Classical Logic Laws

 Here’s the translation of the provided laws of logic into English:

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### **1. Classical Logic Laws (Aristotelian)**  

- **Law of Identity:**  

  Every entity is identical to itself:  

  \( A \equiv A \).  

  Example: "An apple is an apple."  

- **Law of Non-Contradiction:**  

  A statement cannot be both true and false simultaneously:  

  \( \neg (A \land \neg A) \).  

  Example: "It cannot be both raining and not raining at the same time."  

- **Law of Excluded Middle:**  

  Every statement is either true or false, with no middle ground:  

  \( A \lor \neg A \).  

  Example: "It is either raining or not raining."  

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### **2. Propositional Logic Laws**  

- **De Morgan's Laws:**  

  - \( \neg (A \land B) \equiv \neg A \lor \neg B \).  

  - \( \neg (A \lor B) \equiv \neg A \land \neg B \).  

- **Double Negation:**  

  \( \neg \neg A \equiv A \).  

- **Implication:**  

  \( A \to B \equiv \neg A \lor B \).  

- **Distributive Laws:**  

  - \( A \land (B \lor C) \equiv (A \land B) \lor (A \land C) \).  

  - \( A \lor (B \land C) \equiv (A \lor B) \land (A \lor C) \).  

- **Absorption Laws:**  

  - \( A \land (A \lor B) \equiv A \).  

  - \( A \lor (A \land B) \equiv A \).  

- **Commutative Laws:**  

  - \( A \land B \equiv B \land A \).  

  - \( A \lor B \equiv B \lor A \).  

- **Associative Laws:**  

  - \( (A \land B) \land C \equiv A \land (B \land C) \).  

  - \( (A \lor B) \lor C \equiv A \lor (B \lor C) \).  

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### **3. Rules of Inference**  

- **Modus Ponens:**  

  If \( A \to B \) is true and \( A \) is true, then \( B \) must be true.  

  \[

  \frac{A \to B \quad A}{B}

  \]  

- **Modus Tollens:**  

  If \( A \to B \) is true and \( B \) is false, then \( A \) must be false.  

  \[

  \frac{A \to B \quad \neg B}{\neg A}

  \]  

- **Hypothetical Syllogism:**  

  If \( A \to B \) and \( B \to C \), then \( A \to C \).  

  \[

  \frac{A \to B \quad B \to C}{A \to C}

  \]  

- **Disjunctive Syllogism:**  

  If \( A \lor B \) is true and \( \neg A \) is true, then \( B \) must be true.  

  \[

  \frac{A \lor B \quad \neg A}{B}

  \]  

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### **4. Predicate Logic Laws**  

- **Universal Generalization:**  

  If a property holds for an arbitrary element, it holds for all elements.  

  Example: If \( P(x) \) is true for any \( x \), then \( \forall x \, P(x) \).  

- **Universal Instantiation:**  

  If \( \forall x \, P(x) \) is true, then \( P(a) \) is true for any object \( a \).  

- **Contrapositive:**  

  \( \forall x (P(x) \to Q(x)) \equiv \forall x (\neg Q(x) \to \neg P(x)) \).  

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### **5. Non-Classical Logic Systems**  

- **Intuitionistic Logic:**  

  Rejects the Law of Excluded Middle (\( A \lor \neg A \)) in certain cases.  

- **Fuzzy Logic:**  

  Allows degrees of truth between "true" and "false" (e.g., 0.7 true).  

- **Paraconsistent Logic:**  

  Handles contradictions without system collapse.  

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These laws form the foundation of reasoning in mathematics, computer science, philosophy, and artificial intelligence.