True Frogs
Help the frog cross the "truth table pond" to meet its friends. Watch out because any lily pad in a False square of the truth table will sink!
Help the frog cross the "truth table pond" to meet its friends. Watch out because any lily pad in a False square of the truth table will sink!
Symbolic logic involves the study of statements (essentially, any claim that can be true or false, which is called its truth value.)
Statements can be connected to create new statements. For instance, use "and" to connect the false statement "Every cat is a truck" with the true statement "1 + 1 = 2" and you get a new statetment: "Every cat is a truck and 1 + 1 = 2." Remember, even if something is false it can still be a statement.
The terms that connect statements are called connectives. These include "and", "or", "but", and more. The new statements are called compound statements. We say a statement is a simple statement if it is not compound statement. We usually use a lowercase letter, like p, q, r, to represent a simple statement. Connectives have special symbols, as shown in the table on the next page.
| Symbol | English | Truth Value |
|---|---|---|
| ~ | not | ~p has the opposite truth value as p |
| ∧ | and | p ∧ q is true only when both p and q are true (otherwise it is false) |
| ∨ | or | p ∨ q is only false when both p and q are false (otherwise it is true) |
| → | if... then | p → q is only false when p is true but q is false (otherwise it is true) |
| ↔ | ... if and only if... | p ↔ q is only true when both p and q have the same truth value (otherwise it is false) |
| ⊕ | "exclusive or" | p ⊕ q is only false when both p and q have the same truth value (otherwise it is true) |
We can draw up truth tables to look at all possible combinations of truth values for the simple statements we are considering. Using these combinations we can determine the truth values for columns defined by compound statements. Below is a small example.
| p | ~p |
|---|---|
| T | F |
| F | T |
This game is about identifying truth values in truth tables. Be sure you have at least a basic understanding of truth tables before you start.