Truth Tables and Conditional Statements

Fig 1: Example of a truth table for one proposition P.

Fig 2: Examples of Truth Tables for two propositions P and Q.

Think of P as some statement that is verifiable, such as My dog is barking. This is easy to check and has a definite answer. P is either true (T) or false (F). We can also talk about the opposite case where My dog is not barking. We can call that not P, and it is obvious that whenever P is true, not P is false and vice versa. If we write this information down for each possibility, we get a truth table for P and not P (Fig 1). Easy!

We can bring in other statements too. Like Q: My neighbour is awake. Again, easy to check and a definite true or false. If we assess P and Q at the same time there are 4 possibilities. Both P and Q are true, P is true and Q is false, P is false and Q is true, and both P and Q are false. We can write these possibilities in a truth table again (Fig 2a). Now we can look at various conditions for P and Q (the condition that both P and Q must be true) and P or Q (the condition that either P or Q is true). We can do the same with not P and not Q.

Another thing we can do is create a conditional relationship between two propositions. My dog is barking therefore my neighbour is awake. P => Q. And here’s the part my tutees (and I) needed a bit of help with. The truth table for P => Q. We’ll think about it in words. 

  • We will consider the conditional statement under each of the possible circumstances where P and Q are true and false.
  • We consider the conditional statement to be true unless it is explicitly shown to be false by the circumstances.
  • We consider the conditional statement false when it is explicitly disproved by the circumstances.

From Fig 2b, the first case is P and Q both true. i.e. My dog is barking and my neighbour is awake. In this case we cannot say there is no causal link between the two truths, so the conditional statement is considered true because it is not proven false.

In the second case P is true and Q is false. So my dog is barking and my neighbour is asleep. In this case the conditional statement is disproven. If there was some link between my dog barking and my neighbour waking up, she would be awake any time my dog barked.

The third case is the tricky one. P false, Q true. My dog is not barking and my neighbour is awake. This circumstance doesn’t disprove the conditional statement. The neighbour could have been woken by something else. Nothing in the conditional statement says that can’t happen. So P => Q is still possibly true.

The fourth case is P and Q are both false. My dog is quiet and the neighbour is asleep. Again, nothing here disproves the conditional statement so we assume it to be true.

So logical truth is like innocence. It is assumed until proven false. Only one set of circumstances prove the conditional statement false in the dog barking/neighbour waking case. For this reason we must be careful when applying logic or we risk confusing correlation with causation. What if the P and Q statements were something like P: Broccoli is on sale and Q: My window is broken. Now there is no reason to make a connection between these two things, but if we put them in a truth table then we could have predicted the great broccoli riots of ‘97.

Or we could make P: Vaccination rates are rising and Q: Autism diagnosis rates are rising. Pure logic would tell us that, unless we have a year when vaccine rates rise and autism diagnosis rates fall, there’s no logical reason that the conditional statement P => Q is false. And yet it is.

For more on the Causation/Correlation logical fallacy see:

But far from being dumb and useless, Keaton’s heroines are often smart and brave: the girl in Sherlock Jr. solves the mystery by herself; Virginia in Our Hospitality sets out on her own to save Willie McKay’s life (she’s not much help, but it’s the thought that counts); the girl in Steamboat Bill Jr. defies her father to see Buster. The Mountain Girl in Battling Butler has a wonderful rock-throwing tirade, but soon  shows she’s not the sort to hold grudges.
—  Imogen Sara Smith, Buster Keaton: The Persistence of Comedy