80-110: The Nature of Mathematical Reasoning
Spring 1998
Test 2
10 Questions, 10 points each.
1) Deductive validity
a) Define what it is for an argument to be deductively valid using possible worlds.
b) 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, using possible worlds.
The argument is valid ___________
The argument is invalid___________
I can't tell___________
c) Suppose I give you an argument with just a conclusion, and tell you that 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) Identify which are atomic sentences. If they are not atomic, say why they are not.
a) Ö 2 is irrational.
b) If 5 is even, then 5
c) x = 7
d) The number (14 – (9-7)2) is odd.
e) The set of natural numbers N is an infinite set.
3) a) What is the definition of a tautology?
b) Is p v ~p a tautology, according to this definition? Demonstrate that it is or is not with a truth table.
4) Show that the argument from
premise: ~(p v q)
to conclusion: ~p & ~q
is valid
a) with a truth table
b) with a proof in natural deduction system ND.
5) What is the difference between validity and truth? Give examples to illustrate your points.
6) Fill in the missing justifications in the following argument.
6) Ö 2 Ö 2 is rational ® $ irrational b,c s.t. bc is rational
8) Let b = Ö 2 Ö 2 and let c = Ö 2 stipulation
9) Then by algebra, bc = 2 Algebra
So $ irrational b,c s.t. bc is rational
7) Definitions
a) Explain the role of definition in mathematics.
b) What is the definition of an irrational number?
c) What does it mean for two definitions to be equivalent?
8) An argument can be shown to be invalid with a counterexample. What is a counterexample? Draw a Tarski's World like grid that is 3x3, and using the Tarksi's world version of FOL, create a world which provides a counterexample to the following argument.
Premise 1) a is a small cube that is in back of b
Premise 2) if b is large then c is a small dodecahedron
Premise 3) b is a small cube
Conclusion) c is a large
9) Suppose you had a programmer to employ who knew the innards of Tarksi's world perfectly.
a) Precisely describe a procedure for this programmer to implement that would take as inputs a set of TW sentences designated as the premises and a TW sentence designated the conclusion, such that the procedure would output "valid" whenever the argument from the premises to the conclusions was valid, and "invalid" when it was not.
b) Suppose Tarski’s World was modified to allow an infinity of different sized cubes? Now describe a procedure with the same inputs and outputs as in part a, but that would only accept sentences without variables or quantifiers, and that would always return an answer in finite time.
10) Use a truth table to uncover the identity of A and B on the Island of Knight's and Knaves:
A says: I am a Knave, but B isn't.
9) Draw a Tarski's World like grid that is 3x3. Using the Tarksi's world version of FOL, create a world in which sentences 1 - 3 are true but 4 is false.
1) a is a large cube that is in front of b
2) if b is large then c is small
3) b is a small cube
4) c is a large tetrahedron.
10) St. Anselm, St. Thomas Aquinas, and Spinoza all constructed "deductive" arguments for the existence of God. Choose one of these, and then briefly describe the argument and why it is fallacious.