What rules of inference are used in this argument? \"No man is an island. Manhat

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There is no single rule of inference that can get us from (1) No man is an island, and (2) Manhattan is an island, to (3) Manhattan is not a man. Rather, we'll need to employ a number of different rules, each taking us part of the way. Different proof systems use different rules, but here is a pretty standard proof. 'Mx' = 'x is a man', 'Ix' = 'x is an island', and 'm' denotes Manhattan Your argument is of the following form: ~(Ex)(Mx & Ix), Im /- ~Mm We can derive the conclusion, ~Mm, from the two premises, ~(Ex)(Mx & Ix) and Im, as follows. (1) 1. ~(Ex)(Mx & Ix) Premise (2) 2. Im Premise (3) 3. Mm Assumption (2,3) 4. Mm & Im 2,3 &I (2,3) 5. (Ex)(Mx & Ix) 4 EI (2) 6. Mm > (Ex)(Mx & Ix) 3,5 CP (1,2) 7. ~Mm 1,6 MT This proof employs four rules of inference: Conjunction Introduction (&I): from P and Q, you may infer P & Q Existential Introduction (EI): from Fa, you may infer (Ex)Fx Conditional Proof (CP): if you have shown that P and Q jointly entail R, then from P, you may infer Q > R Modus Tollens (MT): from P > Q and ~Q, you may infer ~P We could also express your argument's first premise as a universally quantified conditional, rather than as a negated existentially quantified conjunction. If we went for this translation, the proof would go as follows. (1) 1. (Vx)(Mx > ~Ix) Premise (2) 2. Im Premise (1) 3. Mm > ~Im 1 UE (2) 4. ~~Im 2 DNI (1,2) 5. ~Mm 3,4 MT Universal Elimination (UE): from (Vx)Fx, you may infer Fa Double Negation Introduction (DNI): from P, you may infer ~~P It may well be that the rules of inference that you've been taught are slightly different from these, but my guess is that whatever proof system you use, you'll need to employ more than one rule of inference to deduce your argument's conclusion from its two premises.

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