PHL309 Logic, Language, and Thought
Professor: Craig DeLancey
Midterm exam in class. Here are some study questions.
- Reconstruct one of Galileo's arguments that we cannot have an
actual infinity or that we cannot have actual infinitesimals.
Make your reconstruction an explicit reductio ad absurdum
argument, in which you make clear the contradiction, and the
premise we reject because of the contradiction.
- Give an example of a sentence for each of Kant's four kinds:
a priori, a posteriori, synthetic, analytic. Give an example of
a sentence for each of Kant's complex kinds: analytic a priori,
synthetic a priori, synthetic a posteriori.
- How do you and I have knowledge about geometry, according to
Kant? Why is our knowledge about non-Euclidean geometry a problem
for Kant's account?
- Answer some basic questions about set membership, subsets, powersets,
the definition of cardinality. What is a powerset? Be able to apply
- What is Cantor's Claim (about some proper subsets of infinite
sets)? How we can use Cantor's Claim (assuming it works) to
answer some of Galileo's arguments?
- Reconstruct Cantor's Diagonal Argument to prove that the cardinality
of the reals is greater than the cardinality of the natural numbers.
- What is Cantor's Theorem? Prove it.
- How does Cantor's Theorem, and the claim that a set exists if we
can determine its members, result in Cantor's Antinomy?
- What is Russell's contradiction? (AKA Russell's Paradox)
Why does it arise? What does it mean for Frege's logicism?
- What are consistency, completeness, and decidability?
Tentative Assignments (subject to revision)
Read "Intuitionism and Formalism" by L. E. J. Brouwer!
- How does Brouwer characterize the different approaches
to "mathematical exactness" (on page 83)?
- Where does he find the origin of intuitionism? (The "old
form" of intuitionism.)
- What do you think Brouwer means by "consciousness of delight"?
Why does he think the formalist does away with it?
- What does Brouwer think was the most serious blow to