Are the following arguments valid by virtue of their propositional form? Why or why not? 1. Dona

Are the following arguments valid by virtue of their...

Key Concepts

Conjunction Elimination

Conjunction elimination is a valid inference rule allowing one to infer any individual conjunct from a compound statement. In other words, from a statement of the form 'A and B', one can validly infer either 'A' or 'B'.

Inference Rules

Inference rules are the accepted patterns of reasoning in logic that dictate how one can correctly derive a conclusion from premises. These include rules like conjunction elimination and introduction which preserve truth from premises to conclusion.

Addition of Unrelated Components

A valid argument must not arbitrarily introduce additional components in the conclusion that are not supported by the premises. Adding extra statements that cannot be deduced from the given premises is not a valid logical move.

Validity

Validity in propositional logic refers to an argument structure where, if all the premises are true, the conclusion must also be true. This is independent of the content of the statements—it is solely a matter of the argument’s form.

Propositional Form

The propositional form of an argument is its abstract logical structure, where specific statements are replaced by variables and logical connectives. This abstraction allows evaluation of an argument’s validity based only on its form rather than its subject matter.

Conjunction Introduction

Conjunction introduction is an inference rule that permits combining two true statements into one compound statement of the form 'A and B'. This rule is valid only when both components are independently established.

Transcript

00:01 So we're asked to explain which rules of inference are used for each step for each of the arguments given to us.

00:09 So we'll start with a.

00:12 So a, we're given the argument.

00:16 Linda, a student in this class, owns a red convertible.

00:19 Everyone who owns a red convertible has gotten at least one speeding ticket.

00:23 Therefore, someone in this class has gotten a speeding ticket.

00:28 So let's start assigning functions to different state.

00:33 So we could assign c of x could be x is a student in this class.

00:53 We could say r of x, x owns a red convertible.

01:08 And finally we could say t of x, x has gotten a speeding ticket.

01:26 Speeding ticket.

01:30 So we could start off by writing the premise that we have, which we know linda is a student in the class.

01:46 So, see, linda.

01:51 We know linda has a red convertible, r.

01:55 Linda, and we know that everyone who has a red convertible has gotten a speeding ticket.

02:07 So, universal quantifier.

02:11 If you own a red convertible, then you've gotten a speeding ticket.

02:20 And we see that the conclusion is that someone in this class has gotten a speeding ticket.

02:34 So there is an x where they're in this class and they have a speeding ticket.

02:44 So now let's write down the steps and reasons for each step.

02:51 So let's do step here.

02:56 So step one, we'll pick one of the premise.

03:00 So let's pick everyone who has a red convertible has gotten a speeding ticket.

03:09 R of x, then t of x.

03:17 Now the reason that we add this step is because it's the premise.

03:26 Now step two, let's make this not as universal.

03:36 So linda, linda owns a red car convertible.

03:47 So linda has gotten a ticket.

03:53 And the reason for this is universal instantiation of step one.

04:02 Step three, we know that linda owns a red convertible.

04:11 That's a premise.

04:15 Step four, since step two says, if linda owns a red convertible, then linda owns a has gotten a speeding ticket.

04:28 And we know that linda owns a red convertible.

04:32 We now know that linda has gotten a speed.

04:35 Speeding ticket.

04:38 And we can thank modus ponens of two and three.

04:51 Now let's throw in the last premise.

04:56 Linda is a student in this class.

05:01 Premise.

05:06 Let's keep on going here.

05:08 Step reason.

05:15 Step six.

05:19 We know that linda, the student in the class, and she's gotten a ticket.

05:27 The reason, reason, simplification, not simplification, conjunction, conjunction, a form of steps four and five.

05:49 And this leaves us with the conclusion, there is a student in this class that's gotten a speeding ticket and that's thanks to existential generalization of step six.

06:16 So this is how we showed that the argument is true or is valid.

06:25 For a.

06:27 Now let's get rid of this so we could do b.

06:36 So for b, we're given each of five roommates, melissa, aaron, ralph, venetia, and kishan has taken a course in discrete mathematics.

06:47 Every student has taken a course in discrete, every student who has taken a course in discrete mathematics and take a course in algorithms.

06:54 Therefore, all five roommates can take a course in algorithms next to year.

07:01 So again, let's start by assigning some functions to each of them.

07:08 So we could do r of x.

07:16 X is one of the five roommates.

07:26 R of x.

07:30 X has taken discrete mathematics.

07:38 So discrete mathematics.

07:42 A of x, x can take a course in algorithms.

07:55 Now that's write down the premise and the conclusion that we find in the argument.

08:01 So premise, we know that all the roommates have taken discrete mathematics.

08:11 So if x is a roommate, then they've taken discrete mathematics.

08:21 And we know that all students who have taking discrete mathematics can take a course in algorithms.

08:31 So all students have taken discrete mathematics can take a course in algorithms.

08:40 Now our conclusion is that all the roommates can take a course in algorithms.

08:52 So if you're a roommate, then you could take a course in algorithms.

08:57 That's what this expression says.

09:00 Now how do we prove this? how do we show that it's valid? step.

09:09 Reason.

09:13 So step one, we choose a premise.

09:17 So let's choose all the roommates have taken a course in discrete mathematics.

09:24 If you're a roommate, then you've taken a course of course in discrete mathematics.

09:32 That's a premise.

09:35 Step two, let's make it a little more direct.

09:42 So roommate c has taken a course in discrete mathematics.

09:50 So this is a this is thanks to universal instantiation of step one.

09:56 So step three, we'll put the other premise.

10:03 All students who have taken a course in discrete mathematics can also take a course in algorithms...

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