Logic 102

September 19, 2018

So far, we have two ways of testing for validity and invalidity

- the Truth Table (Boole)
- Tarski’s World

These methods have some advantages and some drawbacks

**Pros**

- Super easy (mechanical, mindless)
- Always gives a definitive yes or no answer to the validity question: you’re never left hanging

**Cons**

- Gets huge quickly (8 unique atomics ⇨ 256 rows!)
- Only tests for tt-validity, which will reject some logically valid arguments
- Super boring (mechanical, mindless)

**Pros**

- A little more interesting, but not too tough. May require creativity.
- Gives you a definitive “No” answer: if there is a counterexample, you know the argument is invalid.
- Tests for TW-validity, so it won’t rule out the logically valid arguments that the truth table ruled out.
- For example, it won’t give you a counterexample to the argument “a is a cube. Therefore, it is not a dodec.”

- For example, it won’t give you a counterexample to the argument “a is a cube. Therefore, it is not a dodec.”
- Doesn’t get exponentially bigger with more atomics.

**Cons**

- More challenging. Warning: May require creativity!
- Tests for TW-validity, which will miss counterexamples for logically invalid arguments.
- For example: “a is neither a cube nor a dodec. Therefore, a is a tet.” No TW counterexamples!

- Only gives a definitive “No” answer to the validity question. It can’t demonstrate the validity of an argument.
- So: When you can’t find a counterexample world, you’re left to wonder: is it because I just haven’t found one yet, or is it because the argument is actually valid?

**Pros**

- Can be used to test for
*Logical*validity, not TT or TW. Woot. - Follows the patterns of reasong and argument we use in real life.
- Gives a definitive “Yes” answer to the validity question: if there’s a proof, you know for certain that the argument is valid.
- Doesn’t get exponentially bigger with more atomics.

**Cons**

- More challenging. Requires mastering some rules and practice!
- Only gives a definitive “Yes” answer to the validity question. It can’t demonstrate the
**in**validity of an argument. - So: When you can’t find a proof for the argument, you’re left to wonder: is it because I just haven’t found one yet, or is it because the argument is actually invalid?

From now on, we won’t be using Truth Tables for our exercises. (If you want to use them to help you understand an argument better, you are permitted to do so.)

Instead, when we’re not sure whether an argument is valid or invalid, we’re going to use a pincer attack on an argument to attack it from both sides.

Talk it through and take your best guess: Valid or Invalid?

- If you think it’s invalid, try to construct a counterexample world in TW.
- If you think it’s valid, try to construct a formal proof.

A proof is an ordered sequence of sentences, starting from the premises and ending in the conclusion, where each step after the premises comes from applying a truth-preserving formal rule to earlier steps in the sequence.

“Truth-preserving” means that when the rule is applied to true sentences, it never allows us to write down a sentence that could be false.

**“Elim”-rules** are applied to a step with that connective as its main connective.

**“Intro”-rules** are applied to allow us to write down a new step that has that connective as its main connective.