Define logical terms like proposition, fallacy, contradiction, contingency, argument and give illustrations. Verify tautologies/contradictions.

Hey Invincible, good to see you again. Here are the solutions to your questions: 1) Define the following terms and give illustrations where necessary: a. Proposition: A declarative sentence that is either true or false, but not both. Illustration:* "The sky is blue" is a proposition. "Is the sky blue?" is not. b. Fallacy: A mistaken belief or an error in reasoning that renders an argument invalid. Illustration:* The argument "All cats have four legs. My dog has four legs. Therefore, my dog is a cat" is a fallacy of the undistributed middle. c. Contradiction: A compound proposition that is always false, regardless of the truth values of its constituent propositions. Illustration:* P P is a contradiction. d. Contingency: A compound proposition that is neither a tautology nor a contradiction; its truth value depends on the truth values of its constituent propositions. Illustration:* P Q is a contingency, as its truth value can be true or false depending on P and Q. e. Argument: A set of statements, one of which is designated as the conclusion and the others as premises, intended to provide support for the conclusion. Illustration:* "All men are mortal. Socrates is a man. Therefore, Socrates is mortal" is an argument. 2) Verify whether each of the following statements is a tautology or contradiction: i) [(A B) (C B)] A This statement is a contingency. Explanation:* If A is True, B is True, and C is True, then (T T) (T T) is T T, which is True. The implication becomes T T, which is T F, which is False. Since it can be false, it is not a tautology. If A is False, then A is True, making the entire implication True regardless of the antecedent. Since it can be both true and false, it is a contingency. ii) (A B) (A B) This statement is a contradiction. Explanation:* (A B) (A B) Using De Morgan's Law, (A B) ( A B): (A B) ( A B) Rearranging terms (associativity and commutativity of ): (A A) (B B) Since A A is always False and B B is always False: F F F The statement is always false. iii) [(A B) A] A This statement is a tautology. Explanation:* [(A B) A] A Using the equivalence P Q P Q: [( A B) A] A Using the Distributive Law: [( A A) (B A)] A Since A A is always False: [F (B A)] A (B A) A Using De Morgan's Law: ( B A) A