Translate the following sentences to propositional logic, where atomic sentences are represented by uppercase letters

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Introduction to Logic Homework assignment

1. (30 points) Translate the following sentences to propositional logic, where atomic sentences are represented by uppercase letters. Do not forget to provide the translation key (one translation key for the whole exercise).

2. • There will be more accidents if we do not drive

   • If we drive slower or we build less houses, then there will be less nitrogen

   • Either there will be more accidents or there will be less nitrogen

3. (30 points) Translate the following sentences to first-order logic. Do not forget to provide the translation key (one key for the whole exercise).

• Ludwig Wittgenstein is a scientist only if Ada Lovelace is a

• David Pinsent and Ludwig Wittgenstein both live in Norway, or David Pinsent lives in England but Ludwig Wittgenstein does not live in

• Ludwig Wittgenstein and Ada Lovelace are both scientists, but Ludwig Wittgen- stein admires David Pinsent more than he admires Ada Lovelace

3. (40 points) Give formal proofs for the following inferences. Do not forget to provide justifications. You may only use the Introduction and Elimination rules and the Reiteration rule:

d = a a = b c = b

d = c

(¬A → A) ↔ A A → B

¬(C ∧ B)

¬(A ∧ C)

(A ∨ (⊥ ∨ B)) (A ∨ B)

4. (Bonus exercise, 10 points) Give a formal proof for:

Do  not  forget  to provide justifications.You may Elimination rules and the Reiteration rule.

¬(A ↔ ¬B)

A ↔ B

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