Semantics
Assignment
• The logical symbol for ‘and’ is ∧, but you can also use the symbol & (shift 7)
• The logical symbol for ‘not’ is ¬, but you can also use either the tilde symbol ∼ (found in different places depending on your keyboard), or simply a dash, as in –P
• The logical symbol for ‘or’ is ∨, but you can also use lower case ‘v’
• Both the conditional (→) and bidirectional (↔) can be replaced by typing dashes and angle brack-ets –> and <–> respectively
1 What are equivalences in PredL good for?
Directions: For each of the equivalences in PredL below,
1. Give a suitable paraphrase in English of the logic.
2. Give a more natural translation into English.
3. Provide a context or fragment of conversation that shows how these equivalences could function
in a natural setting. You might find that same are easier or more natural than others. (Hint: usually it’s to make a contrast, and sometimes you need to add other words to make it seem more natural.)
(1) ¬ X(P X → ¬QX) ⇔ ∃X(P X ∧ QX)
(2) ¬ X(P X → QX) ⇔ ∃X(P X ∧ ¬QX)
(3) ¬ X(P X ∧ QX) ⇔ ∀X(P X → ¬QX)
(4) ¬ X(P X ∧ ¬QX) ⇔ ∀X(P X → QX)
2 Restricted quantification
Directions: Translate the English sentences into PredL. Work it out in steps.
(5) If every class is cancelled, John is happy (every class is cancelled → John is happy)
(6) If every class is cancelled, all students are happy (every class is cancelled → all students are happy)
(7) There is no one who can solve every problem ¬ X(P X ∧ X can solve every problem)
(8) Anyone who is smarter than Einstein deserves the Nobel Prize ∀X(X is smarter than Einstein → X deserves the Nobel Prize)
(9) Every book that Margaret wrote got a negative review
(10) People who don’t live in New York City hate it
