# What rules of inference are used in this famous argument? “All men are mortal. Socrates is a man. Therefore, Socrates is mortal.”

As in the previous examples, first, we need to know the rules of inferences.

In this example, you need to realize that you must use predicates instead of propositional variables. Because of that, we also need the rules of inference for quantified statements.

Let’s use predicates to represent the argument, then we will apply the rules of inference.

Let P(x)=“x is mortal” and Q(x)=”x is a man”.

Now we can rewrite the argument as follows:

(1)∀x (Q(x)->P(x))            Premise (all men are mortal, or, if x is man, then x is mortal)

(2) Q(Socrates)                     Premise (Socrates is a man)

Therefore,

P(Socrates)                           Conclusion  (Socrates is mortal)

Let’s start applying the rules of inference.

(3) Q(Socrates) -> P(Socrates)      Universal instantiation using (1)

(4) P(Socrates)                                 Modus ponens using (2) and (3)

A common mistake I see students make in this type of exercise is using propositional variables instead of predicates. You need to know that if you don’t use predicates, your solution will be wrong. So, probably the most important thing you need to do to get this type of exercise right is to identify that you need to use predicates instead of propositional variables.

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