1. Testing for Validity and Invalidity

2. Proving an Argument Invalid

3. Validity, defined (again)
A valid argument: one where it is impossible for the premises to be true, and the conclusion to be false, together.

4. Test for Invalidity
See if you can, without self- contradiction, conceive a scenario where the premises are true and the conclusion is false at the same time.
If so, the argument is invalid.
counterexample

5. invalid My friend’s name is Susan.
Therefore, I have at least one female friend.
invalid
Counterexample: this friend is a male named Susan.
unlikely, but not logically impossible

6. Counterexample: ice melts at 45ºF.
x is a cube of ice.
Therefore, x will melt at 32ºF.
invalid
Counterexample: ice melts at 45ºF.

7. invalid All As are Bs. Most Bs are Cs. Therefore, most As are Cs.
Counterexample:
A = whale
B = mammal
C = four-legged
(Few or no whales are four-legged, but most mammals are four-legged and aren’t whales.)

8. Proving an Argument Valid

9. Validity, defined (again)
A valid argument: one where it is impossible for the premises to be true, and the conclusion to be false, together.

10. Test for validity
See if conceiving the conclusion to be false, while the premises are true, forces you into a contradiction.
If so, the argument is valid.

11. valid John is a bachelor. Therefore, John is unmarried.
Negated conclusion: John is married.
But then John can’t possibly be a bachelor.

12. valid Socrates is a man. All men are mortal.
Therefore, Socrates is mortal.
valid
Negated conclusion: Socrates is not mortal.
Socrates is a man, so not all men are mortal.
All men are mortal, so Socrates is not a man.
One of the premises must be false, if conclusion is false!

13. valid Most As are Bs. All Bs are Cs. Therefore, most As are Cs.
Most bankers have gills.
Everything with gills has wings.
Therefore, most bankers have wings.
For concreteness:
More than half of As are Bs, and at least that many are Cs; it’s impossible that fewer than half of As are MCs.
More than half the bankers have gills, and at least that many have wings; it’s impossible that fewer than half of the bankers have wings.