Proving Arguments with Natural Deduction

1. The Building Blocks: Introduction and Elimination Rules

Natural deduction systems are built around a beautifully simple principle: for each logical connective, there is a pair of rules.

• An introduction rule allows us to create a formula with that connective.
• An elimination rule allows us to use a formula with that connective to derive something else.

This structure provides a systematic way to handle each part of a logical statement. To get an overview of the standard set of rules we'll be using, let's turn to a clear video explanation.

2. A Closer Look at Key Rules and Sub-proofs

Let's consolidate and formalize the most important rules from the video. We will write proofs in a line-by-line format, where each line has a number, a formula, and a justification (the rule used and the line numbers it depends on).

Modus Ponens (→E): This is the elimination rule for implication. It's one of the most fundamental rules of reasoning.
If you have proven $P\to Q$ and you have also proven $P$, you can derive $Q$.

Conditional Proof (→I): This is the introduction rule for implication. To prove a statement like $P\to Q$, you can assume $P$ for the sake of argument and show that $Q$ follows. This temporary assumption opens a sub-proof. Once you derive $Q$, you can "close" the sub-proof and conclude $P\to Q$, discharging the assumption.

The vertical line represents the scope of the assumption.

The rule for $\to I$ cites the entire range of the sub-proof.

Reductio ad Absurdum (¬I): This is the introduction rule for negation. To prove $\neg P$, you assume $P$ and show that it leads to a contradiction (⊥).
A contradiction arises when you derive a formula and its negation (e.g., $Q$ and $\neg Q$). The rule to formalize this is Negation Elimination (¬E).

This form of argument—assuming something to show it leads to an absurdity—is a cornerstone of mathematical and philosophical reasoning.

For a formal summary of all the rules, the following document is an excellent reference. Don't worry about the different notation (e.g., Ô ⇒ for →); the underlying structure is the same.

3. The Art of the Proof: A Strategic Approach

Knowing the rules is one thing; knowing how to assemble them into a valid proof is another. A common mistake is to start with the premises and randomly apply rules, hoping to stumble upon the conclusion. This "top-down" approach is inefficient. Your experience with designing algorithms and simulations suggests a more goal-oriented approach would be superior.

A much more effective strategy is to work bottom-up. Start by looking at your target conclusion and ask: "Which introduction rule would give me this formula?" This tells you what your penultimate step should be and, in turn, what new, simpler sub-goals you need to prove.

The bottom-up strategy transforms proof-finding from a random walk into a targeted search. You break the main goal into smaller, manageable sub-goals, recursively, until you reach formulas you can derive directly from your premises.

4. A Worked Example: Proof Construction in Action

Let's apply this strategy to derive the conclusion in the argument ¬P ∨ Q ⊢ P → Q.

1. Analyze the Goal: Our conclusion is P → Q. The main connective is →. The introduction rule for → is Conditional Proof (→I). This tells us our proof should have the following structure:

1. ¬P ∨ Q     premise
...
?. | P          assumption
   | ...
?. | Q
?. P → Q       →I

Our new, simpler goal is to derive Q inside a sub-proof that assumes P.

2. Analyze the Resources: Inside the sub-proof, we have the premise ¬P ∨ Q (from line 1) and the assumption P available to us. Our premise is a disjunction. The elimination rule for disjunction (∨E) is proof by cases. It requires us to show that our goal (Q) follows from both disjuncts.

So, we need two sub-proofs:

• Case 1: Assume ¬P and derive Q.
• Case 2: Assume Q and derive Q.

3. Construct the Proof:
The second case is trivial: if we assume Q, we have derived Q.
In the first case, we assume ¬P. But we also have P available from our outer assumption. This is a contradiction! From a contradiction, we can derive anything, including Q. The rule for this is often called Ex Falso Quodlibet (from falsehood, anything follows) or Contradiction Elimination (⊥E).

The following resource provides a formal, step-by-step solution to this exact problem, laid out in a common proof-checking format.

The final, complete proof looks like this:

This example demonstrates how the rules for sub-proofs (assumptions) can be nested within each other to systematically break a problem down until a solution is reached.

Conclusion

In this lesson, we transitioned from the semantic method of truth tables to the syntactic method of natural deduction. This system provides a way to prove an argument's validity by applying a set of truth-preserving rules of inference, much like constructing a mathematical proof.

Key Takeaways:

• Natural deduction is a proof system based on introduction and elimination rules for each logical connective.
• Key rules like Modus Ponens (→E) and Reductio ad Absurdum (¬I) form the backbone of many logical arguments.
• More complex rules like Conditional Proof (→I) and Proof by Cases (∨E) rely on creating temporary sub-proofs with assumptions that are later discharged.
• The most effective way to construct a proof is to work backward from the conclusion, using the introduction rule for its main connective to determine your sub-goals.

Preview of the Next Lesson:

Now that you have a powerful proof system at your disposal, we can use it to explore the relationships between different logical statements. In the next lesson, we will focus on logical equivalence. You will learn how to use natural deduction to formally prove that two different-looking formulas are, in fact, logically the same, establishing foundational laws like De Morgan's laws and contraposition from first principles.