1. Idea of Rules of Inference
Rules of inference are standard logical patterns that allow us to derive a valid conclusion from given statements. They are the building blocks of correct reasoning. When a rule of inference is applied properly, the conclusion obtained is guaranteed to be true if the premises are true.
These rules ensure that arguments follow a valid and accepted logical form.
1.1. Why rules are needed
Rules of inference help avoid logical errors by providing reliable templates for reasoning. They show how new conclusions can be constructed from known facts.
2. Modus Ponens (Rule of Detachment)
Modus Ponens is one of the most commonly used inference rules. It allows us to conclude Q when we know that "If P, then Q" is true and P itself is true.
2.1. Form
\(P \Rightarrow Q\)
\(P\)
Therefore, \(Q\)
2.2. Example
Premises:
1. If it rains, then the ground is wet.
2. It rains.
Conclusion: The ground is wet.
3. Modus Tollens
Modus Tollens lets us conclude that P is false when we know "If P, then Q" is true and Q is false.
3.1. Form
\(P \Rightarrow Q\)
\(\neg Q\)
Therefore, \(\neg P\)
3.2. Example
Premises:
1. If a number is divisible by 4, then it is even.
2. The number is not even.
Conclusion: The number is not divisible by 4.
4. Hypothetical Syllogism
This rule links two implications together. If P implies Q and Q implies R, then P implies R.
4.1. Form
\(P \Rightarrow Q\)
\(Q \Rightarrow R\)
Therefore, \(P \Rightarrow R\)
4.2. Example
Premises:
1. If a figure is a square, then it is a rectangle.
2. If a figure is a rectangle, then it is a quadrilateral.
Conclusion: If a figure is a square, then it is a quadrilateral.
5. Disjunctive Syllogism
Disjunctive Syllogism uses the OR (disjunction) connective. If we know that "P or Q" is true and one of them is false, the other must be true.
5.1. Form
\(P \lor Q\)
\(\neg P\)
Therefore, \(Q\)
5.2. Example
Premises:
1. A number is either even or odd.
2. The number is not even.
Conclusion: The number is odd.
6. Simplification
This rule allows us to extract a single statement from a conjunction when "P and Q" is known to be true.
6.1. Form
\(P \land Q\)
Therefore, \(P\)
6.2. Example
Premise: "The number is even and greater than 10."
Conclusion: The number is even.
7. Conjunction
If two statements P and Q are both known to be true independently, this rule allows us to combine them into a single AND statement.
7.1. Form
\(P\)
\(Q\)
Therefore, \(P \land Q\)
7.2. Example
Premises:
1. "x is even."
2. "x is a multiple of 3."
Conclusion: "x is even and x is a multiple of 3."
8. Addition
The addition rule allows introducing an OR into a statement. If P is true, then "P or Q" is also true for any Q.
8.1. Form
\(P\)
Therefore, \(P \lor Q\)
8.2. Example
Premise: "7 is a prime number."
Conclusion: "7 is a prime number or 100 is even."
9. Notes and Observations
Important points about rules of inference:
- They provide guaranteed truth-preserving reasoning patterns.
- Each rule has a fixed structure that must be followed.
- Arguments built using these rules are automatically valid.
- Rules of inference form the foundation of formal proofs and logical deductions.