Confusion About The Proof Of Replacement Theorem (Mendelson's Intro To Logic)

Introduction

In the realm of mathematical logic, the Replacement Theorem is a fundamental concept that plays a crucial role in the development of first-order logic. This theorem, also known as the Substitution Theorem, states that if a formula is provable, then any substitution of terms for the variables in that formula is also provable. In this article, we will delve into the proof of the Replacement Theorem as presented in Mendelson's Introduction to Mathematical Logic (6th Edition), specifically in Section 2.5, part b) of Proposition 2.9.

The Proof of Replacement Theorem

The proof of the Replacement Theorem in Mendelson's book is as follows:

From $\vdash \mathscr{C} \rightarrow \mathscr{A}(\overline{x})$ and $\vdash \mathscr{C} \rightarrow \mathscr{B}(\overline{x})$, we have by the Hypothetical Syllogism that $\vdash \mathscr{C} \rightarrow (\mathscr{A}(\overline{x}) \wedge \mathscr{B}(\overline{x}))$. Now, by the Distributive Law, we have $\vdash \mathscr{C} \rightarrow (\mathscr{A}(\overline{x}) \vee \mathscr{B}(\overline{x}))$. Finally, by the Addition Law, we have $\vdash \mathscr{C} \rightarrow \mathscr{A}(\overline{x}) \vee \mathscr{B}(\overline{x})$.

However, this seems to be a simplified version of the proof, and it's not entirely clear how the Replacement Theorem is being applied. Let's break down the steps and try to understand the reasoning behind the proof.

Step 1: Hypothetical Syllogism

The first step in the proof is to apply the Hypothetical Syllogism, which states that if $\vdash \mathscr{C} \rightarrow \mathscr{A}(\overline{x})$ and $\vdash \mathscr{C} \rightarrow \mathscr{B}(\overline{x})$, then $\vdash \mathscr{C} \rightarrow (\mathscr{A}(\overline{x}) \wedge \mathscr{B}(\overline{x}))$. This step is straightforward, as it simply involves combining the two premises using the conjunction operator.

Step 2: Distributive Law

The next step is to apply the Distributive Law, which states that $\vdash \mathscr{A} \wedge (\mathscr{B} \vee \mathscr{C}) \leftrightarrow (\mathscr{A} \wedge \mathscr{B}) \vee (\mathscr{A} \wedge \mathscr{C})$. However, in this case, we are not applying the Distributive Law to the conjunction of $\mathscr{A}(\overline{x})$ and $\mathscr{B}(\overline{x})$. Instead, we are applying it to the entire formula $\mathscr{C} \rightarrow (\mathscr{A}(\overline{x}) \wedge \mathscr{B}(\overline{x}))$. This is where things start to confusing.

Step 3: Addition Law

The final step in the proof is to apply the Addition Law, which states that $\vdash \mathscr{A} \vee \mathscr{B} \rightarrow \mathscr{A} \vee \mathscr{C}$. However, in this case, we are not applying the Addition Law to the disjunction of $\mathscr{A}(\overline{x})$ and $\mathscr{B}(\overline{x})$. Instead, we are applying it to the entire formula $\mathscr{C} \rightarrow \mathscr{A}(\overline{x}) \vee \mathscr{B}(\overline{x})$. Again, this is where things start to get confusing.

The Problem with the Proof

The problem with the proof is that it seems to be applying the Hypothetical Syllogism, Distributive Law, and Addition Law in a way that is not entirely clear. The steps seem to be jumping around and not following a logical sequence. Additionally, the proof is not providing any clear justification for why the Replacement Theorem is being applied.

A Possible Explanation

One possible explanation for the proof is that it is trying to show that if $\vdash \mathscr{C} \rightarrow \mathscr{A}(\overline{x})$ and $\vdash \mathscr{C} \rightarrow \mathscr{B}(\overline{x})$, then $\vdash \mathscr{C} \rightarrow \mathscr{A}(\overline{x}) \vee \mathscr{B}(\overline{x})$. However, this is not the Replacement Theorem. The Replacement Theorem states that if $\vdash \mathscr{C} \rightarrow \mathscr{A}(\overline{x})$, then $\vdash \mathscr{C} \rightarrow \mathscr{B}(\overline{x})$ for any term $\mathscr{B}(\overline{x})$ that is obtained by substituting terms for the variables in $\mathscr{A}(\overline{x})$.

Conclusion

In conclusion, the proof of the Replacement Theorem in Mendelson's Introduction to Mathematical Logic (6th Edition) is not entirely clear. The steps seem to be jumping around and not following a logical sequence. Additionally, the proof is not providing any clear justification for why the Replacement Theorem is being applied. A possible explanation for the proof is that it is trying to show that if $\vdash \mathscr{C} \rightarrow \mathscr{A}(\overline{x})$ and $\vdash \mathscr{C} \rightarrow \mathscr{B}(\overline{x})$, then $\vdash \mathscr{C} \rightarrow \mathscr{A}(\overline{x}) \vee \mathscr{B}(\overline{x})$. However, this is not the Replacement Theorem.

Further Reading

For further reading on the Replacement Theorem, I recommend checking out the following resources:

• Mendelson, E. (2015). Introduction to Mathematical Logic (6th ed.). Chapman and Hall/CRC.
• Enderton, H. B. (2001). A Mathematical Introduction to Logic (2nd ed.). Academic Press.
• Boolos, G., & Jeffrey, R. (2002).ability and Logic (4th ed.). Cambridge University Press.

References

• Mendelson, E. (2015). Introduction to Mathematical Logic (6th ed.). Chapman and Hall/CRC.
• Enderton, H. B. (2001). A Mathematical Introduction to Logic (2nd ed.). Academic Press.
• Boolos, G., & Jeffrey, R. (2002). Computability and Logic (4th ed.). Cambridge University Press.
  Frequently Asked Questions about the Replacement Theorem ===========================================================

Q: What is the Replacement Theorem?

A: The Replacement Theorem is a fundamental concept in mathematical logic that states that if a formula is provable, then any substitution of terms for the variables in that formula is also provable.

Q: What is the purpose of the Replacement Theorem?

A: The purpose of the Replacement Theorem is to provide a way to substitute terms for variables in a formula without changing the truth value of the formula.

Q: How does the Replacement Theorem work?

A: The Replacement Theorem works by allowing us to substitute terms for variables in a formula, as long as the substitution is done in a way that preserves the truth value of the formula.

Q: What are some common applications of the Replacement Theorem?

A: The Replacement Theorem has many applications in mathematical logic, including:

• Substitution: The Replacement Theorem allows us to substitute terms for variables in a formula, which is useful for simplifying complex formulas.
• Proofs: The Replacement Theorem is used in many proofs to substitute terms for variables in a formula, which helps to simplify the proof and make it more readable.
• Model theory: The Replacement Theorem is used in model theory to study the properties of models of a theory.

Q: What are some common mistakes to avoid when using the Replacement Theorem?

A: Some common mistakes to avoid when using the Replacement Theorem include:

• Not preserving the truth value: When substituting terms for variables, it's essential to preserve the truth value of the formula.
• Not following the correct substitution rules: The Replacement Theorem has specific rules for substitution, and it's essential to follow these rules to avoid errors.
• Not checking for validity: Before using the Replacement Theorem, it's essential to check that the formula is valid and that the substitution is correct.

Q: How can I apply the Replacement Theorem in my own work?

A: To apply the Replacement Theorem in your own work, follow these steps:

1. Identify the formula: Identify the formula that you want to substitute terms for variables in.
2. Check for validity: Check that the formula is valid and that the substitution is correct.
3. Substitute terms: Substitute the terms for variables in the formula, following the correct substitution rules.
4. Check the result: Check that the resulting formula is valid and that the substitution has preserved the truth value.

Q: What are some resources for learning more about the Replacement Theorem?

A: Some resources for learning more about the Replacement Theorem include:

• Textbooks: There are many textbooks on mathematical logic that cover the Replacement Theorem, including Mendelson's Introduction to Mathematical Logic and Enderton's A Mathematical Introduction to Logic.
• Online resources: There are many online resources available, including lecture notes, videos, and tutorials.
• Research papers: There are many research papers on the Replacement Theorem that provide a deeper understanding of the theorem and its applications.

Q: Can I use the Replacement Theorem in other areas of mathematics?

A: Yes, the Replacement Theorem can be used in other areas of mathematics, including:

• Algebra: The Replacement Theorem can be used in algebra to simplify complex expressions and to prove theorems.
• Geometry: The Replacement Theorem can be used in geometry to prove theorems about geometric shapes and to simplify complex expressions.
• Analysis: The Replacement Theorem can be used in analysis to prove theorems about functions and to simplify complex expressions.

Q: What are some common challenges when using the Replacement Theorem?

A: Some common challenges when using the Replacement Theorem include:

• Difficulty in applying the theorem: The Replacement Theorem can be difficult to apply, especially in complex situations.
• Difficulty in checking for validity: Checking for validity can be challenging, especially when dealing with complex formulas.
• Difficulty in following the correct substitution rules: Following the correct substitution rules can be challenging, especially when dealing with complex formulas.

Q: How can I overcome these challenges?

A: To overcome these challenges, follow these steps:

1. Practice: Practice applying the Replacement Theorem in different situations.
2. Check for validity: Check for validity carefully, especially when dealing with complex formulas.
3. Follow the correct substitution rules: Follow the correct substitution rules carefully, especially when dealing with complex formulas.
4. Seek help: Seek help from a teacher or a colleague if you are having difficulty applying the Replacement Theorem.