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Mathematics 5
Winter Term 2000
The World According to Mathematics

Dwight Lahr and Josh Laison

Friday Discussion: Week #3

Part 1: Conditional Statements and Valid Arguments

1.

Here are five statements taken from a book written by Lewis Carroll, author of Alice
in Wonderland.

1.
2.
3.
4.
5.

No kitten that loves fish is unteachable.
No kitten without a tail will play with a gorilla.
Kittens with whiskers always love fish.
No teachable kitten has green eyes.
No kittens have tails unless they have whiskers.

(a)
(b)
(c)

Rewrite each of the statements in the If __, then __.form.
Write the statements in (a) in symbolic form p‡q.
Using the Law of Syllogism, [(p‡q) and (q‡r)] implies (p‡r), reorganize the
symbolic statements in (b) and deduce the one conclusion that follows from
these statements. [For example, if two of your statements are p‡q and q‡r,
then you can deduce that p‡r. Continue in this way with the other statements.]
Write your symbolic answer in (c) in words again.

(d)

2.

Explain why the following method may be used to show that an argument is valid.
Determine all cases in which the conclusion is false, and show that in each case at
least one premise is false.

Part 2: Linguistic Paradoxes

1.

Liar's Paradox:
Explain why the following sentence is self-contradictory, neither true nor false:

This statement is false.

2.

The Richard Paradox:
Here is a logical paradox formulated by Jules Richard (a Frenchman) in 1903:

Suppose we want to make a list of the properties of the counting numbers, that is,
of the whole numbers. First, we would have to list the
characteristics—characteristics such as even, odd, multiple of 7, or perfect square.