Logic: Symbols, Laws, Quantifiers & Inference Rules

• Study notes on logic, including symbols, laws, quantifiers, and inference rules are represented.

Logical Symbols Table

• Provides the symbols list, their meanings, and examples.
• $\neg$: Negation, e.g., $\neg P$
• $\land$: Conjunction (and), e.g., $P \land Q$
• $\lor$: Disjunction (or), e.g., $P \lor Q$
• $\rightarrow$: Conditional (if... then), e.g., $P \rightarrow Q$
• $\leftrightarrow$: Biconditional (if and only if), e.g., $P \leftrightarrow Q$

Logical Laws

• Covers logical laws like idempotence, associativity, commutativity, distributivity, identity, negation, double negation, DeMorgan's laws, absorption, conditional, and biconditional.

• Idempotence:

  • $P \land P \equiv P$
  • $P \lor P \equiv P$

• Associativity:

  • $(P \land Q) \land R \equiv P \land (Q \land R)$
  • $(P \lor Q) \lor R \equiv P \lor (Q \lor R)$

• Commutativity:

  • $P \land Q \equiv Q \land P$
  • $P \lor Q \equiv Q \lor P$

• Distributivity:

  • $P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)$
  • $P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R)$

• Identity:

  • $P \land \top \equiv P$
  • $P \lor \bot \equiv P$

• Negation:

  • $P \land \neg P \equiv \bot$
  • $P \lor \neg P \equiv \top$

• Double Negation:

  • $\neg (\neg P) \equiv P$

• DeMorgan's Laws:

  • $\neg (P \land Q) \equiv \neg P \lor \neg Q$
  • $\neg (P \lor Q) \equiv \neg P \land \neg Q$

• Absorption:

  • $P \land (P \lor Q) \equiv P$
  • $P \lor (P \land Q) \equiv P$

• Conditional:

  • $P \rightarrow Q \equiv \neg P \lor Q$

• Biconditional:

  • $P \leftrightarrow Q \equiv (P \rightarrow Q) \land (Q \rightarrow P)$

Quantifiers

• $\forall$: "for all"
  • Example: $\forall x (P(x))$ - For all $x$, $P(x)$ is true.
• $\exists$: "there exists"
  • Example: $\exists x (P(x))$ - There exists an $x$ such that $P(x)$ is true.

Inference Rules

• Modus Ponens:
  • Premises: $P \rightarrow Q$, $P$
  • Conclusion: $\therefore Q$
• Modus Tollens:
  • Premises: $P \rightarrow Q$, $\neg Q$
  • Conclusion: $\therefore \neg P$
• Hypothetical Syllogism:
  • Premises: $P \rightarrow Q$, $Q \rightarrow R$
  • Conclusion: $\therefore P \rightarrow R$
• Disjunctive Syllogism:
  • Premises: $P \lor Q$, $\neg P$
  • Conclusion: $\therefore Q$
• Addition:
  • Premise: $P$
  • Conclusion: $\therefore P \lor Q$
• Simplification:
  • Premise: $P \land Q$
  • Conclusion: $\therefore P$
• Conjunction:
  • Premises: $P$, $Q$
  • Conclusion: $\therefore P \land Q$
• Resolution:
  • Premises: $P \lor Q$, $\neg P \lor R$
  • Conclusion: $\therefore Q \lor R$