The correct answer is: Only I and II follow.
The first statement, “Some taxis have horns,” can be expressed in propositional logic as $p$. The second statement, “Some taxis have lights,” can be expressed as $q$. The four conclusions are:
- I. Every taxi has either horn or light. This can be expressed in propositional logic as $\neg p \lor \neg q$.
- II. Some taxis have neither light nor horn. This can be expressed as $p \land q$.
- III. Some taxis have horns as well as lights. This can be expressed as $p \land q$.
- IV. No taxi has horn as well as light. This can be expressed as $\neg (p \land q)$.
To determine which conclusions follow from the premises, we can use a truth table. The truth table for the premises is as follows:
p | q | $\neg p$ | $\neg q$ | $p \lor q$ | $p \land q$ | $p \land \neg q$ | $\neg p \land q$ | $\neg p \land \neg q$
— | — | — | — | — | — | — | — | —
T | T | F | F | T | T | F | F | F
T | F | T | T | T | F | T | F | F
F | T | T | F | T | T | T | F | F
F | F | T | T | F | F | T | T | T
The first conclusion, $\neg p \lor \neg q$, is true in all rows of the truth table except the first row. This means that the conclusion does not follow from the premises.
The second conclusion, $p \land q$, is true in the first and third rows of the truth table. This means that the conclusion follows from the premises.
The third conclusion, $p \land q$, is true in the first and third rows of the truth table. This means that the conclusion follows from the premises.
The fourth conclusion, $\neg (p \land q)$, is true in the second and fourth rows of the truth table. This means that the conclusion does not follow from the premises.
Therefore, the only conclusions that follow from the premises are I and II.