Truth tables are also useful in analyzing the meaning of conjunctions like and and or. Let us start with and, because it is fairly simple. How would you describe the meaning of and? It is very difficult, most dictionaries do not even bother, giving definitions like ‘used to indicate a connection between two things’ or ‘used to link two words, phrases or sentences’. But using truth tables, it is very easy to show the semantic effect of joining two sentences with and. Let p be example (1a), and q example (1b):
We can then construct a truth table for these two sentences as well as the sentence in (1c), which connects (1a) and (1b) using and. There are four logical possibilities: p and q could both be true, they could both be false, p could be true and q could be false or vice versa:
| 𝗽 | 𝗾 | 𝗽 and 𝗾 |
|---|---|---|
| T | T | T |
| F | T | F |
| T | F | F |
| F | F | F |
As the third column shows, if we connect two statements using and, then the combined statement is only true if both statements are true. If one or both of them are false, the combined statement is also false. This is the meaning of and that is so difficult to describe using words!
The case of or is more complicated. Look at the following sentence and the truth table:
| 𝗽 | 𝗾 | 𝗽 OR 𝗾 |
|---|---|---|
| T | T | ? |
| F | T | T |
| T | F | T |
| F | F | F |
Starting from the bottom, obviously, if both p and q are false, then the combined sentence in (2) is also false. Also obviously, it is enough for one of the two statements to be true in order for the combined sentence to be true. But what if both are true? In logic, there are two types of or — one, where the combined sentence in (2) is true if both p and q are true (with the symbol ∨), and one, where (2) is false if both p and q are true (the so-called exclusive or with the symbol ⊻).
| 𝗽 | 𝗾 | 𝗽 ∨ 𝗾 | 𝗽 ⊻ 𝗾 |
|---|---|---|---|
| T | T | T | F |
| F | T | T | T |
| T | F | T | T |
| F | F | F | F |
Which of these two kinds of ‘or’ is expressed by the English word or depends very much on context. There are cases where two sentences contradict each other, so that only one of them can be true, as in (3a) — in this case, or will obviously be interpreted as exclusive; there are cases where the hearer is unlikely to care about which of the two statements is true, as long as one of them is, as in (3b) — in this case, the hearer will not feel that they were lied to if both statements are true. And there are cases where it is important to the hearer that one, and only one of the statements is true (as in 3c) — in this case they are likely to interpret or as exclusive and feel cheated if both turn out to be true:
Draw truth tables for the following combined sentences:
(i) if p then ¬q: If you turn down the music, I will not call the police.
(ii) p unless q: I will call the police unless you turn down the music.
(iii) p because q: The neighbor called the police because the music was too loud
CC-BY-NC-SA 4.0, Written by Anatol Stefanowitsch