Propositional Logic (2)

Rule #2: modus tollens

1. P -> Q
2. ¬Q
3. ¬P

Examples

1. If it is raining (P), there are clouds in the sky (Q)
2. There are no clouds in the sky (¬Q)
3. It is not raining (¬P)

1. If the match is lit (P), then there is oxygen in the room (Q)
2. There is no oxygen in the room (¬Q)
3. The match is not lit (¬P)

This rule works by denying the consequent of the first premise. The consequent is the statement after the '->' (implies) symbol. The antecedent is the statement before the '->'. 

Necessary and sufficient conditions, and framing a premise correctly

This rule of logic brings up an important point. 

The antecedent is a sufficient condition of the consequent. If the antecedent is true, then that is sufficient to say that the consequent is true (i.e if it is raining, then we have enough to be able to say that there are clouds in the sky). 

The consequent is a necessary  condition of the antecedent. The antecedent is never true without the consequent. If Q is not true, then P is not true either, because Q is necessary for P to be true.

Therefore, when you write premises in the form of P->Q, make sure that the necessary condition is Q, and the sufficient condition is P. For example:

"The light is on only if there is a current flowing through the circuit"

Now the necessary part is "there is a current flowing through the circuit", since this must be fulfilled for the light to be on. The sufficient part is "The light is on". So we frame it like this:

1. If the light is on (P) -> there is a current flowing through the circuit (Q)
2. There is no current flowing through the circuit (¬Q)
3. The light is not on (¬P)

Logical Fallacy: Affirming the consequent

Using the previous example, what is wrong with this argument:

1. If the light is on (P), there is a current flowing through the circuit (Q)
2. There is a current flowing through the circuit (Q)
3. The light is on (P)

It is not good argument. Just because the current is flowing in the circuit, it doesn't mean the light is necessarily on. What if the filament in the light bulb was broken? Where the argument falls, is that is in the fallacy of 'Affirming the consequent'. You cannot affirm Q and then take P to be true. For an argument of the form modus ponens and modus tollens, you can either:

• Affirm the antecedent (say that P - the sufficient condition - is true)
• Deny the consequent (say that Q - the necessary condition - is false)