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I would appreciate a critique of the my proofs. Thank you

Analyze the logical forms of the following statements. Use $A$ to represent “Amanda has a dog,” $B$ to represent “Bill has a dog,” and $C$ to represent “Cathy has a cat” to write each as a symbolic statement.

a. Either Amanda or Bill has a dog.

This is a disjunctive statement. Disjunctive connectives are are symbolized by a “$\lor$” which means “or.”

It is given that $A$ represents “Amanda has a dog” and $B$ represents “Bill has a dog.” This statement is therefore symbolized $A \lor B$.

b. Neither Amanda nor Bill has a dog, but Cathy has a cat.

The atomic statement “Amanda has a dog is” is symbolized by $A$. The atomic statement “Bill has a dog” is symbolized by $B$. The atomic statement “Cathy has a cat” is symboized by $C$. Since it is the case that the first statement is false, the fact that Alice does not have a dog is symbolized via the negation of the original statement. That is, $¬ A$.

Since it is the case that the second atomic statement is false, the fact that Bill does not have a dog is symbolized via the negation of the original statement. That is, $¬ B$.

The third statement is true, so the fact that Cathy has a cat therefore retains its original symbol, $C$.

Since it is not true that Amanda has a dog and since it is not true that that Bill has a dog, we have a conjunct statement because the logical connective “and” makes it so. The symbol for “and” is $\lor$. This statement is therefore symbolized as $(¬ A \land ¬ B)$.

By DeMorgan’s Law, which states that when it is not the case that $A$ and it is not the case that $B$, then it is neither the case that $A$ nor is it the case that $B$. This is symbolized as $¬ (A \lor B)$.

We still must account for the fact that Cathy exists within the universe of Amanda and Bill and that she has a cat. Since she has been included in the original statement, if we view the compound statement, $¬ (A \lor B)$, as being one statement, and Cathy’s ownership of a cat, $C$, as being another statement, we can combine them in symbolic language by using adding the logical connective “and.” Hence we have $¬(A \lor B) \land C$.

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Absolutely correct! In general, "Neither $P$ nor $Q$" is quite directly translated to "$\neg ( P \lor Q )$", but your method of reasoning actually is the right way to do it, namely you analyze the English to see what it means, before you construct a sentence in first-order logic to express that same meaning.

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