Question

Teachers in the Middle Ages supposedly tested the realtime propositional logic ability of a student via a technique known as an obligato game. In an obligato game, a number of rounds is set and in each round the teacher gives the student successive assertions that the student must either accept or reject as they are given. When the student accepts an assertion, it is added as a commitment; when the student rejects an assertion its negation is added as a commitment. The student passes the test if the consistency of all commitments is maintained throughout the test.

Suppose that in a four-round obligato game, the teacher first gives the student the proposition \(\neg \left( {p \to \left( {q \wedge r} \right)} \right)\), then the proposition \(p \vee \neg q\), then the proposition \(\neg r\), and finally, the proposition \(\left( {p \wedge r} \right) \vee \left( {q \to p} \right)\). For which of the 16 possible sequences of four answers will the student pass the test?

Short answer

The required sequences are,

• Sequence 1: F, T, F, T
• Sequence 2: T, T, T, T
• Sequence 3: T, T, F, T
• Sequence 4: F, F, F, F
• Sequence 5: F, F, T, F
• Sequence 6: F, T, T, T

Step-by-step solution

Truth Table

A conjunction\(p \wedge q\)is true, if both (sub)propositions (pand q) are true.

A disjunction\(p \vee q\)is true, if either of the (sub)propositions (por q) are true.

A negation\(\neg p\)is true, if the (sub)propositionpis false.

A conditional statement\(p \to q\)is true, if pis false, or if both (sub)propositions are true.

A bi-conditional statement\(p \leftrightarrow q\)is true, if both (sub)propositions are true or if both (sub)propositions are false.

Find the 16 possible sequences of four answers for which the student will pass the test

Make the truth table as follows.

\(p\)	\(q\)	\(r\)	\(q \wedge r\)	\(p \to \left( {q \wedge r} \right)\)	\(\neg q\)	\(p \wedge r\)	\(q \to p\)	\(\neg \left( {p \to \left( {q \wedge r} \right)} \right)\)	\(p \vee \neg q\)	\(\neg r\)	\(\left( {p \wedge r} \right) \vee \left( {q \to p} \right)\)
\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)
\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)
\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)
\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)
\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)
\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)
\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)
\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{F}}\)	\({\rm{T}}\)	\({\rm{T}}\)	\({\rm{T}}\)

The student will pass the test, if the system is consistent. The system is consistent, if none of the statement contradicts each other.

The student will then pass the test, if he/she uses one of the\(8\)answer options in the last\(3\)columns of the given table.

Note: sequence 1 occurs four times in the truth table and sequence 2 twice.

• Sequence 1: F, T, F, T
• Sequence 2: T, T, T, T
• Sequence 3: T, T, F, T
• Sequence 4: F, F, F, F
• Sequence 5: F, F, T, F
• Sequence 6: F, T, T, T

Here, \({\rm{T}} = {\rm{accepts}}\), and \({\rm{F}} = {\rm{rejects}}\).