1
$\begingroup$

"With every dinner I have three rules":

  1. If I don't drink wine, then I eat soup
  2. If I eat soup and drink wine, then I'll have some pudding
  3. If I have pudding or don't drink wine, then I'll skip the soup

I have to answer these questions:

  • Do I drink wine with every dinner?
  • Do I eat soup with every dinner?

I have to figure out the solution with the help of truth tables.

I have the following translation keys:

  • Drink wine = w
  • Eat soup = s
  • Eat pudding = p

The propositions then, are translated as follows:

  1. ~w -> s
  2. (s ^ w) -> p
  3. (p v ~w) -> ~s

I have one truth table with only the proposition letters w en s, and two with w, p, and s.

Now, here's where I can't really decide what I should do. Should I just look at the truth tables? The ones for 2 and 3 are not easy to figure out just by looking at them. Should I extend the truth tables, and include e.g. for the first question a column for w, and check which models of w are also models for the propositional sentences?

$\endgroup$
1
$\begingroup$

First of all you must "model" your problem in the correct way.

I think that the translations of the three assumptions (call the three sentences $\sigma_1$, $\sigma_2$ and $\sigma_3$) are correct.

But you must model also the answer to the two questions :

Do I drink wine with every dinner?

Do I eat soup with every dinner?

in terms of the propositional variables : $w, s$ and $p$; call the supposed answers $\alpha_1$ and $\alpha_2$.

Then we must work with a logical relation between these sentences; I imagine that your problem is about tautological implication :

$\Gamma \vDash_{TAUT} \varphi$

where : $\Gamma = \{ \sigma_1, \sigma_2, \sigma_3 \}$ and $\varphi$ must be in turn each of the proposed answer to the two above question, i.e. $\alpha_1$ and $\alpha_2$.

Only at this point you will introduce truth-table, building a tt for the three propositional variables $w, s$ and $p$ and the calculations for the three formulas $\sigma_i$ and for $\alpha_1$ and $\alpha_2$.

In order to show that there is a relation of tautological implication between $\Gamma$ and $\alpha_i$, you must check that for each row in the tt such that all three formulas $\sigma_i \in \Gamma$ have the value true, also $\alpha_i$ has true (in that row).

$\endgroup$
  • $\begingroup$ I think I get what you're saying. I must show that $\alpha_1$ and $\alpha_2$ logically follow from $\{ \sigma_1, \sigma_2, \sigma_3 \}$. I'm thinking $\alpha_1$ could then be $w$ or $s$. I'm guessing that if all models make $\Gamma$ true, then it should also make $w$ true, and in that case, yes, wine is drunk during dinner. $\endgroup$ – Garth Marenghi Mar 6 '14 at 10:41
  • $\begingroup$ @GarthMarenghi - I agree; the "formalization" of the answers to the two questions amount simply at $w$ and $s$. So, you must check if $\Gamma \vDash w$ and if $\Gamma \vDash s$. Populating the tt, I've seen (if I do not made a mistake) that in conclusion you have only two rows of the tt where all three formulas (the $\sigma_i$) are all true; in both rows $w$ is true: this validate the conclusion that $\Gamma \vDash w$. $\endgroup$ – Mauro ALLEGRANZA Mar 6 '14 at 10:47
  • $\begingroup$ I came to the same conclusion. You're awesome, thanks! $\endgroup$ – Garth Marenghi Mar 6 '14 at 11:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.