Direct proofs in propositional logic are step-by-step arguments that start with premises and end with a conclusion. Each step is justified by rules of inference or replacement, creating a logical chain from given statements to the final result.
Constructing a direct proof involves applying logical rules to derive new statements until reaching the conclusion. Key components include premises, conclusion, and proof steps. Justification and evaluation ensure the proof's correctness and completeness.
Direct Proof Method in Propositional Logic
Structure of direct proofs
- Sequence of steps starting with premises and ending with the conclusion
- Each step justified by a rule of inference or replacement
- Components include:
- Premises: Given statements assumed to be true (If it is raining, then the ground is wet)
- Conclusion: Statement to be proved (Therefore, the ground is wet)
- Proof steps: Logical deductions leading from premises to conclusion (It is raining, so based on the premise, the ground must be wet)
Construction of direct proofs
- Begin with the premises of the given statement
- Apply rules of inference and replacement to derive new statements
- Proceed until the conclusion is reached (Premise: If x > 3, then x > 2; Conclusion: x > 2)
- Ensure each step is logically valid and follows from previous steps
- Example proof of $p \rightarrow (q \rightarrow r)$ from premises $p \rightarrow q$ and $q \rightarrow r$:
- $p \rightarrow q$ (Premise)
- $q \rightarrow r$ (Premise)
- $p$ (Assumption)
- $q$ (Modus Ponens, steps 1 and 3)
- $r$ (Modus Ponens, steps 2 and 4)
- $p \rightarrow r$ (Conditional Proof, steps 3-5)
- $p \rightarrow (q \rightarrow r)$ (Exportation, step 6)
Justification in direct proofs
- Rules of inference justify steps:
- Modus Ponens: If $p$ and $p \rightarrow q$ are true, then $q$ is true (If it is raining and if it is raining, then the ground is wet, therefore the ground is wet)
- Modus Tollens: If $\neg q$ and $p \rightarrow q$ are true, then $\neg p$ is true (If the ground is not wet and if it is raining, then the ground is wet, therefore it is not raining)
- Hypothetical Syllogism: If $p \rightarrow q$ and $q \rightarrow r$ are true, then $p \rightarrow r$ is true (If I study, I will pass the test. If I pass the test, I will graduate. Therefore, if I study, I will graduate)
- Disjunctive Syllogism: If $p \lor q$ and $\neg p$ are true, then $q$ is true (Either it is a car or a truck. It is not a car. Therefore, it is a truck)
- Constructive Dilemma: If $p \rightarrow q$, $r \rightarrow s$, and $p \lor r$ are true, then $q \lor s$ is true (If I go to the store, I will buy milk. If I go to the bank, I will deposit a check. I will either go to the store or the bank. Therefore, I will either buy milk or deposit a check)
- Rules of replacement also justify steps:
- Double Negation: $p \equiv \neg \neg p$ (It is true that the sky is blue $\equiv$ It is not true that the sky is not blue)
- De Morgan's Laws: $\neg (p \land q) \equiv \neg p \lor \neg q$ and $\neg (p \lor q) \equiv \neg p \land \neg q$ (It is not the case that I will buy a car and a house $\equiv$ I will not buy a car or I will not buy a house)
- Commutative Laws: $p \land q \equiv q \land p$ and $p \lor q \equiv q \lor p$ (I will have cake and ice cream $\equiv$ I will have ice cream and cake)
- Associative Laws: $(p \land q) \land r \equiv p \land (q \land r)$ and $(p \lor q) \lor r \equiv p \lor (q \lor r)$
- Distributive Laws: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$ and $p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$
- Each proof step must cite the appropriate rule and the steps it is applied to
Evaluation of direct proofs
- Correctness verified by:
- Each step following logically from previous steps
- Rules of inference and replacement applied correctly
- Conclusion matching the statement to be proved
- Completeness verified by:
- All necessary steps included
- Proof starting with premises and ending with conclusion
- No essential steps or justifications missing
- A correct and complete proof satisfies all the above criteria (All steps are logically valid, rules are applied correctly, and the proof is comprehensive from start to finish)