Exercise 1 (2 points each): For each of the following, either give an example in SL or
explain why there can’t be any example:
1. A conditional sentence that is a tautology and whose antecedent is also a
tautology.
2. A conditional that is inconsistent and whose antecedent is also inconsistent.
Exercise 2 (2 points each): Symbolize each of these sentences in SL, using the following
symbolization key:
A: I will go on vacation
B: You will go on vacation
C: I need to work
D: You need to work
E: I will stay home
F: You will stay home
1. I will go on vacation as long as I don need to work.
2. If neither you nor I need to work, then we will go on vacation, but if at least
one of us needs to work, we will stay home.
3. Unless I need to work, I will go on vacation.
4. Either you or I will go on vacation even though I do need to work.
5. I will go on vacation only if you don.
Exercise 3 (4 points each): Formalize each of the following arguments in SL (remember
that to do this, you need to first list the argumentàpremises and its conclusion and
specify a symbolization key), then use the ndirect method4o determine whether the
argument is valid in SL.
1. If the top card is an ace, then, unless the second card is also an ace, you’ll win $50. If
the top card and the second card are both aces, you’ll win $100. Therefore, if the top
card is an ace, you’ll win either $50 or $100.
2. If Daria skips lunch, then she, be tired or cranky. And while Daria will not be cranky if
she doesn skip lunch, she will be tired. So Daria will be tired only if she is also cranky.
Bonus exercise (6 points):
A set of sentential connectives is said to be xpressively complete7hen any wff (i.e.,
well-formed formula) of SL (which, recall, contains 5 different sentential connectives) is
logically equivalent to a formula that only uses connectives in this set. For example, we
know that {~, v, &, ?} is expressively complete because any biconditional sentence of SL
&or ex., ?B) is logically equivalent to a sentence that only uses connectives from
this set – for ex., ?B))s logically equivalent to (A?B) &(B ?A))ŠAre each of the following sets of sentential connectives expressively complete? Explain
why or why not.
1. {~, v, &}
2. {~, v}
3. {v, &, ?, ?}

Purchase answer to see full
attachment