
Complete the logical proof for the following argument.

(p Vr) → q q→ t

↑

..t

Step

Proposition

Justification

1

p

Hypothesis

2

pVr

Addition, 1

3

(pVr)→q

Hypothesis

4

q

Modus Ponens, 2, 3

сл

5

q t

Hypothesis

6

t

Modus Ponens,4,3

Note that the justification for each step is either hypothesis or it would include both the name of the law or rule and the step(s) to which it is applied to.

Copy and paste the logical operators when filling in the blanks: V, A, →,~, V, 3.

lote: You can earn partial credit on this problem.

Let's complete the proof step by step and include the missing details. The goal is to show the argument's conclusion step by step using logical rules.

We are given:

1. \((p \lor r) \rightarrow q\) (Hypothesis)
2. \(q \rightarrow t\) (Hypothesis)
3. We need to prove \(t\).

### Proof:

| Step  | Proposition               | Justification                           |
|-------|---------------------------|-----------------------------------------|
| 1     | \(p\)                     | Hypothesis                              |
| 2     | \(p \lor r\)              | Addition (from 1)                       |
| 3     | \((p \lor r) \rightarrow q\) | Hypothesis                              |
| 4     | \(q\)                     | Modus Ponens (from 2 and 3)             |
| 5     | \(q \rightarrow t\)        | Hypothesis                              |
| 6     | \(t\)                     | Modus Ponens (from 4 and 5)             |

### Explanation:

- In **step 2**, we used **Addition** to infer \(p \lor r\) from \(p\). This is a valid step because \(p \lor r\) is true if \(p\) is true.
- In **step 4**, we used **Modus Ponens** on steps 2 and 3. Since we know \(p \lor r\) is true (from step 2) and we have \((p \lor r) \rightarrow q\), we can conclude that \(q\) is true.
- In **step 6**, we again used **Modus Ponens** on steps 4 and 5. Since \(q\) is true and \(q \rightarrow t\), we conclude that \(t\) is true, which is the conclusion of the argument.

Would you like further details on any step?

### Follow-up questions:
1. Can you explain why Modus Ponens is used in this proof?
2. What is the purpose of the Addition rule in step 2?
3. Why is \(p \lor r\) valid from the hypothesis of \(p\)?
4. How does the structure of this proof ensure the conclusion \(t\)?
5. What would happen if we reversed the hypotheses of \(q\) and \(t\)?

### Tip:
When working with logical proofs, always identify which statements are your hypotheses and carefully apply rules like Modus Ponens or Addition based on earlier steps.

Complete the logical proof for the following argument.

(p Vr) → q
q→ t

..t

Step

Proposition

Justification

1 p Hypothesis

2 pVr Addition, 1

3 (pVr)→q Hypothesis

4 q Modus Ponens, 2, 3

5 q → t Hypothesis

6 t Modus Ponens, 4, 5

Explore a step-by-step solution to complete a logical proof using Modus Ponens and Addition. This problem covers key propositional logic concepts, suitable for undergraduate logic courses or students interested in formal proofs.

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