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ATS2866 - Symbolic logic - S2 2025

ATS2866 - Symbolic logic - S2 2025

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Consider a Dunn-Belnap valuation w where w(p) = T, w(q) = B, w(r) = N, and w(s) = F. What is $w((\neg q \wedge p) \vee (s \vee \neg\neg r))$ w((\neg q \wedge p) \vee (s \vee \neg\neg r))?

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For this question, think in terms of Boolean valuations. Which of the following sentences takes the value T on every valuation v where v(p) = T and v(q) = F, and takes the value F on every other valuation?

$\neg (q \wedge p)$ \neg (q \wedge p)

❌

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

✅

$\neg (q \vee p)$ \neg (q \vee p)

❌

$\neg q \vee p$ \neg q \vee p

❌

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Consider a Boolean valuation v where v(p) = T, v(q) = F, and v(r) = T. What is $v(\neg (p \wedge (q \vee r)) \vee q)$ v(\neg (p \wedge (q \vee r)) \vee q)?

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A logic is a:

set of arguments

✅

a connective (the main connective) applied to the right number of formulas (the components)

❌

set of formulas (the premises) together with a formula (the conclusion)

❌

set of formulas

❌

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Which of the following claims is always true, no matter what logic X is?

Both the empty logic and the universal logic are stronger than X.

❌

Both the empty logic and the universal logic are weaker than X.

❌

The empty logic is weaker than X, and the universal logic is stronger than X.

✅

The empty logic is stronger than X, and the universal logic is weaker than X.

❌

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Which of the following is an argument? Choose all answers that are arguments, and none of the others.

$p \wedge q, p \rightarrow q \;/\; r \vee s$ p \wedge q, p \rightarrow q \;/\; r \vee s

✅

$/\; \bot$ /\; \bot

✅

the set of all formulas $\;/\; r \rightarrow (p \wedge r)$ \;/\; r \rightarrow (p \wedge r)

33%

$\bot$ \bot

the set of all formulas that are conjunctions

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Which of the following sets are subsets of the set $\{p, q\}$ \{p, q\}? Choose all of the responses that are subsets, and none of the others.

$\{p, r\}$ \{p, r\}

The set of all atomic formulas

$\{p, q\}$ \{p, q\}

$\{p, q, r\}$ \{p, q, r\}

$\{p\}$ \{p\}

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The following sentences are from a language with atomic sentences p, q, r and all the connectives covered in reading 1. For each, say what its main connective is.

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

$p \wedge (\bot \vee r)$ p \wedge (\bot \vee r)

$(\neg q) \rightarrow p$ (\neg q) \rightarrow p

$\bot$ \bot

$(q \wedge \bot) \vee \neg (p \rightarrow r)$ (q \wedge \bot) \vee \neg (p \rightarrow r)

$\neg(q \rightarrow p)$ \neg(q \rightarrow p)

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The following sentences are taken from a language with atomic sentences p, q, r and all the connectives discussed in reading 1. Which of these sentences are conjunctions? Choose all the conjunctions, and none of the others.

$(p \vee q) \wedge (r \rightarrow \bot)$ (p \vee q) \wedge (r \rightarrow \bot)

✅

$(\bot \wedge \bot) \vee ((\bot \wedge \bot) \wedge (\bot \vee \bot))$ (\bot \wedge \bot) \vee ((\bot \wedge \bot) \wedge (\bot \vee \bot))

❌

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

❌

$((\bot \wedge \bot) \wedge (\bot \vee \bot)) \wedge (\bot \wedge \bot)$ ((\bot \wedge \bot) \wedge (\bot \vee \bot)) \wedge (\bot \wedge \bot)

✅

$\neg (p \wedge q)$ \neg (p \wedge q)

❌

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Consider a language with atomic sentences p, q, r and connectives $\wedge, \neg$ \wedge, \neg. Which of the following are sentences of this language? There might be more (or less) than one correct answer: make sure to choose *all* correct answers.