15.Write the arguments and check the validity of each of the following

A)	Let P≡ price is high  and q≡ items are available 

“ price is high. Items are available is necessary price not to be high. 

   Therefore items are not available”

B) Let P≡ Production is low  and q≡ rain continues 

“Production is high if rain continues. rain does not continues . 

   Therefore either Production is low or rain continues”

C) Let P≡ he love her and q≡ she will not marry him 

“ if he does not love her she will marry him. He love her

Therefore  she will not marry him


Let's analyze and check the validity of each argument by formalizing them in symbolic logic. We'll determine whether each argument is logically valid.

### A) "Price is high. Items are available is necessary for the price not to be high. Therefore, items are not available."

**Let:**
- \( P \equiv \) "Price is high"
- \( q \equiv \) "Items are available"

**Argument:**
1. \( P \) (Price is high)
2. \( \neg P \rightarrow q \) ("Items are available is necessary for the price not to be high," which means "If the price is not high, then items are available.")
3. \( \therefore \neg q \) (Therefore, items are not available.)

**Validity Check:**

The argument is not valid. The reasoning is flawed because from \( \neg P \rightarrow q \) (if the price is not high, then items are available), and given \( P \) (price is high), you cannot logically conclude \( \neg q \) (items are not available). The conclusion does not logically follow from the premises.

### B) "Production is high if rain continues. Rain does not continue. Therefore, either production is low or rain continues."

**Let:**
- \( P \equiv \) "Production is low"
- \( q \equiv \) "Rain continues"

**Argument:**
1. \( \neg P \rightarrow q \) (Production is high if rain continues, which is equivalent to "If production is not low, then rain continues.")
2. \( \neg q \) (Rain does not continue)
3. \( \therefore P \vee q \) (Therefore, either production is low or rain continues.)

**Validity Check:**

This argument is valid. Here's why:
- From \( \neg P \rightarrow q \), the contrapositive is \( \neg q \rightarrow P \) (If rain does not continue, then production is low).
- Given \( \neg q \) (rain does not continue), we can infer \( P \) (production is low).
- Therefore, \( P \vee q \) (either production is low or rain continues) is true because \( P \) is true.

### C) "If he does not love her, she will marry him. He loves her. Therefore, she will not marry him."

**Let:**
- \( P \equiv \) "He loves her"
- \( q \equiv \) "She will not marry him"

**Argument:**
1. \( \neg P \rightarrow \neg q \) (If he does not love her, she will marry him, which can be rewritten as "If he does not love her, then it is not the case that she will not marry him.")
2. \( P \) (He loves her)
3. \( \therefore q \) (Therefore, she will not marry him.)

**Validity Check:**

This argument is not valid. The reasoning is flawed because the premises do not logically lead to the conclusion. From \( \neg P \rightarrow \neg q \) and \( P \), you cannot conclude \( q \). The argument does not hold because the conclusion doesn't follow from the premises provided.

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### Summary of Validity:

- **A)** Not Valid
- **B)** Valid
- **C)** Not Valid

Would you like more details on any of these points, or do you have any further questions?

### 5 Related Questions:
1. What are the basic rules of logical inference used to determine validity?
2. How do you identify the contrapositive of a conditional statement?
3. What is the difference between necessary and sufficient conditions in logic?
4. Can you provide examples of valid and invalid arguments?
5. How does one use truth tables to check the validity of an argument?

### Tip:
Always double-check the logical structure of arguments by translating them into symbolic form, as it helps to avoid common reasoning errors.

Learn how to analyze logical arguments for validity using symbolic logic. Explore examples A, B, and C with detailed explanations and validity checks. Suitable for advanced level students and enthusiasts of logic and reasoning.

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