Conditional The truth table of the conditional is one of the more abstract to intuitively comprehend Let p be the statement You leave a tooth under your pillow and q be the statement The Tooth Fairy will give you $5 Consider the statement If p, then q. Is the statement true when: p is true and q is true? p is true and q is false? p is false and q is true? (most difficult to comprehend) p is false and q is false? – See excellent discussion on pgs 102 – 3 in the textbook
Conditional (Continued) Hypothesis: if part of the conditional Conclusion: then part of the conditional Conditional: False when hypothesis is true and conclusion is false; otherwise true – Remember that the F T T case does not indicate a broken promise pqp → q TTT TFF FTT FFT
Biconditional Consider the statement p if and only if q A biconditional is a conditional that goes both ways – i.e. Biconditional: True when both hypothesis and conclusion have the same truth value; false otherwise pqp ↔ q TTT TFF FTF FFT
Conditional & Biconditional (Example) Ex 1: Construct a truth table: a) b)
There are three other common forms in which we can rewrite a conditional Let p be the statement You leave a tooth under your pillow and q be the statement The Tooth Fairy will give you $5 Conditional: p q – e.g. Write the conditional in English Converse: q p – Switch the hypothesis and conclusion of the conditional – e.g. Write the converse of the above conditional in English
Other Forms of Conditionals (Continued) Inverse: ~p ~q – Negate the hypothesis and conclusion, but do NOT switch their order – e.g. Write the inverse of the above conditional in English Contrapositive: ~q ~p – Negate the hypothesis and conclusion, but SWITCH their order – e.g. Write the contrapositive of the above conditional in English What happens when we construct a truth table of a conditional, converse, inverse, and contrapositive? – Which forms are logically equivalent?
Other Forms of Conditionals (Example) Ex 2: Write the inverse, converse, and contrapositive of each in the indicated form: a) If you are filing a joint return, then include your spouse’s income (in English) b) (symbolically; HINT: Use DeMorgan’s Laws!)
Summary After studying these slides, you should know how to do the following: – Understand the truth tables for conditional & biconditional – Write a conditional as a converse, inverse, or contrapositive in either English or symbols Additional Practice: – See the list of suggested problems for 3.3 Next Lesson: – Verifying Arguments (Section 3.4)