# Logic: Problem 1

Claim: For every dinosaur D, if D eats meat, then D has sharp teeth and D is not huge.

### Correct Solution

Negation: There is a dinosaur D, such that D eats meat, but D does not have sharp teeth or D is huge.

Alternate negation: There is a dinosaur D, such that D eats meat, and D has blunt teeth or D is huge.

Contrapositive: For every dinosaur D, if D does not have sharp teeth or D is huge, then D does not eat meat.

### Buggy Solutions

Buggy negations:

• There is a dinosaur D, such that if D eats meat, then D does not have sharp teeth or D is huge.
• There is a dinosaur D, such that D eats meat, then D does not have sharp teeth or D is huge.
• There is a dinosaur D, such that D eats meat, but D does not have sharp teeth or D is not huge.
• There is a dinosaur D, such that D eats meat, but D does not have sharp teeth and D is huge.
• D eats meat, but D does not have sharp teeth or D is huge.

Buggy contrapositives

• There is a dinosaur D, such that if D does not have sharp teeth or D is huge, then D does not eat meat.
• If D does not have sharp teeth or D is huge, then D does not eat meat.

### Self Check

Main structure

• Were your answers written entirely in words (except for the variable D)? (That's what the problem specifically asked for.)
• Did your answers include the quantifier, e.g. "For every dinosaur D"? If so, did you choose the correct quantifier? Notice that you keep the same quantifier in forming the contrapositive but it changes when you're negating.
• Now, look at the overall structure of the expression after the quantifier. The negation of an if/then statement should be an AND: the hypothesis is true and the conclusion is false. The contrapostive of an if/then statement will be a (different) if/then statement.

Details

• Check the individual predicates to make sure that you have a "not" on the correct ones.
• When a predicate has a well-known opposite (e.g. sharp vs. blunt), you can use this to simplify your answer, as in the alternate negation.
• The meanings of "and" and "but" are very similar. For many examples, either word can be used.
• Now look through the buggy answers and make sure you see what's wrong with each one.