Question

What rules of inference are used in this argument? “No man is an island. Manhattan is an island. Therefore, Manhattan is not a man.”

Short answer

The inference rules such as universal instantiation, double negation and Modus tollens are used in the famous argument like “No man is an island. Manhattan is an island.”

Step-by-step solution

Definition of Argument    

Theargument is a collection of propositions that have premises and a conclusion.

Rules of inference used in given argument

Suppose \(P\left( x \right)\) be the statement “\(x\) is a man,” whereas \(Q\left( x \right)\) be the statement “\(x\) is an island.”

The premise \(1\) is \(\forall x\left( {P\left( x \right) \to \neg Q\left( x \right)} \right)\)and premise \(2\) is \(Q\left( {Manhattan} \right)\).

Now, apply universal instantiation rule for \(1\).

\(\left( {P\left( {Manhattan} \right) \to \neg Q\left( {Manhattan} \right)} \right)\)

Follow double negation law for premise\(2\).

\(\neg \left( {\neg Q\left( {Manhattan} \right)} \right)\)

Apply modus tollens law to the result ofuniversal instantiation and double negation.

\(\neg P\left( {Manhattan} \right)\)

Therefore, the rules of inference such as universal instantiation, double negation and Modus tollens are used in the famous argument like “No man is an island. Manhattan is an island.”