1. Universally Quantified Sentence (For All)

Imagine you have a box of cookies. If I say, “Every cookie in the box is chocolate,” that means every single cookie inside the box is chocolate—no exceptions.

In logic, we use a symbol that looks like this: ∀. This symbol stands for “for all.”

So, if we say ∀ cookies (they are chocolate), it means every single one is chocolate.

2. Existentially Quantified Sentence (There Exists)

Now, let’s say you have a basket of fruits. If I say, “There is at least one apple in the basket,” that means you might have a lot of different fruits, but there’s at least one apple in there.

In logic, we use this symbol: ∃. It stands for “there exists.”

So, if we say ∃ fruit (it is an apple), it means there is at least one apple in the basket.

Recap:

• Universally quantified sentence: “All cookies are chocolate” (∀ means “for all”).
• Existentially quantified sentence: “There is at least one apple” (∃ means “there exists”).

7.3 Exercises

Given:

p(a) → 1

p(b) → 0