Answering questions with truth tables "With every dinner I have three rules":


*

*If I don't drink wine, then I eat soup

*If I eat soup and drink wine, then I'll have some pudding

*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:


*

*~w -> s

*(s ^ w) -> p

*(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?
 A: 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).
