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)

Propositional Logic

The 9 Rules of Propositional Logic

Propositional logic is the most basic level of logic, dealing with inferences based on sentential connectives, such as "if...., then", "or", and "and". There are 9 rules of inference. Equipped with these, you will be able to assess the validity of most of the arguments you will ever here. Remember that a good deductive argument is one which:

1. Is formally and informally valid (i.e the conclusion(s) must follow from the premises in accord with the rules of logic)
2. Has premises which are more plausible than their negation

I'm going to post each of the 9 rules here in 9 separate posts, with examples and perhaps a short discussion on each. I'll start with the first, "modus ponens". Knowing these rules is vital in the task of christian apologetics. First we must know that our arguments and reasons for believing in the existence of god and the christian gospel are reasonable. Secondly, we must be able to point out any logical inconsistencies to refute and correct others (with gentleness and respect - 1 Peter 3:15)

Rule #1: modus ponens

1. P -> Q
2. P
3. Q

Examples

1. If John studies hard, he will get a good grade in logic
2. John studies hard
3. John will get a good grade in logic

1. If it is a Sunday, the library is closed
2. It is a Sunday
3. The library is closed