Reading (~30 mins)

How To Be Right by Dr. Eugenia Cheng
DMOI, “Beyond Propositions” (it’s a short section).

Videos (~30 mins)

Predicates and Their Truth Sets (6:03)
Universal and Existential Quantifiers (9:32)
Negating Universal and Existential Quantifiers (8:03)

Warmup (~45 mins)

Problem 1

Simplify the statements below (so negation appears only directly next to predicates). In these statements without domains of quantification, you may assume that the domain is the set \(\mathbb{Z}\) of integers. This implies that \(\lnot(x < y)\equiv (y \leq x)\).

\(\lnot \exists x \forall y(\lnot O(x) \lor E(x))\)
\(\lnot \forall x \lnot \forall y \lnot ((x < y) \land \exists z((x < z)\lor (y < z)))\)

Note: this problem is the first two parts of 3.16 in DMOI, and it has an online solution. I recommend that you try this problem on your own first, carefully show all your steps, and then check against the solution.

Problem 2

Note

While completing one of the parts of this problem, you’ll need to use a predicate with multiple arguments. Some examples:

\(P(x, y)\): \(x\) is the parent of \(y\).
\(Q(x, y, z)\): \(x\), \(y\), and \(z\) are in a project group.
\(R(x, y)\): \(x\) is the number the results from multiplying \(y\) by 3; that is, \(x = 3y\).

Problem Statement

For each of the English statements below:

Express the statement using quantified predicate logic. Explicitly define any predicates you use, as well as your domain of quantification. If you make any interpretations of English language, state what they are and why. Your “translation” may not be perfect, but it’s ok as ong as you state your assumptions and interpretations.
Negate the expression, simplifying the result far enough so that negation \(\lnot\) symbols appear only directly before predicates (so, \(\lnot P(x)\) is ok but \(\lnot (\exists x \;:\; P(x))\) is not).
Translate the negated version back into English.
Comment: in your view, is the original statement or its negation correct?

No one likes Mondays.
Everyone in Vermont loves skiing!
Nobody’s perfect.
Students are successful in classes with active learning.
- for this problem, please use the predicate \(A(s, c)\) to mean that student \(s\) is successful in class \(c\). Define two appropriate domains of quantification.

Finally, decide which if any of these statements express a “kernel of truth” or a feeling that resonates with you. What would Dr. Cheng suggest with those statements?