Problem 15 - Discrete Mathematics with Applications

Question

Give the simplest possible conclusion in each argument. Assume each premise is true. $$\begin{aligned} &p \rightarrow \sim q\\\ &\sim r \rightarrow q\\\ &p \end{aligned}$$

Step-by-Step Solution

Answer

The simplest possible conclusion is \(r\) is true.

Step 1: Write down the premises

We are given the following premises: 1. \(p \rightarrow \sim q\) 2. \(\sim r \rightarrow q\) 3. \(p\)

Step 2: Apply Modus Ponens on Premise 1 and 3

Since we have \(p\) as a premise, we can apply Modus Ponens on premise 1 and 3: If \(p\) is true, then \(\sim q\) is true. So, now we know that \(\sim q\) is true.

Step 3: Apply Modus Tollens on Premise 2 and \(\sim q\)

Now that we have deduced that \(\sim q\) is true, we can apply Modus Tollens on premise 2 and \(\sim q\): If \(\sim q\) is true, then \(\sim (\sim r)\) is true, which implies that \(r\) is true. So, now we know that \(r\) is true.

Step 4: State the simplest possible conclusion

The simplest possible conclusion we can draw from the given premises is that \(r\) is true.

Key Concepts

Modus Ponens

Modus Ponens is a fundamental rule of inference in propositional logic. It allows us to conclude a statement when two initial conditions are satisfied:

• An implication of the form \( p \rightarrow q \)
• The premise \( p \) is true

Once both these conditions hold, we can conclude that \( q \) must also be true. This inference is straightforward and is widely used in logical reasoning.
For instance, in our exercise, we applied Modus Ponens using the premise \( p \rightarrow \sim q \) and the fact \( p \) is true. By Modus Ponens, we can infer that \( \sim q \) is true. This simple yet powerful rule facilitates progression in logical arguments and helps deduce new facts from given premises.

Modus Tollens

Modus Tollens is another essential rule of inference in propositional logic. It comes into play when:

• There is an implication of the form \( p \rightarrow q \)
• The negation of the conclusion \( \sim q \) is true

If these conditions are met, we can deduce that the negation of the premise \( \sim p \) must also be true. Modus Tollens allows us to "work backwards" from the conclusion to the premise.
In the given exercise, after determining \( \sim q \) is true by Modus Ponens, we used Modus Tollens on the premises \( \sim r \rightarrow q \) and \( \sim q \). By inferring that \( \sim(\sim r) \) is true, we concluded that \( r \) is true. Modus Tollens is particularly useful when the result of a condition does not occur, leading us to negate the initial assumption.

Logical Reasoning

Logical reasoning is the process of using a structured and methodical approach to deduce new information or make decisions based on existing knowledge. In propositional logic, logical reasoning employs rules like Modus Ponens and Modus Tollens to draw conclusions from a set of premises.
In our exercise, logical reasoning involved identifying valid inference rules and applying them to the premises provided. We started with known implications and premises, then made logical deductions to arrive at a conclusion.
Good logical reasoning requires clear understanding of the relationships between propositions and ensures that each step is both valid and logically sound. This skill is invaluable in problem-solving across various disciplines, from mathematics and computer science to everyday decision-making.