Define proposition, fallacy, contradiction, contingency, and argument, and provide illustrations. Also, verify tautologies/contradictions.

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: $P \land \sim P$ 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: $P \land Q$ 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) $[(A \Leftrightarrow B) \lor (C \Leftrightarrow B)] \Rightarrow \sim A$ This statement is a contingency.

  • Explanation: If $A$ is True, $B$ is True, and $C$ is True, then $(T \Leftrightarrow T) \lor (T \Leftrightarrow T)$ is $T \lor T$, which is True. The implication becomes $T \Rightarrow \sim T$, which is $T \Rightarrow F$, which is False. Since it can be false, it is not a tautology. If $A$ is False, then $\sim A$ is True, making the entire implication True regardless of the antecedent. Since it can be both true and false, it is a contingency.

• ii) $(A \land B) \land \sim (A \lor B)$ This statement is a contradiction.

  • Explanation: $$(A \land B) \land \sim (A \lor B)$$ Using De Morgan's Law, $\sim (A \lor B) \equiv (\sim A \land \sim B)$: $$(A \land B) \land (\sim A \land \sim B)$$ Rearranging terms (associativity and commutativity of $\land$): $$(A \land \sim A) \land (B \land \sim B)$$ Since $A \land \sim A$ is always False and $B \land \sim B$ is always False: $$F \land F$$ $$F$$ The statement is always false.

• iii) $[(A \Rightarrow B) \land A] \Rightarrow A$ This statement is a tautology.

  • Explanation: $$[(A \Rightarrow B) \land A] \Rightarrow A$$ Using the equivalence $P \Rightarrow Q \equiv \sim P \lor Q$: $$\sim [(\sim A \lor B) \land A] \lor A$$ Using the Distributive Law: $$\sim [(\sim A \land A) \lor (B \land A)] \lor A$$ Since $\sim A \land A$ is always False: $$\sim [F \lor (B \land A)] \lor A$$ $$\sim (B \land A) \lor A$$ Using De Morgan's Law: $$(\sim B \lor \sim A) \lor A$$