Assignment 2

Instructions:

Answer all of the following questions. Type up your answers and/or write up and photograph your answers, compile them all into one document or pdf, and submit them to Moodle. This assignment is due on November 30^th, by 11:59 PM. No late submissions will be accepted.

Total Points: 178

Part I:

1. Think of a claim, or sentence, that many people might consider to be a logical truth, but is really only a logical possibility. Explain why the claim, despite its appearing to only possibly be true, could, despite appearances, be false. (6 Points)

• Truth Tables (simpliciter)

Construct truth tables for the following questions drawn from our textbook. (3 Points Each)

a) 4.2

b) 4.6

c) 4.7

• Translations and Truth Tables:

Translate the following sentences into FOL and then construct truth tables for them. If a sentence is ambiguous, provide a joint truth-table covering including both possible FOL construals. (5 Points Each)

• Steven lunched with Dan, but didn’t dine with Ethan.

• Kyle threw rocks at the Principle’ or He didn’t and it was Sally.

Part II:

1. Explain, using examples, why Logical Truth is a broader category than that of Tautology. (5 Points)

• Tautological Equivalence:

Produce truth tables for the following questions drawn from our textbook. (3 Points Each)

a) 4.12

b) 4.15

c) 4.16

d) 4.17

e) 4.18

• Tautological Equivalence Again:

Construct truth tables for the following sets of sentences demonstrating that they are tautologically equivalent. (10 Points)

(RanTo(j,r) Ù RanTo(r,j)) Ú (ØRanTo(j,r) Ù RanTo(r,j)) and

Ø((ØRanTo(j,r) Ú ØRanTo(r,j)) Ù (RanTo(j,r) Ú ØRanTo(r,j)))

• In what ways are the concepts of validity, logical consequence, and tautological consequence related and/or similar? In what ways are they different? Explain using an example of a valid argument you have created. (8 Points)

• Tautological Consequence:

Construct truth tables demonstrating tautological consequence in these questions drawn from our textbook. (3 Points Each)

a) 4.20

b) 4.21

c) 4.23

d) 4.24

Part III:

• Proofs, Arguments and Audience:

Construct a version of a Gaunilo-style counter-argument to St. Anselm’s proof for the existence of God (which we will cover in lecture). Explain why the argument you’ve constructed is valid, and explain why you think your conclusion is false.

Explain further how this argument is not a straightforward proof on non- consequence.

Present this argument as two different informal proofs. In the first, assume that I am already familiar with Anselm’s argument (10 Points). In the second assume that you are presenting your argument to a person unfamiliar with Anselm’s original argument (20 Points).

• Informal Proofs:

Complete the following questions drawn from the text. (4 Points Each)

a) 5.9 except do not submit a world if the conclusion does not follow. Instead, explain why the conclusion does not follow in a short paragraph.

b) 5.10

c) 5.13

And:

d) 5.19 (5 Points)

• Joe believes that the incoming Conservative Government will: crack down on crime and protect the environment and lower taxes and balance the budget and not borrow money. Show how this conjunctive belief is impossible by offering a formal proof by contradiction. (8 Points)

PART IV:

1. Formal Proofs with Boolean Connectives (From our Text). Write out your answers and then insert photos of your proofs into the word or pdf file you are submitting with your other answers. (4 points each):

6.3

6.5

6.6

6.9

6.18

6.20

6.24

6.27

6.29

6.30

6.40

6.41