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ATS2866 - Symbolic logic - S2 2025

ATS2866 - Symbolic logic - S2 2025

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Which of the following are correctly-formed terms? Select all the terms, and none of the things that are not terms.

$\lambda x . x$ \lambda x . x

100%

xyz

100%

$\lambda (\lambda x.x) . y$ \lambda (\lambda x.x) . y

❌

$(\lambda y.z)(\lambda w. \lambda z . w z)$ (\lambda y.z)(\lambda w. \lambda z . w z)

100%

$\lambda x. x \rightarrow y$ \lambda x. x \rightarrow y

❌

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All of the following are things the word "realism" can mean. Which one of these meanings is connected to debates about intuitionistic vs classical logic?

The belief that there are unknowable truths

✅

The belief that it's important to represent things as they actually are, without idealizations

❌

The belief that world politics is fundamentally competitive and about power

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The belief that science aims to produce, and sometimes succeeds in producing, true beliefs about the world.

❌

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Which of the following arguments is an instance of modus ponens?

$p \rightarrow q, q / p$ p \rightarrow q, q / p

$(p \rightarrow q) \wedge (q \rightarrow r), p / q$ (p \rightarrow q) \wedge (q \rightarrow r), p / q

$p \wedge q, p / q$ p \wedge q, p / q

$(p \wedge q) \rightarrow (p \rightarrow q), p \wedge q / p \rightarrow q$ (p \wedge q) \rightarrow (p \rightarrow q), p \wedge q / p \rightarrow q

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Only one of the following is valid in intuitionistic logic. Which one is it? (One strategy: start trying to build proofs of all of these, and see which one you can succeed on!)

$\neg \neg p / p$ \neg \neg p / p

$/ (p \rightarrow q) \rightarrow q$ / (p \rightarrow q) \rightarrow q

$p / \neg \neg p$ p / \neg \neg p

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I'm thinking of a logic, and I'll only tell you two things about it: first, it obeys conditional proof; and second, the argument $p, q \rightarrow s / r$ p, q \rightarrow s / r is valid in the logic. Which of the following things can you safely conclude from this information only?

$p / q \rightarrow (s \wedge r)$ p / q \rightarrow (s \wedge r) is valid in the logic

$p / r \rightarrow (q \rightarrow s)$ p / r \rightarrow (q \rightarrow s) is valid in the logic

$/ (q \rightarrow s) \rightarrow (p \rightarrow r)$ / (q \rightarrow s) \rightarrow (p \rightarrow r) is valid in the logic

$q \rightarrow s, r / p$ q \rightarrow s, r / p is valid in the logic

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For this question, think of all the logics mentioned as using the language with infinitely many atomic sentences and the connectives $\wedge, \vee, \neg, \fum$ \wedge, \vee, \neg, \fum only: no $\rightarrow$ \rightarrow. With that language in mind, which one of the following claims (and it's just one) is correct?

The Łukasiewicz matrix is the K3 matrix, and the fuzzy order matrix is the FDRM matrix.

❌

The Łukasiewicz matrix is the FDRM matrix, and the fuzzy order matrix is the K3 matrix.

❌

Łukasiewicz logic is K3, and fuzzy order logic is FDRM.

✅

Łukasiewicz logic is FDRM, and fuzzy order logic is K3.

❌

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Which one (and it's just one!) of the following claims is true?

Every counterexample to an argument in the Łukasiewicz matrix is also an order counterexample to that same argument, and vice versa.

Every counterexample to an argument in the Łukasiewicz matrix is also an order counterexample to that same argument, but not always vice versa.

Every order counterexample to an argument is also a counterexample to that same argument in the Łukasiewicz matrix, but not always vice versa.

No fuzzy valuation can at the same time be a counterexample to an argument in the Łukasiewicz matrix and also an order counterexample to that same argument.

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Consider a valuation v in the fuzzy structure with v(p) = 1 and v(q) = .6 and v(r) = .5. In this valuation, which of the following sentences receives a value that is below .5? Select all the sentences that get values below .5 in this valuation, and none of the sentences that get .5 or above.

$q \rightarrow (p \rightarrow r)$ q \rightarrow (p \rightarrow r)

$q \wedge \neg q$ q \wedge \neg q

$p \rightarrow r$ p \rightarrow r

$p \rightarrow \neg q$ p \rightarrow \neg q

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Which of the following are values in our fuzzy structure? Select all the answers that are values in this structure, and none of the ones that are not.

$\frac{\pi}{2}$ \frac{\pi}{2}