## Introduction to Logic, Summer 2020

Instructions: Answer each of the following questions to the best of your ability. Be sure to read the directions to all questions carefully.

Part A: Predicate translations

For each of the following English sentences, translate it by using subject-predicate form and the quantifiers, each when appropriate. (3 points each)

- Thomas is a junior.
- Neither Thomas nor Hope is a senior.
- Everything is sunny.
- There is at least one place where it is raining.
- No junior is graduating.

Part B: Predicate derivations

For each of the following three sets of premises and show-lines, construct a derivation that satisfies the show-line. These proofs require the inference rules of sentential logic, along with the four predicate logic rules that we discussed (∀O, ∃I, ~∀O, and ~∃O). (5 points each)

Derivation 1

- ~Fp [Pr]
- Ha↔Fp [Pr]
- SHOW: ~Ha&~Fp [DD]

Derivation 2

- ∀x(FxvGx) [Pr]
- ~∃Gx [Pr]
- ∀x(Fx→~Ix) [Pr]
- SHOW: Fc&~Ic [DD]

Derivation 3

- Fa [Pr]
- ~Ia [Pr]
- Fa→(IavHa) [Pr]
- Gav~Ha [Pr]
- Show: ∃x(Gx&~Ix) [DD]

Part C: Miscellaneous

True/False Questions: For each of the following sentences, say whether its claim is true or false. (2 points each)

- A sentence is a theorem only if the sentence is provable from true premises.
- The modal operators cannot be defined via the truth table method.
- Deductive logic plays the central role in science.
- Grice argues that sentences like ‘if I am alive, then if I am dead, then I am alive’ are true in ordinary English.

Inductive logic: Say whether the following two arguments are deductively valid or inductively strong. (2 points each)

- Argument 1:

a. Sally is a physics professor

b. Sally has published many articles in her field

c. Many of Sally’s friends are physicists

d. Nobody has forced Sally to do these things

e. Therefore, Sally enjoys physics - Argument 2:

a. Sally is a physics professor

b. Sally has published many articles in her field

c. Many of Sally’s friends are physicists

d. Nobody has forced Sally to do these things

e. Therefore, Sally tolerates physics