16.1 The Conditional: →
BOOL is a powerful tool. It is a complete logical system for conjunction, disjunction and negation: for any argument that is valid on account of those operators, BOOL can prove its validity.
But BOOL is also limited. For example, it doesn't have a natural way of expressing if-then claims, which are used in a lot of the reasoning we do in our everyday lives.
So in this chapter we create a new logical system called PROP, which is an extension of BOOL. It has everything from BOOL plus two new symbols: → and ↔.
→ is called the conditional or arrow or if-then. Here's how it works:
If Pia is guilty, then Quinn is guilty
is translated like this:
Now you try.
If-then sentences like if P then Q are called conditionals because they assert a kind of relationship or conditionality between two things: the second one is true if the first one is. Sometimes people also call them hypotheticals, because they posit a "what-if" scenario, but we'll stick with the term conditionals.
The "if part" of the conditional is called the antecedent, and the "then part" is called the consequent. For the sentence if P, then Q, the antecedent is P and the consequent is Q.
If P then Q is not the only way of expressing conditionality in English. These all say the same thing:
- If P then Q
- Q whenever P
- Q if P
- P only if Q
We translate all of these sentences as P→Q. Scanning those sentences in English from left to right, sometimes the Q comes first and sometimes the P comes first, but P is the antecedent for all of them. The lesson is that you can't just assume that antecedent is whatever comes grammatically first in English: you have to think about what is being claimed; which part is dependent or conditional on the other part.
There is another mistake to avoid when learning conditionals: you cannot assume conditionals are causal. The claim P→Q does not make a causal claim, that P causes Q. If just claims that if P is true, then Q is true too.
For example, say Pia's alias is Crusher. Then I might claim: if Crusher is guilty, then Pia is guilty. But that doesn't mean Crusher caused Pia to be guilty. Crusher and Pia are two ways to designate the same guilty person; one doesn't cause the other.
But sometimes causal claims can create conditionals. Imagine that Uma and Tamar are good friends, so if Tamar wins the race, that will cause Uma to be happy. Then we might claim: If Tamar wins, then Uma will be happy. The causal relationship in the world is what makes that conditional true.
It is important to remember, though, that conditional claims and causal claims are not the same thing. The logical power of conditionals is quite general and is not just about causality.