I would argue that the one type of statement that holds up the most "weight" in terms of logic and how we reason through problems is the conditional, or "if statement". A conditional is of the form, "If A is true, then B is true", where A and B are some statement of any kind that can be true or false. This is denoted in symbols as \( A \to B \).
The guiding idea you should have with logic is that something that's logical might be complicated, but it should always make sense. In other words, you should believe what you're saying when you talk about logic or think about something from the lens of logic. There have been debates about certain logical constructions which we'll hopefully get to cover eventually, and you are free to push the boundaries and see what you think of logic principles in practice, but pretty much everyone fundamentally believes in the construction of the conditional statement. The case of one thing being true implying something else is true is a fundamental idea. When we think about causation, we're thinking about a much more complicated form of an if statement involving time and intention, like "If I didn't go to the store that day, I wouldn't have met Randall." In some sense, to deny the conditional applies to logic is to deny causation, and that's a huge thing to also have to give up.
So, what intuitively makes sense to us about a conditional? Well, if we're still looking at \( A \to B \), \( A \) being true means \( B \) must be true. If we found a situation where \( A \) were true but \( B \) were false, we'd have to conclude then that someone lied when they first told us "If A, then B".
But we still have some questions. What happens when \( A \) is false? Well, we don't really know anything. If \( A \) is always false for one reason or another, then the original statement \( A \to B \) didn't really mean anything to begin with. So, whether it's true or false doesn't really matter; it's vacuously true in that it's true only in the sense that it's a statement that doesn't actually apply to anything and cannot be wrong.
I think it's helpful to think of a conditional statement like a promise, or a contract. When \( A \) is true, \( B \) is true. But a contract doesn't have to cover every case anyone could possibly conceive of; only the cases that it specifically covers, which in this case is the case \( A \) is true. The contract is automatically fulfilled when \( A \) is false.
For people coming from my other writing, it's like copyleft software licenses. If you distribute the software, then you have to follow the license guidelines. If you don't distribute it, the license is being followed by default because the guidelines on distribution don't apply in that case.
An interesting property of the conditional statement is that it is asymmetrical. You cannot reverse the direction of an if statement and arrive at the same thing. That's what's captured with its symbol, \( \to \), which points in a particular direction. In the statement \( A \to B \), "If A, then B", the statement in the A location is called the antecedent, and B is the consequent. That's just a fancy term and it's simply to make it easier to talk about conditional statements. If you remember this term from school, we say that the conditional statement is not commutative, where commutative means an operation can be reversed. For example, in the world of math addition and multiplication are commutative because you can flip them, but subtraction and division are not commutative.
If you could reverse the direction of a conditional, what would happen? Well, then you would have something like \( A \to B \) and \( B \to A \). In this case, notice that A and B have to either both be true, or both be false, because if one were true and the other false, then one of the conditional statements was a lie. Thus, a bidirectional conditional statement is effectively an equality of truth: A and B are tied, where proving or disproving one likewise means you proved or disproved the other. Such a logical statement does exist, and it's called a biconditional, and is denoted \( A \leftrightarrow B \), a shorthand for \( A \to B \) and \(B \to A \). A biconditional is written formally with the English sentence structure "A if and only if B", or in shorthand, "A iff B". As discussed in the introduction, biconditionals come up all the time in ordinary English even if the sentence structure here "if and only if" doesn't get said a lot outside of academic writings. For our purposes though, I will try to be very explicit with if and only if constructions when I mean to invoke one.
An ordinary conditional statement can be reversed in a particular way, though. Say we have "If A, then B". Then, what if B were false? Well, A couldn't be true. If A were true, then B would have to be true and that doesn't make any sense because it's false. So, B being false means A must be false. Thus, "If A, then B" means the same thing as saying "If not B, then not A". This equivalence we have just shown is a special logical construction with a fancy term, modus tollens. If you ever hear modus ponens, that effectively is just talking about a conditional, whereas modus tollens is talking about this flipped construction, not \( B \to \) not \( A \).
I plan on covering truth tables where you can see of this stuff more concretely soon, but I feel like I need to cover something sideways to this discussion first, intuitionism, before I can introduce truth tables. I also have no idea how to draw them well with the current site setup. But look out for that here at some point in the future.
June 5, 2026