# Answering questions with truth tables

"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?

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).

• 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. – Garth Marenghi Mar 6 '14 at 10:41
• @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 $\Gamma \vDash w$. – Mauro ALLEGRANZA Mar 6 '14 at 10:47
• I came to the same conclusion. You're awesome, thanks! – Garth Marenghi Mar 6 '14 at 11:16