Philosophy 103 - Introduction to Logic: Critical Thinking

FURTHER NOTES ON
PROPOSITIONAL LOGIC

I. CONDITIONALS: (Further notes) We can say that Passing the test is necessary in order to pass the class. This means that if you pass the class, then you will pass the test. Or we can say that Passing the test is sufficient for passing the class. This means something quite different, e.g., that if you pass the test, you automatically pass the class.

Thus, the consequent of a conditional is a necessary condition for the antecedent; hence a necessary condition is the consequent of a conditional. Thus the antecedent of a conditional is a sufficient condition of its consequent.

Remember: If P, then Q is equivalent to:

if P, Q
Q if P
P implies Q
P entails Q
P only if Q
P is a sufficient condition for Q.
Q is a necessary condition for P.

CONSTRUCTING TRUTH-TABLES

We will be using a different (and easier!) method for constructing truth-tables than the one used in your text. Here are the steps:

Example: P (Q R)

1. First, list all the possible combinations of truth-values for the simple propositions involved. Everytime we add another simple proposition (or letter symbolizing such a proposition) the number of possible combinations of T and F doubles, and thus so do the number of rows in the truth-table.

2. When constructing a truth-table that contains three or more letters, the rule is to begin with the right-hand column alternateing T's and F's, then alternate pairs of T's and F's, then sets of four T's and F's, then sets of 8 T's and F's, etc. as you move to the left.

3. After constructing the possible truth-value (the columns) simply fill in the spaces underneath the appropriate truth-functional operators, being careful to follow the rules for such operators as defined by their truth-tables (see last handout).