[Math] Rules of Inference problem of Kenneth Rosen 7e -Discrete Mathematics

discrete mathematicspropositional-calculus

How to proceed with the question below ?

For each sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises.

  • "If I take the day off, it either rains or snows."
  • "I took Tuesday off or I took Thursday off."
  • "It was sunny on Tuesday."
  • "It did not snow on Thursday."

I proceeded with the question as:

  • Let $p$ represent "I take the day off"
  • $q$ represent "It rains"
  • $r$ represent "It snows"
  • $s$ represent "I took Tuesday off"
  • $t$ represent "I took Thursday off"
  • $u$ represent "It was sunny on Tuesday"
  • $v$ represent "It snowed on Thursday"

Then argument form is

  1. $p \implies (q \lor r)$
  2. $s \lor t$
  3. $u$
  4. $\neg v$

Now how do I proceed for the conclusions.

Best Answer

You shouldn't be assigning propositional variables to the pieces of the sentences; they should be predicates, so that you can substitute the day of the week.

  • "If I take the day off, it either rains or snows."

Let's call this one $$(\forall d)(T (d) \to (R(d) \vee s(d)),$$ where $T(d)$ is "I take take (day) $d$ off", $R(d)$ is "It rains on (day) $d$", and $S(d)$ is "It snows on (day) $d$".

  • "I took Tuesday off or I took Thursday off."

This now becomes $T({\rm Tuesday}) \vee T({\rm Thursday})$.

  • "It was sunny on Tuesday."

This becomes $S({\rm Tuesday})$.

  • "It did not snow on Thursday."

This becomes $\neg s({\rm Thursday})$. Now you can use the rules of inference (including the ones for quantifiers).

(1) $(\forall d)(T (d) \to (R(d) \vee s(d))$ [Assumption]

(2) $T({\rm Tuesday}) \vee T({\rm Thursday})$ [Assumption]

(3) $S({\rm Tuesday})$ [Assumption]

(4) $\neg s({\rm Thursday})$ [Assumption]

We're also missing some key information. It cannot be both rainy and sunny on day $d$, so we have

(5) $(\forall d)(\neg R(d) \vee \neg S(d))$ [Assumption]

Similarly, it cannot be sunny and snow at the same time. Thus

(6) $(\forall d)(\neg s(d) \vee \neg S(d))$ [Assumption]

Now we can use some rules of inference. It was sunny on Tuesday, so we show that it wasn't raining or snowing on Tuesday.

(7) $\neg R({\rm Tuesday}) \vee \neg S({\rm Tuesday})~~~~$ [Universal Instantiation, (5)]

(8) $\neg R({\rm Tuesday})~~~~$ [Disjunctive Syllogism, (7), (3)]

(9) $\neg s({\rm Tuesday}) \vee \neg S({\rm Tuesday})~~~~$ [Universal Instantiation, (6)]

(10) $\neg s({\rm Tuesday})~~~~$ [Disjunctive Syllogism, (9), (3)]

(11) $\neg R({\rm Tuesday}) \wedge \neg s({\rm Tuesday})~~~~$ [Conjunction, (8), (10)]

(12) $\neg (R({\rm Tuesday}) \vee s({\rm Tuesday}))~~~~$ [Logically equivalent to (11)]

Now we show what days you took off.

(13) $T({\rm Tuesday}) \to (R({\rm Tuesday}) \vee s({\rm Tuesday}))~~~~$ [Universal Instantiation, (1)]

(14) $\neg T({\rm Tuesday})~~~~$ [Modus Tollens, (12), (13)]

(15) $T({\rm Thursday})~~~~$ [Disjunctive Syllogism, (14), (2)]

Now you know you took Thursday off. Now to determine the weather on Thursday.

(16) $T({\rm Thursday}) \to (R({\rm Thursday}) \vee s({\rm Thursday}))~~~~$ [Universal Instantiation, (1)]

(17) $R({\rm Thursday}) \vee s({\rm Thursday})~~~~$ [Modus Ponens, (16), (15)]

(18) $R({\rm Thursday})~~~~$ [Disjunctive Inference, (17), (4)]

Thus it rained on Thursday.

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