translate the following argument into symbolic form and determine whether it's logically correct by constructing a truth table.
-ifthe team wins, then the team remains in the playoffs. If the team stays in the playoffs, then the championship is within reach. Therefore, if the team wins, the championship is within reach.
To translate the argument into symbolic form, we will assign variables to each statement:
Let W represent "the team wins."
Let P represent "the team remains in the playoffs."
Let C represent "the championship is within reach."
The argument can be translated into symbolic form as follows:
W → P
P → C
Therefore, W → C
To determine if the argument is logically correct, we will construct a truth table.
A truth table consists of all possible combinations of truth values for the variables involved. In this case, since we have three variables (W, P, and C), there will be 2^3 = 8 rows in the truth table.
Here is the truth table for this argument:
| W | P | C | W → P | P → C | W → C |
|---|---|---|-------|-------|-------|
| T | T | T | T | T | T |
| T | T | F | T | F | F |
| T | F | T | F | T | T |
| T | F | F | F | T | F |
| F | T | T | T | T | T |
| F | T | F | T | F | T |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
In this truth table, T represents true and F represents false. The columns W → P, P → C, and W → C show the truth values for each implication based on the given statements.
For an argument to be logically correct, the conclusion (W → C) must always be true when the premises (W → P and P → C) are true. By observing the truth values in the W → C column, we can see that it is always true when both premises are true. Therefore, the argument is logically correct.