Answer
See explanation
Work Step by Step
### Problem (a):
Assume the following sentence is a **statement**:
> “**If this sentence is true, then 1 + 1 = 3.**”
We are asked to **prove that \(1 + 1 = 3\)** under that assumption.
---
### ✅ Let’s call the sentence \(S\):
\[
S: \text{If } S \text{ is true, then } 1 + 1 = 3
\]
Now suppose \(S\) is **true**. Then, according to its own content:
\[
\text{If } S \text{ is true, then } 1 + 1 = 3
\Rightarrow \text{So } 1 + 1 = 3
\]
So assuming \(S\) is true, we are forced to conclude \(1 + 1 = 3\), which is **clearly false**.
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### ✅ Final Answer (a):
If the sentence S is assumed to be a true statement, then it logically forces 1 + 1 = 3.
---
### Problem (b):
What can you deduce about the truth of “This sentence is true”?
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Since assuming \(S\) is **true** leads to a falsehood (\(1 + 1 = 3\)), the **only possible conclusion** is that:
> \(S\) **cannot** be true.
If \(S\) is a **statement**, and assuming it's true yields something false, then:
\[
\boxed{S \text{ must be false.}}
\]
But here's the catch: if \(S\) is false, then the **antecedent** ("this sentence is true") is false, making the implication \((S \Rightarrow 1 + 1 = 3)\) **true** (since false implies anything in classical logic).
So the sentence would still be **true**, which is a contradiction.
---
### 🌀 This is **Löb’s Paradox**:
It shows that reasoning about the **truth of self-referential statements** (especially those that assert something about their own truth) can produce **contradictions** unless your logical system is carefully restricted.
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### ✅ Final Answer (b):
S cannot be both a statement and true. Assuming it is true leads to a contradiction. So, S must be false or meaningless.
