Problem Analysis:
The task involves solving a
Single-Additive-Evaluation problem, which appears to be related to formal logic or proof systems. The goal is to evaluate the given statements and determine their validity using specific rules and axioms.
#### Key Components:
1.
Axioms and Rules:
-
Axiom 1: $ \vdash J $
-
Axiom 2: $ \vdash K $
-
Rule of Inference (MP): Modus Ponens (If $ \vdash P \rightarrow Q $ and $ \vdash P $, then $ \vdash Q $).
-
Rule of Inference (US): Universal Specification (If $ \vdash \forall x P(x) $, then $ \vdash P(t) $ for any term $ t $).
2.
Goal: Prove $ \vdash L $.
3.
Given Statements:
- $ \vdash J \rightarrow (K \rightarrow L) $ (Premise 1)
- $ \vdash J $ (Premise 2)
- $ \vdash K $ (Premise 3)
4.
Steps to Solve:
- Use the given premises and rules to derive $ \vdash L $.
---
Solution:
#### Step-by-Step Derivation:
1.
Premise 1: $ \vdash J \rightarrow (K \rightarrow L) $
- This is given as part of the problem statement.
2.
Premise 2: $ \vdash J $
- This is also given as part of the problem statement.
3.
Apply Modus Ponens (MP):
- From Premise 1 ($ \vdash J \rightarrow (K \rightarrow L) $) and Premise 2 ($ \vdash J $), we can use Modus Ponens to infer:
$$
\vdash K \rightarrow L
$$
4.
Premise 3: $ \vdash K $
- This is given as part of the problem statement.
5.
Apply Modus Ponens (MP) Again:
- From the result of Step 3 ($ \vdash K \rightarrow L $) and Premise 3 ($ \vdash K $), we can use Modus Ponens to infer:
$$
\vdash L
$$
---
Final Answer:
$$
\boxed{\vdash L}
$$
---
Explanation:
The solution relies on the application of the Modus Ponens rule twice. First, we use it to derive $ \vdash K \rightarrow L $ from $ \vdash J \rightarrow (K \rightarrow L) $ and $ \vdash J $. Then, we use it again to derive $ \vdash L $ from $ \vdash K \rightarrow L $ and $ \vdash K $. This step-by-step approach ensures that each inference is logically valid based on the given axioms and rules.
Parent Tip: Review the logic above to help your child master the concept of geometry angle addition worksheet.
