For each of these arguments determine whether the argument is correct or incorrect and explain w

For each of these arguments determine whether the argument...

Key Concepts

Affirming the Consequent Fallacy

This fallacy emerges when one mistakenly infers that if 'if P then Q' is true and Q is true, then P must be true. This form of reasoning is invalid because Q can be true for reasons other than P. Awareness of this fallacy is critical in evaluating arguments where conclusions are drawn based solely on the presence of a consequence.

Denying the Antecedent Fallacy

This is a common logical fallacy that occurs when an argument incorrectly assumes that if 'if P then Q' is true, then 'not P' must imply 'not Q.' In reality, the truth of Q may be established by other means, making the inference invalid. Recognizing this fallacy is key to avoiding erroneous reasoning.

Validity in Deductive Arguments

This concept involves determining whether an argument’s conclusion logically and necessarily follows from its premises. In a valid deductive argument, if the premises are true, the conclusion must be true. Understanding validity is essential when assessing any logical reasoning or argument structure.

Modus Tollens

Modus tollens is a valid form of argument in propositional logic. It works by denying the consequent; that is, if one has a statement of the form 'if P then Q' and observes 'not Q,' one can validly conclude 'not P.' This method underpins certain correct forms of inference.

Modus Ponens

Modus ponens is another fundamental and valid form of deductive reasoning. It follows the pattern 'if P then Q' along with the affirmation of P, which together justify the conclusion Q. Recognizing modus ponens helps in identifying valid inferences in categorical arguments.

Transcript

00:01 So in this question we're asked to determine whether each argument is correct or incorrect and explain why.

00:07 So let's do part a.

00:08 In part a, we know that everyone enrolled in the university has lived in the dormitory.

00:13 Mia has never lived in a dormitory.

00:16 Therefore, mia is not enrolled in the university.

00:19 Now first, let's assign some functions to these statements.

00:25 So we could assign e of x is x.

00:33 Enrolled in the university university and we could also sign d of x x has lived in a dorm now the next thing we're gonna do is we're gonna write the premise that we have and we'll write the conclusion that the argument has so the first premise is everyone enrolled in the university has lived in a dormitory so we could rewrite that as all x if x is enrolled in the university, then x has lived in a dormitory.

01:43 We also know that mia has never lived in a dormitory.

01:49 So not d, m for mia.

01:56 And the conclusion is, therefore mia has not enrolled in the university.

02:04 So that'll be not e and m for mia.

02:15 Let's do the steps and the reasons for each step to see if this argument is correct or incorrect.

02:22 So i'll write step and the column for reason.

02:30 So the first step, all x, if x is enrolled in the university, then x has lived in a dorm.

02:45 The reason for this step, it's a premise.

02:51 Now what can we do with this? we could say, if mia is enrolled in the university, then mia has lived in a dorm.

03:05 That's thanks to universal instantiation of step one.

03:11 Step three, we already know that mia has never lived in a dorm.

03:20 That's a premise.

03:25 So step four, if we look at table one, which is the rules of inference, we could see that if we take a look at modus tollens, it's right.

03:37 And is not q, if p then q, therefore not p.

03:44 So in this case, it'll be not enrolled.

03:49 Mia is not enrolled in the university.

03:52 And that is thanks to modus tolens of two and three.

04:03 So since we were able to conclude using rules of inference, we know that this statement or this argument is correct.

04:19 So a is correct.

04:21 Now let's get rid of this so we could do software b.

04:29 So for b, we know a convertible car is fun to drive.

04:41 Isaac's car is not a convertible.

04:43 Therefore, isaac's car is not fun to drive.

04:47 So again, we'll assign some variables to different statements.

04:51 So we could assign c of x where x is a convertible, x is a convertible.

05:03 We could assign f of x where x is fun to drive.

05:21 Now we'll write out the premise that we have and the conclusion that we have.

05:34 So the premise that we have is a convertible car is fun to drive and isaac's car is not a convertible.

05:41 Therefore, all x, if x is a convertible, then it is fun to drive.

05:52 And we have that isaac's car is not a convertible so not c i for isaac and the conclusion we have is isaac's car is not fun to drive so not fun to drive isaac's car now we'll run through the steps and reasons for each of for the premise to see if we could determine if the conclusion is correct or incorrect so step and reason so step one all x if x is a convertible then it is fun to drive reason we have it here it's a premise step two we could say if isaac's car is in convertible then isaac's car is fun to drive this is thanks to universal instantiation of step one step three we know that isaac's car is not a convertible.

07:25 This is a premise.

07:29 So now let's look at table one of the rules of inference to see if we could find any rule of inference that's written in this form.

07:41 After a review, we can't.

07:44 Therefore, we determine that there's no rule of inference that concludes this conclusion.

07:50 So the argument is incorrect.

07:54 So we cannot conclude that isaac's car is not fun to drive.

07:58 Alright, now let's get rid of this so we could solve for c.

08:10 So c, we're given quincy likes all action movies.

08:15 Quincy likes the movie 8 men out.

08:18 Therefore, 8 men out is an action movie.

08:21 So again, we'll assign some functions to different statements...