Problem 1: Back and Forth

Translate each of the natural language propositions into formal logical propositions. Ensure in your answer that you (1) identify the atomic propositions and assign them to variables and (2) write your formal proposition in terms of these variables. Make sure you maximally translate the proposition using as many of the connectives introduced in the reading. In particular, make sure you use the negation logical operator when appropriate in your answers.

1. It rained this morning and it will be sunny this afternoon.
2. If I am forced to stay in-doors this week, I will play Genshin Impact, have fun, and become poor.
3. If I either eat Taco Bell or McDonalds, I will either become swoll, or I will not survive the night.
4. If Jan and Bob both work at the company, then Jan certainly works more than Bob.

Now, consider the following parameterized atomic propositions:

• \(A(x)\) = “I love \(x\).”
• \(B(x)\) = “\(x\) barfs.”
• \(C(x)\) = “I own \(x\).”
• \(D(x)\) = “I’m sending away \(x\).”

As well as their parameterized versions:

Translate each of the formal logical propositions to natural language propositions. Ensure that your natural language propositions clearly indicate the explicit grouping found in the formal logic propositions

1. \(A(\text{cats}) \wedge B(\text{cats})\).
2. \(A(\text{cats}) \rightarrow \neg C(\text{dogs})\).
3. \(A(\text{cats}) \vee (B(\text{cat}) \rightarrow C(\text{dog}))\).
4. \(B(\text{my cat}) \rightarrow D(\text{my cat}) \wedge \neg A(\text{my cat})\).
5. \(A(\text{dogs}) \rightarrow C(\text{dogs}) \wedge D(\text{cats})\).

Note that \((\wedge)\) and \((\vee)\) have higher precedence than \((\rightarrow)\). So, for example,

\[ B(\text{my cat}) \rightarrow D(\text{my cat}) \wedge \neg A(\text{my cat}) \]

is equivalent to:

\[ B(\text{my cat}) \rightarrow (D(\text{my cat}) \wedge \neg A(\text{my cat})). \]

Problem 2: Thinking About Proof

Thinking ahead, we’ve used first-order logic to explore the extends of what we can express as far as propositions go. However, we are studying logic in this course to derive a set of rules for proving a proposition. Because our propositions are defined inductively, i.e., as a finite set of cases, our rules for proving a proposition also follow by case analysis on the different cases.

For each possible form of a proposition given in the reading:

1. Describe at a high-level how you would go about proving that proposition in a few sentences per form.
2. Use a concrete example of a proposition in English for each case to describe your process.

For example, for conjunction, \(p_1 \wedge p_2\), I would instantiate \(p_1\) and \(p_2\) to concrete propositions, e.g.,

• \(p_1\) = “The sky is rainy.”
• \(p_2\) = “It is dark outside.”

And then explain how I would prove \(p_1 \wedge p_2\). Use your example and your intuition about what each logical connective means to arrive at your proving process. In our next class, we’ll firm up your intuition with concrete rules and then look at how we express these rules precisely using mathematical notation.

Problem 3: Flip It

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 4: 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.