1) Deductive Validity

a) Suppose I give you an argument with two premises and a conclusion, and tell you that the premises and conclusion are true. Check one of the following, and explain your answer with a sentence or two.

The argument is valid ___________

The argument is invalid___________

I can't tell___________

b) Suppose I give you an argument with two premises and a conclusion, and tell you that the premises are true and the conclusion is false. Check one of the following, and explain your answer with a sentence or two.

The argument is valid ___________

The argument is invalid___________

I can't tell___________

2) Definitions

Circle the primitive, or undefined terms in the following definitions:

a) G is a causal graph over V if G is a directed acyclic graph over V such that there is an edge from X to Y in G if and only if X is a direct cause of Y relative to V.

b) A number is wonderous if the square of the numberís maximum tetroscopic elixir is less than 3.

c) What (if anything) is wrong with the following definitions:

1) A number p is dapper if there is some other number q s.t. i) q is neat, and ii) p is contained in q.

2) Two numbers are harmonious if the ratio of their squares is 2:1.

4) What is the principle of the excluded middle, and what was the alternative offered to it by Post and Tarski, among others.

5) Subjective vs. Objective

Give two examples of a concept that is objective in theory but not in practice, with at most one of the examples taken from those we discussed in class. Explain in what way the concept is objective in theory, and explain why it is subjective in practice.

6) Proofs and Counterexamples

Suppose I give you the following theory about whole numbers.

Def. A number is gork if it is divisible by 3.

Def. A number is glub if it is divisible by 2.

Assuming normal facts about arithmetic, provide a counterexample to proposition 1.

Proposition 1. There is no number that is both gork and glub.

Prove the following claim:

If a whole number P is odd, then so is its square.

7) In a paragraph, explain the point of having a formal theory of deductive mathematical reasoning.

8) Which of the following are atomic sentences as they are defined in FOL.

a) Scheines is a strange professor

b) 6 is even and 7 is odd.

c) p = 2q.

d) Put down your pencil.

9) What are the limitations of Aristotles logic. Provide an example not covered in class of a simple valid argument that cannot be represented as a valid argument in Aristotle.