Translate the argument into symbolic form. Then determine whether the argument is valid or invalid.
If it rains on Sunday​, then we will go swimming. It does not rain on Sunday. ​Therefore, we will not go swimming.
Let p be​ "it rains on Sunday​" and let q be​ "we will go swimming​." What is the argument into symbolic​ form?

To translate the argument into symbolic form, let's first assign the propositions:

- \( p \): "It rains on Sunday"
- \( q \): "We will go swimming"

The argument presented is:

1. **Premise 1**: If it rains on Sunday, then we will go swimming.
   - In symbolic form: \( p \rightarrow q \)
   
2. **Premise 2**: It does not rain on Sunday.
   - In symbolic form: \( \neg p \)
   
3. **Conclusion**: Therefore, we will not go swimming.
   - In symbolic form: \( \neg q \)

The argument in symbolic form is:
\[ (p \rightarrow q), \, \neg p \vdash \neg q \]

Now, to determine whether the argument is valid, we need to check if the conclusion \( \neg q \) logically follows from the premises \( p \rightarrow q \) and \( \neg p \).

### Validity Check:

For an argument to be valid, if the premises are true, then the conclusion must also be true. The argument uses the form known as "denying the antecedent," which is not a valid form of argument. The validity depends on the logical form, not on the actual truth of the statements.

The premises provided are:

1. \( p \rightarrow q \)
2. \( \neg p \)

To check the validity, consider a situation where \( p \) is false, and \( q \) is true:
- If \( p \) is false, then \( p \rightarrow q \) is true regardless of the truth value of \( q \).
- However, if \( q \) is true, the conclusion \( \neg q \) (we will not go swimming) is false.

Thus, the premises do not guarantee the truth of the conclusion, making the argument invalid.

### Conclusion:
The argument is invalid because it commits the logical fallacy of denying the antecedent.

Would you like a detailed explanation of any part, or do you have any questions?

### Related Questions:
1. What is the truth table for the implication \( p \rightarrow q \)?
2. Can you give an example of a valid argument using \( p \) and \( q \)?
3. What is the contrapositive of the statement \( p \rightarrow q \)?
4. How does denying the antecedent differ from modus tollens?
5. What does it mean for an argument to be valid in logic?
6. Can you explain the difference between a sound argument and a valid argument?
7. How can we use truth tables to verify the validity of an argument?
8. What is the difference between necessary and sufficient conditions in logical statements?

**Tip:** In logical arguments, understanding the difference between sufficient and necessary conditions can help clarify whether conclusions are validly drawn from premises.

Learn how to translate and validate a logical argument using propositional logic. Understand the symbolic form of arguments, logical premises, and conclusions. Explore the concept of denying the antecedent and its impact on argument validity.

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