Section 1.5 Disjunction
Definition 1.5.1.
The disjunction of two statements \(\phi\) and \(\psi\) is the statement \(\quine{\phi\text{ or } \psi}\text{.}\) We will write this as \(\quine{\phi \vee \psi}\text{.}\)
The truth-value of the disjunction operates like this:
\(\quine{\phi \vee \psi}\) is true exactly when either \(\phi\) is true or \(\psi\) is true.
Here is the truth table for the disjunction:
| \(\phi\) | \(\psi\) | \(\quine{\phi \vee \psi}\) |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Note that “or” in this sense is inclusive not exclusive: if \(\phi\) and \(\psi\) are both true, then \(\quine{\phi\vee \psi}\) is true as well.
Question 1.5.2. A quick check.
What is the truth-value of the statement “\(2\) is even or \(57\) is prime”?
It is true, because \(2\) is even.
Once you have these basic connectives, you can combine statements to your heart's content, and you can analyze the results.
Example 1.5.3. Combining or and and.
Let's have a look at the statement \(\quine{(\phi \vee \psi) \wedge \chi}\text{.}\) Let's analyze the truth table by dealing with what's in parentheses first, and then combining that with the statement \(\chi\text{:}\)
| \(\phi\) | \(\psi\) | \(\chi\) | \(\quine{\phi \vee \psi}\) | \(\quine{(\phi \vee \psi) \wedge \chi}\) |
| T | T | T | T | T |
| T | T | F | T | F |
| T | F | T | T | T |
| T | F | F | T | F |
| F | T | T | T | T |
| F | T | F | T | F |
| F | F | T | F | F |
| F | F | F | F | F |
This is the kind of sentence that we don't say too often in natural language (at least, outside of the legal profession): it asserts that both \(\quine{\phi \vee \psi}\) is true and \(\chi\) is true. Maybe a reasonable way to say this in English is “Either \(\phi\) or \(\psi\) is true, and \(\chi\) is true too.”
Example 1.5.4. The importance of being parenthesized.
The previous example brings up a critical issue.
We must write our connectives carefully!For example, consider the statement
“\(57\) is prime or \(2\) is even and \(937\) is prime.”
What do we mean by this statement? As written, it's ambiguous, and that ambiguity affects the truth-value!
Parentheses clarify things: the statement
“(\(937\) is prime or \(2\) is even) and \(57\) is prime”is false, whereas the statement
“\(937\) is prime or (\(2\) is even and \(57\) is prime)”is true.
Here's the truth table for \(\quine{\phi \vee (\psi \wedge \chi)}\text{;}\) notice the differences from the truth table for \(\quine{(\phi \vee \psi) \wedge \chi}\) above:
| \(\phi\) | \(\psi\) | \(\chi\) | \(\quine{\psi \wedge \chi}\) | \(\quine{\phi \vee (\psi \wedge \chi)}\) |
| T | T | T | T | T |
| T | T | F | F | T |
| T | F | T | F | T |
| T | F | F | F | T |
| F | T | T | T | T |
| F | T | F | F | F |
| F | F | T | F | F |
| F | F | F | F | F |