Problem: Flip It

You may have noticed that we did not state any equivalences involving implication in the reading. It turns out that reasoning about such equivalences—in particular with negation—is tricky. We’ll explore such equivalences in this problem.

Consider an implication \(p = A \rightarrow B\) and the following variants on this implication:

• The converse of \(p\), \(B \rightarrow A\).
• The inverse of \(p\), \(\neg A \rightarrow \neg B\).
• The negation of \(p\), \(\neg (A \rightarrow B)\).
• The contrapositive of \(p\), \(\neg B \rightarrow \neg A\).

These variants arise from negating the arguments to the implication in various combinations as well as reversing the roles of the premise and conclusion.

1. Define the following concrete propositions:

   • \(A =\) “I like ice cream.”
   • \(B =\) “Coldstone is heaven.”

   Translate each of the above variants of implication \(p = A \rightarrow B\) into English.

2. In a sentence or two a piece, argue why the converse, inverse, and negation of \(p\) for our concrete choices of \(A\) and \(B\) are not equivalent to \(p\).

3. It turns out that the only equivalence here is between an implication and its contrapositive. In a sentence or two, argue why the contrapositive is equivalent to the original implication for our concrete choices of \(A\) and \(B\).

Problem: Flip It Again

Now, let’s consider how negation works with quantifiers.

Consider the following six variations of propositions involving quantifiers and negations:

• \(\forall x \ldotp p(x)\).
• \(\exists x \ldotp p(x)\).
• \(\neg \forall x \ldotp p(x)\).
• \(\neg \exists x \ldotp p(x)\).
• \(\forall x \ldotp \neg p(x)\).
• \(\exists x \ldotp \neg p(x)\).

1. Define the following parameterized proposition:

   • \(p(x) = \text{\( x \) wants to be a big shot}\).

   Translate each of the propositions into English.

2. Determine which pairs of these propositions are equivalent. For each such pair, argue why the pair of propositions are equivalent.

3. From your discovery, formulate a general rule about how one should reason about negation and quantifiers.