So I’ll try to keep this brief.
Classical Logic, and a few other logics, suffer from the Principle of Explosion. Simply put anything follows from a contradiction. Thus given P and ~P anything follows. This is a great problem especially for the conditional in Classical Logic. To illustrate let us look at the familiar truth tables we all learned in our Symbolic Logic classes. (We will avoid the question of what mode most properly represents the conditional). So…
P → Q
1 1 1
1 0 0
0 1 1
0 1 0
There is the truth table for the conditional in Classical Logic. Notice the third and fourth lines where if the antecedent is false, regardless of the truth-value of the consequent the entire conditional is true. Thus if the moon is made out of cheese, then the President is Barack Obama. Clearly the moon is not made out of cheese, which being the antecedent makes the entire conditional true regardless of the truth-value of the consequent. This is very troublesome indeed. Given that most Logicians accept the conditional as material conditional, these difficulties have come to be known as the ‘paradoxes of material implication’ and while they are not strictly paradoxes, they are still counter-intuitive.
Back to our comic, we should first review the Law of Non-Contradiction. LNC states that P ^ ~P has a truth value of 0, that is false. It is not the case that the Cat is and is not on the Mat at the same time and in the same way. The LNC thus appears to make metaphysical claims of which I shall not discuss here, but they are ones of which I do believe accurately represent the nature of reality. Thus having established that P ^ ~P give a truth value of 0, we can now see how contradictions entail everything. Let us focus on the third line of the truth table once again.
0 1 1
Which corresponds to, ((P ^~P)→Q). Our Classical Logic truth table thus says that this conditional is on the whole true. In natural language, If I am in Jacksonville and not in Jacksonville, then Hydrogen is an element. This conditional appears to lack a notion of relevancy between the antecedent and consequent. What is the connection between I being in Jacksonville and the classification of Hydrogen as an element. Now I happen to be sympathetic to relevance logics, but again that will not be further discussed in this post. The fourth line of the truth table yields similar oddities, If I am in Jacksonville and not in Jacksonville, then Ron Paul is President of the United States. Now while the material conditional does give a truth value of 1 for the entire conditional, the facts of the world sadly do not match up.
What is to be done?
Well, some logics drop bivalence, others deny disjunctive syllogism, while some like Professor Priest bite the painful bullet and accept that there are true contradictions…
I happen to move onto the Strict Conditional, created by the first great Lewis of the previous century, as a more accurate representation of the conditional. But the difficulty is real and apparent within Philosophy of Logic.
The comic is from xkcd.com, always a source of amusement.