When Proving A Conclusion Of The Form P V Q, We Only Have To Prove A Single Disjunct. Is That A Slight Abuse Of Syntax And Exactness Required Of Proof

Introduction

In the realm of propositional logic, we often encounter the task of proving conclusions of the form P v Q, where P and Q are propositions. A common question arises: do we need to prove both disjuncts, P and Q, or can we get away with proving just one of them? In this article, we will delve into the world of propositional logic, explore the concept of disjunctive proofs, and examine whether proving a single disjunct is a slight abuse of syntax and exactness required of proof.

The Basics of Propositional Logic

Before we dive into the world of disjunctive proofs, let's briefly review the basics of propositional logic. Propositional logic is a branch of mathematics that deals with statements that can be either true or false. These statements are called propositions, and they can be combined using logical operators such as conjunction (∧), disjunction (∨), and negation (¬).

Disjunctive Proofs

A disjunctive proof is a type of proof that involves proving a disjunction, i.e., a statement of the form P v Q. In a disjunctive proof, we aim to show that at least one of the disjuncts, P or Q, is true. However, we can often get away with proving just one of the disjuncts, rather than both.

The Rule of Disjunction

In propositional logic, we have a rule of disjunction that states: if we can prove P, then we can prove P v Q. This rule is often represented as:

P ⊢ P v Q

This rule tells us that if we can prove P, then we can prove the disjunction P v Q, without needing to prove Q.

Example: Proving a Disjunction

Let's consider an example from Velleman's "How to Prove It" (page 376). Suppose we want to prove the following disjunction:

P v Q

where P and Q are propositions. We can prove this disjunction by proving just one of the disjuncts, say P. In this case, we would have:

P ⊢ P v Q

This means that if we can prove P, then we can prove the disjunction P v Q, without needing to prove Q.

Is Proving a Single Disjunct an Abuse of Syntax?

Now that we have seen how to prove a disjunction by proving just one of the disjuncts, we may wonder whether this is an abuse of syntax and exactness required of proof. After all, we are not proving both disjuncts, P and Q, but rather just one of them.

The Answer: No

In this case, the answer is no. Proving a single disjunct is not an abuse of syntax and exactness required of proof. The rule of disjunction allows us to prove a disjunction by proving just one of the disjuncts, and this is a perfectly valid way to prove a disjunction.

Why is this the case?

The reason why proving a single disjunct is not an abuse of syntax and exactness required of proof is that the rule of disjunction is a rule of propositional logic. It tells us that if we can prove P, then we can prove the disjunction P v Q, without needing to prove Q. This rule is not a trick or a shortcut, but rather a fundamental property of propositional logic.

Conclusion

In conclusion, proving a conclusion of the form P v Q does not require us to prove both disjuncts, P and Q. We can often get away with proving just one of the disjuncts, and this is a perfectly valid way to prove a disjunction. The rule of disjunction is a fundamental rule of propositional logic, and it allows us to prove a disjunction by proving just one of the disjuncts.

Additional Examples

Let's consider a few more examples to illustrate the concept of disjunctive proofs.

Example 1: Proving a Disjunction with Two Disjuncts

Suppose we want to prove the following disjunction:

P v Q v R

where P, Q, and R are propositions. We can prove this disjunction by proving just one of the disjuncts, say P. In this case, we would have:

P ⊢ P v Q v R

Example 2: Proving a Disjunction with Three Disjuncts

Suppose we want to prove the following disjunction:

P v Q v R v S

where P, Q, R, and S are propositions. We can prove this disjunction by proving just one of the disjuncts, say P. In this case, we would have:

P ⊢ P v Q v R v S

Example 3: Proving a Disjunction with Four Disjuncts

Suppose we want to prove the following disjunction:

P v Q v R v S v T

where P, Q, R, S, and T are propositions. We can prove this disjunction by proving just one of the disjuncts, say P. In this case, we would have:

P ⊢ P v Q v R v S v T

Conclusion

In conclusion, proving a conclusion of the form P v Q does not require us to prove both disjuncts, P and Q. We can often get away with proving just one of the disjuncts, and this is a perfectly valid way to prove a disjunction. The rule of disjunction is a fundamental rule of propositional logic, and it allows us to prove a disjunction by proving just one of the disjuncts.

References

• Velleman, D. (2006). How to Prove It: A Structured Approach. Cambridge University Press.

Further Reading

• Enderton, H. B. (2001). A Mathematical Introduction to Logic. Academic Press.
• Mendelson, E. (1997). Introduction to Mathematical Logic. Chapman and Hall/CRC.
  Frequently Asked Questions (FAQs) about Disjunctive Proofs in Propositional Logic =====================================================================================

Q: What is a disjunctive proof in propositional logic?

A: A disjunctive proof is a type of proof that involves proving a disjunction, i.e., a statement of the form P v Q. In a disjunctive proof, we aim to show that at least one of the disjuncts, P or Q, is true.

Q: Why can we often get away with proving just one of the disjuncts in a disjunctive proof?

A: We can often get away with proving just one of the disjuncts because of the rule of disjunction, which states that if we can prove P, then we can prove P v Q. This rule allows us to prove a disjunction by proving just one of the disjuncts.

Q: Is proving a single disjunct an abuse of syntax and exactness required of proof?

A: No, proving a single disjunct is not an abuse of syntax and exactness required of proof. The rule of disjunction is a fundamental rule of propositional logic, and it allows us to prove a disjunction by proving just one of the disjuncts.

Q: Can we always prove a disjunction by proving just one of the disjuncts?

A: No, we cannot always prove a disjunction by proving just one of the disjuncts. There are cases where we need to prove both disjuncts in order to prove the disjunction.

Q: What are some examples of disjunctive proofs?

A: Here are a few examples of disjunctive proofs:

• Proving a disjunction with two disjuncts: P v Q
• Proving a disjunction with three disjuncts: P v Q v R
• Proving a disjunction with four disjuncts: P v Q v R v S

Q: How do we know when to prove both disjuncts in a disjunctive proof?

A: We know when to prove both disjuncts in a disjunctive proof when the disjunction is not a simple disjunction, but rather a disjunction of multiple statements. In such cases, we need to prove all the statements in order to prove the disjunction.

Q: Can we use disjunctive proofs to prove more complex statements?

A: Yes, we can use disjunctive proofs to prove more complex statements. Disjunctive proofs can be used to prove statements that involve multiple disjunctions, conjunctions, and negations.

Q: What are some common mistakes to avoid when using disjunctive proofs?

A: Here are a few common mistakes to avoid when using disjunctive proofs:

• Assuming that we can always prove a disjunction by proving just one of the disjuncts.
• Failing to recognize when we need to prove both disjuncts in a disjunctive proof.
• Not using the rule of disjunction correctly.

Q: How can we practice using disjunctive proofs?

A: We can practice using disjunctive proofs by working through examples and exercises in a textbook or online resource. We can also try to come up with our own examples of disjunctive proofs and see if we can prove them using the rule of disjunction.

Q: What are some resources for learning more about disjunctive proofs?

A: Here are a few resources for learning more about disjunctive proofs:

• Velleman, D. (2006). How to Prove It: A Structured Approach. Cambridge University Press.
• Enderton, H. B. (2001). A Mathematical Introduction to Logic. Academic Press.
• Mendelson, E. (1997). Introduction to Mathematical Logic. Chapman and Hall/CRC.

Conclusion

In conclusion, disjunctive proofs are a powerful tool in propositional logic that allow us to prove disjunctions by proving just one of the disjuncts. By understanding the rule of disjunction and how to use it correctly, we can prove more complex statements and avoid common mistakes. With practice and experience, we can become proficient in using disjunctive proofs to prove a wide range of statements in propositional logic.