Logic - How to say "Not only but also".

Question

I am trying to translate an English sentence into propositional logic.

"To limit the loss of our company, not only the economical and statistical researches are necessary, but also a change in our spending patterns."

L: Limiting the loss of our company

E: Economical researches

S: Statistical researches

C: Change in our spending patterns

$ (E \land S \land C ) \Rightarrow L$

I think that this conveys the meaning that we need to do all 3 to be able to limit the loss, but I am not sure it conveys the meaning that only E and S won't be enough. Do I have to find a way to include them into the formula, and if so how could I do that?

Answer 1

It is in fact the other way around. If you want to say "$\psi$ is necessary for $\varphi$", then this translates to $\varphi \rightarrow \psi$ (see https://en.wikipedia.org/wiki/Necessity_and_sufficiency for the distinction between the terms "necessary" and "sufficient"). This means that your answer is almost correct, but you need to change the order of implication: $$L\rightarrow (E \land S \land C)$$

Edit: One way to see it in this example is to use the following interpretation of implication: $\varphi \rightarrow \psi$ means that "either $\varphi$ does not hold or $\psi$ holds". In your example, this translates to the following:

"Either we do not limit the loss of our company, or we perform economical and statistical researches, and change our spending patterns."

This is equivalent to the English sentence you started with because "not only but also" is equivalent to "and".

Answer 2

Technically, "for $A$ it is necessary that $B$" means $A\implies B$ and not $B\implies A$. So $(E\land S\land C)\implies L$ conveys the meaning that if $E,S,C$ are all true, then $L$ is true, but not that for $L$ to be true, then $E,S,C$ must all be true. So the proper translation should be $L\implies (E\land S\land C)$.

Answer 3

Things you can say include:

1. $(E \land S \land C) \implies L$: if we do all three things, then we will limit the losses.
2. $L \implies (E \land S \land C)$: to limit the losses, we need to do all three things.
3. $L \implies C$: to limit the losses, we need change.
4. $(E \land S \land \neg C) \implies \neg L$: if we do the first two things but not the third, we will definitely not limit the losses.

It depends on what you want to say. My personal interpretation is "$1 \land 3$", but I can see how some people might read your statement as "$1 \land 2$"; a pessimist might exclude $1$.

There's some relationships between these. Statement 1 stands on its own as the only one that guarantees that it's possible to limit the losses: the others are consistent with the possibility that no matter what you do, losses will just happen. You may or may not want to include this (but it's probably not all of what you mean, either way).

Statements 2, 3, and 4 are in order of decreasing strength (and statement 2 implies 3, which implies 4). Statement 2 says that there's only one way to limit losses: do all three things. Statement 3 merely says that we need $C$ to limit losses, and you're not making claims about $E$ and $S$. (But in particular, $E$ and $S$ alone are not enough.)

Finally, statement 4 is a bit weird: it says that in the case where you do both kinds of research, but make no change, we will definitely not limit losses, but it doesn't say anything about what happens when $E$ or $S$ is false. (For example, maybe statistical research is counterproductive: just $E$ on its own would be enough to get $L$, but with $E$ and $S$ together, you also need $C$.)