Determining necessary conclusions from logical statements

Question

I am working on a logic problem involving quantified statements, and I need help determining which conclusion necessarily follows.

The premises are:

1. All lemon cookies are tasty.
2. Some tasty cookies are expensive.

We need to evaluate which of the following conclusions is necessarily true:
a) All lemon cookies are expensive.
b) Some lemon cookies are expensive.
c) Some expensive cookies are lemon.
d) Some lemon cookies are not expensive.
e) None of the above (n.a.).

---

My attempt at a solution:

I represent the reasoning symbolically using predicate logic. Let $ U $ be the set of all cookies. Define the following propositional functions:

• $ L(x) $: $ x $ is a lemon cookie.
• $ T(x) $: $ x $ is tasty.
• $ E(x) $: $ x $ is expensive.

The premises become:

1. $ \forall x \in U, L(x) \Rightarrow T(x) $
2. $ \exists x \in U, T(x) \land E(x) $

We want to check each conclusion by constructing a model where the premises hold true:

• Let $ U = \{ A, B, C \} $.
• Suppose $ A $ is a lemon cookie that is tasty but not expensive.
• $ B $ is a tasty, expensive cookie that is not lemon.
• $ C $ is neither lemon, tasty, nor expensive.

Now evaluate each conclusion:

• a) $ \forall x \, (L(x) \Rightarrow E(x)) $: False, since $ A $ is a lemon cookie that is not expensive.
• b) $ \exists x \, (L(x) \land E(x)) $: This is not necessarily true.
• c) $ \exists x \, (E(x) \land L(x)) $: This does not follow from the premises.
• d) $ \exists x \, (L(x) \land \neg E(x)) $: While possible, it is not a necessary conclusion.

Since none of the options are guaranteed, the correct answer is $ \text{e) n.a.} $.

Is my solution correct? I am not sure if I reached the correct answer, and I also wonder if I applied the counterexample method correctly. Any feedback or suggestions on how to improve my approach would be greatly appreciated!

Answer 1

We want to check each conclusion by constructing a model where the premises hold true

Did you mean that you want to check that each conclusion is false? If so, then your method indeed correctly reaches the answer.

On an orthogonal note, exhibiting a model for the premises and conclusion doesn't demonstrate that an argument is valid. For instance, the singleton universe {tasty expensive lemon cookie} satisfies the two given premises as well as conclusion b, but, as you've already demonstrated, the former don't jointly entail the latter.

Answer 2

Your conclusion and reasoning are valid. There is a model where none of those are true and yet satisfy the premises (i.e., none are a logical consequence of the axioms).

We could also use modal logic ($\square$=it is necessary that, $\Diamond$=it is possible that..) to see what the issue is:

$\square(L(x) \implies T(x))$ : It is always the case that if something is a lemon cookie, it is tasty.

$\Diamond (T(x) \wedge E(x))$ : It is possibly the case the a cookie is tasty and expensive

From these, we try to derive something about expensive lemon cookies (assuming lemon cookies are not impossible (i.e., $\square \neg L(x)$)

(1) $\square(L(x) \implies T(x))$
(2) $\Diamond (T(x) \wedge E(x))$
(3) $\Diamond L(x)$
(4) $\Diamond (L(x) \wedge T(x))$ (by 1 + 3)
(5) $\Diamond (T(x) \wedge E(x)) \wedge \Diamond (L(x) \wedge T(x))$ conjunction of 2 and 4
(6) $\Diamond T(x) \wedge \Diamond E(x)) \wedge \Diamond (L(x)$ Possibility distributes over a conjunction

However, it doesn't follow that because there is a world where each of the conjuncts is true that there is a world where two or more are true (e.g. it is sunny vs it is raining -- each can be true in one world but not both in any world)

We need some connecting assumption (C), like the following:

(C) $\Diamond (L(x) \wedge E(x))$ we need to assert that they are not logically contradictory possibilities like prime vs non-prime number

But each of these connecting clauses are exactly what each of the options are, hence they are logically independent of the other premises.