How to Check if a Proof is Correct in a Defined System

In mathematics and computer science, a proof serves as a formal argument that verifies the truth of a statement based on axioms, assumptions, and logical deductions. However, not all proofs are created equal. To determine whether a proof is correct, it must adhere to the logical framework within which it was constructed. In this article, we'll explore how to check the correctness of a proof by verifying its adherence to the system it’s defined in.

What Makes a Proof Correct?

A proof is only valid if it meets the following criteria:

1. Adheres to a defined logical framework: A proof must be constructed within a specific system, such as propositional logic, set theory, or first-order logic. The rules, axioms, and methods of inference must be well-defined.
2. Valid assumptions: Any assumptions made within the proof must be consistent with the system. They must not contradict any existing theorems or axioms.
3. Logical consistency: Each step in the proof must follow from the previous one using a valid rule of inference.
4. Clear conclusion: The proof must ultimately demonstrate the statement's truth without logical fallacies or contradictions.

Steps to Verify a Proof

Here are the steps to check the correctness of a proof:

1. Identify the logical framework or system:
   Understand which formal system the proof belongs to, such as propositional logic, set theory, or any other domain-specific system. This defines the rules and methods that should be followed.

2. Verify the assumptions:
   Ensure that all assumptions used in the proof are valid and consistent with the system. Assumptions are often the starting point of a proof and need to be logically sound.

3. Check the logical steps:
   Review each step of the proof to ensure it logically follows from the previous steps. Each inference or deduction should be justified using valid rules of inference (e.g., modus ponens, conjunction introduction).

4. Ensure valid transitions:
   Make sure that the transition between steps is justified by the system's rules. If a lemma or theorem is used, it must be valid and previously established.

5. Evaluate the conclusion:
   Check if the final conclusion logically follows from the premises. The proof should demonstrate the truth of the statement without introducing contradictions.

6. Use automated tools:
   If available, use automated theorem proving tools (such as Coq, Lean, or Isabelle) to verify the correctness of the proof. These tools can help ensure the proof adheres to the rules of the system.

Example: Proving ( p \land q \rightarrow p ) in Propositional Logic

Let's work through a simple example of a proof in propositional logic to demonstrate how to verify correctness.

Statement to prove: If ( p ) and ( q ) are true, then ( p ) is true. This can be written as ( p \land q \rightarrow p ).

Proof:

1. Assume the premise: Start by assuming that ( p \land q ) is true.

[
\text{Assume: } p \land q \quad \text{(Assumption 1)}
]

1. Apply conjunction elimination: From ( p \land q ), we can deduce that ( p ) is true using conjunction elimination. This rule states that if ( p \land q ) is true, then both ( p ) and ( q ) must individually be true.

[
\text{From assumption 1, we can deduce: } p \quad \text{(Conjunction Elimination)}
]

1. Conclude the proof: Since ( p ) is true, we have shown that if ( p \land q ) is true, then ( p ) must be true.

[
p \land q \rightarrow p \quad \text{(Implication Introduction)}
]

Thus, we have proven that ( p \land q \rightarrow p ).

Verifying the Proof:

Now let's verify if this proof is correct.

1. Identify the logical system: The proof is conducted in propositional logic, which is a formal system that uses logical connectives like AND (( \land )), OR (( \lor )), and IMPLIES (( \rightarrow )).

2. Verify the assumptions: The assumption ( p \land q ) is valid in propositional logic. It’s a standard assumption where we assume the conjunction of two propositions.

3. Check the logical steps: The proof uses two valid rules:

   • Conjunction elimination: If ( p \land q ) is true, then ( p ) is true.
   • Implication introduction: Once we have shown that ( p ) follows from the assumption, we can conclude ( p \land q \rightarrow p ).

4. Ensure valid transitions: The transitions are valid:

   • From ( p \land q ), we correctly deduced ( p ) using conjunction elimination.
   • We correctly applied implication introduction to conclude ( p \land q \rightarrow p ).

5. Evaluate the conclusion: The final conclusion is correct and logically follows from the premises. We have shown that if ( p \land q ) is true, then ( p ) must be true.

6. Automated tools: If we wanted, we could use a theorem-proving tool like Coq to verify this proof, but in this case, manual verification suffices since it’s a simple example.

Conclusion

A proof is only valid if it adheres to the rules and structure of the logical system it’s built within. By following the steps outlined above—identifying the logical system, verifying assumptions, checking logical steps, ensuring valid transitions, and evaluating the conclusion—you can ensure the correctness of any proof.

For more complex proofs, automated tools such as Coq or Lean can assist in verifying the correctness by rigorously checking each step according to the system's rules. By mastering these steps, you can confidently validate the correctness of both simple and advanced proofs.