your head, and it's a mathematical inferential link to see if it's mathematical reasoning.
15:57
Samantha Emswiler: Again, a syllogism is a two-premise, one-conclusion deductive argument. Samantha Emswiler: Here's an example. If it is alive, it is made of carbon. It is not made of carbon, therefore it is not alive. Samantha Emswiler: Okay? This is an example of, conditional reasoning again.
16:18
Samantha Emswiler: Now I want you to practice, conceptualizing these different types of deductive reasoning. Samantha Emswiler: It's either disjunctive, categorical, hypothetical, mathematical, or by definition. So, one of those five types.
16:35
Samantha Emswiler: If you… if you're unsure, feel free to go back and look at the earlier, indicator words and indicators of inferential links.
16:48
Samantha Emswiler: If feet had wings. Samantha Emswiler: Pigs could fly. If pigs could fly, I'd own a farm called Where Pigs Fly. Thus, if feed had wings, I'd own a farm called Where Pigs Fly.
17:05
Samantha Emswiler: Next one? Samantha Emswiler: If I wish to withdraw from a class, then I'll complete a registration schedule change form. I will not complete a registration schedule change form, therefore I do not wish to withdraw from a class.
17:25
Samantha Emswiler: Some cats are not gray. Samantha Emswiler: All gray things are colored. Ergo, some cats are not colored.
17:38
Samantha Emswiler: A circle is not the same type of shape as a square. Though both are geometric figures. Samantha Emswiler: A circle is not a polygon, while a square is one.
17:53
Samantha Emswiler: Okay, next one. Samantha Emswiler: Again, your possibilities, math, definitions, categorical, disjunctive, or hypothetical. Samantha Emswiler: Bernie or Amir cook tonight? Bernie does not cook tonight, so Amir cooks tonight.
18:09
Samantha Emswiler: All garlic is tasty. Samantha Emswiler: Some plants are not tasty, hence not all plants are garlic.
18:20
Samantha Emswiler: Given that a citizen is a natural-born resident of a locale, and a denizen is an alien who is made into a citizen through some process. Samantha Emswiler: Tabor is a denizen of Spain, but not yet a citizen.
18:40
Samantha Emswiler: Okay, next, if there was a hurricane, then there is no school today. Samantha Emswiler: There was a hurricane, therefore there is no school today.
18:52
Samantha Emswiler: Because triangle A is congruent with triangle B, and triangle A is isosceles, it follows that triangle B is isosceles.
19:04
Samantha Emswiler: Next, Adele trespassed. Samantha Emswiler: So, she intentionally entered someone's land or property without consent or necessity.
19:18
Samantha Emswiler: Now, we talked a little bit so far about rebuttals. Samantha Emswiler: And we're just gonna go over two very quick, sort of, intuitive rebuttal methods. And by intuitive, I mean creative, okay? Samantha Emswiler: And by creative, I mean that you come up with an example Samantha Emswiler: That goes against an argument by either demonstrating that the argument has true premises with a false conclusion. Samantha Emswiler: Given the way the argument is, itself.
19:51
Samantha Emswiler: Or you can do what's called a counter-argument method, wherein you create an argument of the same form to demonstrate that it's possible the premises to be true while the conclusion falls.
20:03
Samantha Emswiler: So these are rebuttal methods that offer quick ways to assess deductive strength. Samantha Emswiler: by testing invalidity. So if you can find an example where the same structure of argument Samantha Emswiler: or the same argument can't have true premises while you have a false conclusion and you've demonstrated invalidity. Thus, this is not a good deductive argument.
20:28
Samantha Emswiler: So, let's talk about the counterexample method first. I just want to say, you kind of intuitively do this when people make claims. Samantha Emswiler: So if I… I said something like, all ravens are black, you might immediately want to look for a counter instance in the world, look for an instance or an example that is contrary to my general claim.
20:51
Samantha Emswiler: So if you say, okay, well, here's an albino parrot, that's a counterexample method. You are demonstrating the falsity of the claim. Samantha Emswiler: By saying, if this were true, then this particular occlusion, conclusion would, Samantha Emswiler: would also be true, and it's not. The particular conclusion being this. Samantha Emswiler: Raven is also black. When this raven is not black, it's albino.
21:16
Samantha Emswiler: Okay, so here, counterexample method tests to see if premises can be true, while conclusion false, by assuming the logically possible situation that fits. Samantha Emswiler: And this is where it's creative. You have to come up or create your own instance of when the conclusion is false and the premises is true. So here's an example. Samantha Emswiler: Argument. If it has… if it is a square, then it has four sides.
21:44
Samantha Emswiler: And let's say it has 4 sides, so assume those are both true.
21:51
Samantha Emswiler: The conclusion is, thus it is a square. So, what you want to do is you want to assume it's true that if it were a square, it has four sides. Samantha Emswiler: And you also know it's four sides. Samantha Emswiler: But in this case, it's not a square. Samantha Emswiler: So, one thing you can imagine would be a rectangle, something that has four sides, but isn't necessarily a square. Samantha Emswiler: I should say non-square or practical there, to be specific.
22:17
Samantha Emswiler: Sup! Samantha Emswiler: I want you to identify a correct counterexample. Samantha Emswiler: If there's a hurricane, then the school is closed. The school is closed, therefore, there's a hurricane.
22:34
Samantha Emswiler: So let's go through the possible ones. A, there is no hurricane, but school is closed for the holidays.
22:42
Samantha Emswiler: B, there is no hurricane. Samantha Emswiler: But the school is closed because of flooding. Samantha Emswiler: See, there is a hurricane, but the school is closed because of flooding.
22:54
Samantha Emswiler: A and B. Samantha Emswiler: Or A, B, and C.
23:02
Samantha Emswiler: Okay, I'm gonna stop and give you the answer for this one. Samantha Emswiler: Okay, to make sure that you understand why this is the case. Samantha Emswiler: If you picked D, A, and B, you are correct. Oop! Samantha Emswiler: Clicked on it by accident, my apologies. Samantha Emswiler: So, here we go, this is why.
23:23
Samantha Emswiler: The conclusion says there is a hurricane, so in order to create your counterexample, you have to assume the falsity of the conclusion, so you have to assume there is no hurricane. Samantha Emswiler: Okay? Samantha Emswiler: But you also have to assume that first, if there is a hurricane, school's closed, which is… Samantha Emswiler: Okay, because there wasn't one. Samantha Emswiler: And you also have to assume the school is closed.
23:44
Samantha Emswiler: So, in A and B, you assume both that there is no hurricane, so you assume the falsity of the conclusion. Samantha Emswiler: While preserving the truth of the premises, and coming up with a way that it makes sense that it could be the case, that's possible logically.
24:02
Samantha Emswiler: Now, the counterexample method Samantha Emswiler: I'm just going to introduce it here, but we won't use it necessarily as much as we will when we're actually working with deductive arguments in full later. This is just an introduction to deduction, and thus this is an introduction to a technique that we'll use throughout. Samantha Emswiler: So, for counter-argument method.
24:26
Samantha Emswiler: What we're gonna do is we're gonna look for the structure of the argument, and we're gonna learn for different types of arguments, different structures. We call our structures forms. Sometimes we call them arrangements, but we can just call them forms right here, okay? Samantha Emswiler: So… If we are doing a counterargument, what we're trying to do is we're trying to look for Samantha Emswiler: The underlying form or arrangement of the parts of the argument.
24:55
Samantha Emswiler: So, here's an invalid argument that sounds valid, okay? Some fruit are green. Samantha Emswiler: Some fruit are apples, thus, some fruit are green. Samantha Emswiler: Apples. Samantha Emswiler: Okay, they're both green and apples. Samantha Emswiler: Because those three sentences are all true in the natural world. Samantha Emswiler: It's hard to conceptualize, using that, how it couldn't… could be possible that an argument of this form has true premises but a false conclusion.
25:26
Samantha Emswiler: So, we're gonna keep fruit in there, and we're gonna switch green and apples to bananas and pears. Samantha Emswiler: So here, this is true. Some fruit are bananas. Samantha Emswiler: And it's also true some fruit are pears. Samantha Emswiler: conclusion, some fruit are bananas and pears, that's false. So here we have the same arrangement of the parts. Samantha Emswiler: We could call it, SUM, ARB, some ARC, some ARB and C.
25:56
Samantha Emswiler: Okay, that would be the way to do it, to show the form. Samantha Emswiler: Using letters to substitute in for the possible different terms.
26:07
Samantha Emswiler: Given, this term, I want you to create your own counterargument for this form. Samantha Emswiler: using this idea, some X are Y, some X are Z, conclusion, some X are Y and Z. So the goal is to get a true premise. Samantha Emswiler: True premise, and then a false conclusion. Samantha Emswiler: Good luck with that
26:29
Samantha Emswiler: Again, this is creative work, and you might not hit it on the first time, that's okay, this is the, for me, fun part about logic is to do creative conceptualizations.
26:40
Samantha Emswiler: So, given this lecture, you should be able to define and identify types of deductive arguments, define and identify validity and validity soundness for deductions, define, identify, and create counterexamples and counter-arguments, and define and identify necessary and sufficient conditions.
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