Weird Conditionals

Truth Functional Logic and the Weird Conditionals
(I will use an arrow instead of a horseshoe)

The logic we are studying in our course is truth-functional.  This is a very useful thing and it applies to a huge number of actual arguments and situations we encounter every day. Of course, there are some limitations but those need not disturb us because the types of statements that are not truth-functional can still be analyzed using the tools you will acquire plus some other ones. 

As I said in class, the most controversial truth-functional analysis of a logical connective is the analysis of conditionals. Conditionals are if…, then… statements. In ordinary language, we assume that when someone states a conditional he intends the antecedent to be related to the consequent in some way. However, our analysis of conditionals allows some of them to be true even when there is no relationship between the antecedent and the consequent. Here’s the truth-functional analysis of conditionals:

“If p, then q” is true whenever either the antecedent is false or the consequent is true; it is false otherwise

If you can grasp the above way of putting it, you are in good shape (note that if you cannot yet grasp the above way of putting it, it does not follow that you are not in good shape; that’s fallacious reasoning)

P Q // P àQ
T T //      T
T F //      F
F T //      T
F F //      T

Because of this, some logicians have attempted to develop logics that avoid this so-called paradox. Let’s look at the paradox a bit more closely. 

1.     If there is such a thing as morality, then God exists. 

The above conditional tells us that the existence of morality (e.g. some moral claims are true independent of our psychological states—beliefs, desires, etc) is sufficient for the existence of God. So, morality requires God according to 1. But the following seems quite odd:

2.     If there is such a thing as morality, then Genevieve Alexander exists. 

However, given the truth-functional analysis 2 is true. But Genevieve’s existence is not a necessary condition for the existence of morality. If Genevieve never existed or ceases to exist, morality will be just fine. So, the fact that conditionals are true whenever their consequent is true seems to be weird. Now consider this one:

3.     If there is no such thing as morality, then Genevieve Alexander exists

But the antecedent is false, and thus on our analysis the whole conditional is true. Now it looks like Genevieve’s existence is a necessary condition for the non-existence of morality. Indeed, 3 implies that when the world failed to contain Genevieve (what an awful world that was J) morality existed. Yikes! 

There are a few different ways to think about the weird situation we find ourselves in with respect to conditionals. First, we might abandon the analysis altogether. I think this is a bad route to take. Second, we might accept the analysis and simply remember that it applies to a wide range of conditionals (all those conditionals where there really is a connection between the antecedent and the consequent) but fails to apply to weird ones; perhaps one’s we’d never be inclined to think or especially say anyway. Third, we might accept the analysis and argue that it applies to all indicative conditionals just fine. 

Extra credit (a decent amount) will be given to any student who pursues this topic further. You must chat with me about what to read and what you need to do to demonstrate to me that you have in fact done the work. 

(Please remember that this is an informal blog and I am leaving out all sorts of technicalities)





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