Presentation 2c: Introduction to Validity and Soundness (Phil 1230: Reasoning and Critical Thinking)

Summary

This lecture introduces the concepts of validity and soundness in arguments. Validity is explained as an argument where it's impossible for the premises to be true and the conclusion false, meaning the truth of the premises guarantees the truth of the conclusion. Soundness is when an argument is both valid and has true premises, ensuring the conclusion is also true. Several examples and exercises are provided to illustrate these concepts, and the distinctions between validity, soundness, truth, and falsity are highlighted.

Takeaways

• 📘 Validity means an argument is valid if it's impossible for the premises to be true and the conclusion to be false at the same time.
• 🧐 Logical impossibility in this context means there is no conceivable way for the premises to be true while the conclusion is false.
• ✅ A valid argument guarantees that if the premises are true, the conclusion must also be true.
• 🐴 Example of a valid argument: 'If horses are mammals, then they are warm-blooded. Horses are mammals. Therefore, horses are warm-blooded.'
• 🚫 An invalid argument allows for the possibility that the premises are true while the conclusion is false.
• 🤔 Example of an invalid argument: 'Joseph is tall. Manuela is taller than Joseph. Therefore, Manuela is happy.' This is invalid because the premises do not guarantee the conclusion.
• 🔍 Soundness means an argument is both valid and has all true premises.
• 📜 A sound argument always leads to a true conclusion because it combines valid reasoning with true premises.
• ❌ An argument with false premises, even if valid, is not sound. For example: 'Guinea pigs like to eat hay. If guinea pigs like hay, then humans like hay. Therefore, humans like hay.'
• 🔗 Arguments are valid or sound, while statements are true or false. Attributing validity or soundness to statements is incorrect.

Q & A

What is the definition of a valid argument?

An argument is valid if and only if it is logically impossible for the premises to be true while the conclusion is false.

Why is logical impossibility important for understanding validity?

Logical impossibility means that there is absolutely no situation where the premises could be true and the conclusion false. It emphasizes the strongest form of impossibility, which ensures the conclusion must be true if the premises are true.

Can an argument be valid if the conclusion is true by coincidence?

No, an argument’s validity depends on the logical relationship between the premises and the conclusion. Even if the conclusion happens to be true, the argument is invalid if it’s possible for the premises to be true and the conclusion false.

What is an example of a valid argument from the script?

One example of a valid argument is: If horses are mammals, then horses are warm-blooded. Horses are mammals. Therefore, horses are warm-blooded.

What does it mean for an argument to be invalid?

An argument is invalid when it is possible for all the premises to be true, but the conclusion could still be false. This indicates that the conclusion does not logically follow from the premises.

What is the definition of a sound argument?

An argument is sound if it is both valid and all its premises are true.

Why do we care about the soundness of an argument?

We care about soundness because it ensures not only that the argument is valid, but also that its premises are true. A sound argument guarantees a true conclusion, making it useful for reasoning.

Is a valid argument always sound?

No, a valid argument is only sound if all of its premises are true. A valid argument can still have false premises and thus be unsound.

Can a statement be valid or sound?

No, validity and soundness are properties of arguments, not statements. A statement can be true or false, but an argument can only be valid or sound.

What happens if an argument is valid but has false premises?

If an argument is valid but has false premises, it is unsound. The argument would only show that if the premises were true, then the conclusion would be true as well, but with false premises, this reasoning is not useful.

Outlines

00:00

📝 Introduction to Validity and Soundness

This section introduces the two core concepts of validity and soundness. Validity is defined as the logical impossibility of a situation where the premises are true, but the conclusion is false. An argument is valid if the truth of the premises guarantees the truth of the conclusion. An example of a valid argument is given: 'If horses are mammals, then they are warm-blooded; horses are mammals; therefore, horses are warm-blooded.' The key point is that in a valid argument, the truth of the premises necessitates the truth of the conclusion.

05:03

💡 Understanding Validity with Examples

This section further explains validity by providing examples. It contrasts valid and invalid arguments. A valid argument is one where the premises guarantee the truth of the conclusion, as seen in the 'Joseph is tall' and 'Manuela is taller' example. However, the argument 'Joseph is tall; Manuela is taller than Joseph; therefore, Manuela is happy' is invalid because it's possible for the premises to be true while the conclusion is false. This illustrates that validity is about whether the premises can make the conclusion necessarily true.

🔍 Introduction to Soundness

The focus shifts to soundness, which requires both validity and truth of the premises. Soundness is important because a valid argument with false premises isn’t useful in reality. An argument is unsound if it’s either invalid or has one or more false premises. The section also explains that in everyday language, 'sound' can mean 'good,' but in logic, it specifically refers to arguments that are both valid and have true premises. A sound argument, by definition, will always have a true conclusion.

✅ Examples of Soundness

In this final part, examples are provided to illustrate soundness. One example is a sound argument about Western University being a good school. The argument is sound because the premises are true, and the argument is valid. In contrast, an argument involving guinea pigs and humans liking to eat hay is unsound because one of the premises is false, even though the argument is valid. The section also emphasizes the importance of distinguishing between properties of statements (truth or falsity) and properties of arguments (validity or soundness).

Keywords

💡Validity

Validity refers to a key concept in logic where an argument is valid if it is impossible for the premises to be true while the conclusion is false. In the video, validity is introduced as a central topic, emphasizing that valid arguments ensure that true premises lead to a true conclusion. An example from the script is the argument about horses being mammals and warm-blooded.

💡Soundness

Soundness is the concept that an argument is both valid and has true premises. In the video, soundness is described as a crucial property of good arguments because not only must the conclusion logically follow from the premises (validity), but the premises themselves must also be true. For example, the argument about Western being a good school is considered sound.

💡Premise

A premise is a statement or proposition from which a conclusion is drawn in an argument. The video explains that premises are critical because the truth of a conclusion depends on the truth of its premises. For example, the premise 'Horses are mammals' is used to establish the conclusion that 'Horses are warm-blooded.'

💡Conclusion

The conclusion is the statement in an argument that logically follows from the premises. The video illustrates that the truth of the premises guarantees the truth of the conclusion in valid arguments. For example, the conclusion in the argument 'Manuela is tall' follows logically from the premises that 'Joseph is tall' and 'Manuela is taller than Joseph.'

💡Invalidity

Invalidity occurs when an argument's premises do not logically guarantee the conclusion. The video contrasts invalid arguments with valid ones, showing that even if the premises are true, the conclusion could still be false. An example given is the argument that 'Manuela is taller than Joseph, therefore, Manuela is happy,' which is invalid because the conclusion does not logically follow from the premises.

💡Logical Impossibility

Logical impossibility refers to a situation where it is impossible for all premises to be true while the conclusion is false. The video introduces this as a core idea in understanding validity. For instance, it is logically impossible for horses to be mammals and not warm-blooded if the premises about horses are true.

💡True Premises

True premises are statements that reflect facts or reality, which are essential for an argument to be sound. In the video, true premises are discussed in relation to soundness. An example is 'Western is a good school,' which is assumed to be a true premise in a sound argument about taking a course at Western.

💡False Premises

False premises are statements that are not true, which can make an otherwise valid argument unsound. The video provides an example with guinea pigs, where the premise 'If guinea pigs like to eat hay, then humans like to eat hay' is false, leading to an unsound argument even though it is valid.

💡Guarantee

In logic, a guarantee means that the truth of the premises ensures the truth of the conclusion in a valid argument. The video emphasizes that in valid arguments, the premises must guarantee the conclusion. For example, if it is true that 'Horses are mammals' and 'Mammals are warm-blooded,' then it is guaranteed that 'Horses are warm-blooded.'

💡Category Mistake

A category mistake is when properties that apply to one type of thing are mistakenly attributed to another. The video explains that attributing validity or soundness to statements rather than arguments is a category mistake because these properties apply to arguments, not individual statements. This concept clarifies that arguments and statements have different logical roles.

Highlights

Introduction to two key concepts in logic: validity and soundness.

Definition of validity: an argument is valid if it is logically impossible for the premises to be true while the conclusion is false.

Simplified explanation: the truth of the premises guarantees the truth of the conclusion.

Example of a valid argument: 'If horses are mammals, then horses are warm-blooded. Horses are mammals. Therefore, horses are warm-blooded.'

Explanation of invalid arguments: an argument is invalid if it is possible for the premises to be true and the conclusion false.

Example of an invalid argument: 'Joseph is tall. Manuela is taller than Joseph. Therefore, Manuela is happy.'

Definition of soundness: an argument is sound when it is valid and all its premises are true.

Importance of soundness: a valid argument with false premises is not helpful, as it does not lead to a true conclusion.

The connection between soundness and truth: a sound argument must have a true conclusion since its premises are true and the conclusion follows from them.

Exercise on soundness: 'Western is a good school. You are taking this course at Western. Therefore, you are taking this course at a good school.' This argument is sound.

Exercise on unsoundness: 'Guinea pigs like to eat hay. If guinea pigs like to eat hay, then humans like to eat hay. Therefore, humans like to eat hay.' This argument is unsound due to a false premise.

Emphasis on the distinction between arguments and statements: arguments can be valid or sound, while statements are true or false.

A valid argument ensures that if the premises are true, the conclusion cannot be false.

Clarification: validity pertains to the structure of the argument, not the truth of its premises.

Conclusion: validity and soundness are essential in evaluating the strength and truthfulness of an argument.

Transcripts

Welcome to Lecture 2c: Introduction  to Validity and Soundness.  

This presentation introduces two of the most  important concepts we will encounter in this  

course, the concepts of validity and soundness. Next time, we will consider these  

notions in a bit more detail. For now, this is just a brief introduction.  

We’ll start with validity.  Then we’ll turn to soundness.  

Here is the definition of validity. An argument is valid if and only if there  

is no logically possible situation  where all the premises are true and  

the conclusion is false at the same time. This is the definition from the textbook.  

Here is a simplified way of  saying exactly the same thing.  

An argument is valid if and only  if it is logically impossible  

for the premises to be true  while the conclusion is false.  

We will talk about precisely what logical  impossibility is in a later presentation,  

but for now, we can think of it intuitively as  impossibility in the strongest sense---for an  

argument to be valid, there must be absolutely  no way that the premises could possibly be true  

and the conclusion fail to also be true. Put otherwise, when an argument is valid,  

the truth of the premises (if they are true)  would guarantee the truth of the conclusion.  

If the premises are true, the  conclusion has got to be true as well.  

Here is an example of a valid argument. If horses are mammals,  

then horses are warm-blooded. Horses are mammals.  

Therefore, horses are warm-blooded. It is impossible for the premises to be  

true---for it to be true that if horses  are mammals, then they are warm-blooded  

and that horses are mammals---without  the conclusion also being true.  

So, if the premises are true, then the conclusion  has got to be true, too. The argument is valid.  

When an argument is not valid,  we say that it is invalid.

The notion of validity is supposed to capture  the intuitive idea of an argument’s conclusion  

following from its premises When an argument is valid,  

its conclusion follows from its premises. Here is another example of a valid argument:  

Joseph is tall. Manuela is taller than Joseph.  

Therefore, Manuela is tall. The conclusion follows from the premises.  

This is just another way of saying  that if the premises are all true,  

then the conclusion has got to be true, too.

Let’s do an exercise. Is the following argument valid?  

Joseph is tall. Manuela is taller than Joseph.  

Therefore, Manuela is happy. No, it is not valid.  

That is because it is possible for the premises  to be true without the conclusion being true.  

It might be the case that Joseph is tall,  Manuela is taller, and Manuela is not happy.

Note that all that matters for the argument’s  being invalid is that it is possible for the  

premises to be true and the conclusion false. Even if Manuela is in fact happy,  

the argument is nonetheless invalid. Intuitively, the truth of the premises does  

not guarantee the truth of the conclusion---even  if the conclusion happens to in fact be true.

Let’s now turn to another important  property of arguments, soundness.

An argument is sound when it is valid  and all its premises are true.  

Why should we be interested in this notion? Well, a valid argument whose premises are false  

is of little use to us: it shows that  the conclusion would be true if the  

premises were true, but if the premises  are not true, that’s not very helpful.  

This is why we normally care about  the soundness of an argument,  

which requires both that the it be  valid and that its premises be true.  

When an argument is not sound,  we say that it is unsound.

Note that the term “sound” is sometimes used  in a way that means something close to “good”  

in everyday language. For example, a sound  plan is a good plan. The term “sound” used  

in critical thinking, logic, and philosophy  has the more specific meaning given here,  

which somewhat corresponds to the idea of  a good argument but is not quite the same.  

We will say a little bit more about what makes  for a good argument in a future presentation.

Must a sound argument have a true conclusion? Yes. Since a sound argument has true premises  

and the conclusion follows from the premises,  the conclusion must be true as well.  

So, if we know that an argument is sound,  then we know that the conclusion is true.  

Let’s do an exercise together. We are asked to evaluate whether  

this argument is sound. Western is a good school.  

You are taking this course at Western.  

Therefore, you are taking this  course at a good school.  

The argument is indeed sound. It is valid---i.e., it’s impossible for the  

premises to be true and the conclusion false. If Western is a good school and you’re taking  

this course at Western, it just follows that  you’re taking this course at a good school.  

And the premises are true. Western is a  good school, and, presumably, you are taking  

this course at Western. Is this argument sound?  

Guinea pigs like to eat hay. If guinea pigs like to eat hay,  

then humans like to eat hay. Therefore, humans like to eat hay.  

No. The argument is valid---if  the premises are true,  

then the conclusion has got to be true, too. But the premises are not true. In particular,  

the second premise is false. So the argument is unsound.

It is important to keep clear that validity  and soundness are properties of arguments.  

A statement cannot be valid or sound---attributing  these properties to a statement would be a  

category mistake. It just wouldn’t make sense,  because statements are not the kinds of things  

that can have these properties. Likewise, truth and falsity are properties  

of statements. An argument cannot be true or  false. It’s premises or conclusion can be true  

or false, but the argument itself---the set of  sentences---can be neither of those things.

That’s it for now. Thanks for watching!