# What's the intuition behind difference of $\neg(S\land E)$ and $\neg S \land \neg E$?

Let $S$ represent the English statement "Sales will go up."And let $E$ represent the English statement "Expenses will go up." Now my problem is that I have trouble telling the difference between these two statements: $\neg(S\land Q)$ and $\neg S \land \neg E$. When these two statements are translated into English they sound,to me, the same. However, when you look at their truth tables they are different.

Here is my best translation of these two statements:

1) $\neg(S\land E)$ represented in English, "Both sales and expenses will not go up."(Correct me if I translated it wrong)

2)$\neg S \land \neg E$ represented in English, "Expenses will not go up and sales will not go up."(Again, correct me if I'm wrong)

I still do not understand the difference between these two statements in the English representation. They both sound the same thing to me.

Once more, I have the same trouble I have with understanding with the or-connective( $\lor$ ). Again let E and S be the same statements as before.Tell me if I might have translated them wrong.

1) $\neg (S \lor E)$ represented in English is "Neither sales nor expenses will go up."

2)$\neg S \lor \neg E$ represented in English is "Sales will not go up, or Expenses will not go up."

Yet I still have trouble understanding the difference between the English translation.

In short, can you give me some alternative example that better explains the difference, something intuitive that gives me a clear picture of the difference of the meaning between the statements I showed with the connective-or and connective-and. Thank you.

• You translated 1) wrongly: it is "Sales and expenses will not both go up" (allowing also for the possibility that only one of them will go up) Sep 20, 2013 at 16:07
• @Peter Let me understand what you are saying. When we use the word both at the very beginning in the first statement, we mean that one will go up and the other won't go up. However when we put the word both at the near the end of the statement the alternative meaning is that Sales and expenses at the same time are affected? Tell me if I might be wrong or something else.
– user93971
Sep 20, 2013 at 16:11
• I'm not a linguist, but I don't think the issue is the position of "both" at the beginning or the end that makes the difference. "Both X and Y do Z" and "X and Y do both Z" mean the same, viz. "X does Z and Y does Z". However, the position of the "not" matters: "X and Y do not both do Z" v.s "X and Y both do not do Z". Sep 20, 2013 at 16:19
• It might be clearer anyway to just translate 1) into English as "It it not the case that both sales and expenses go up", which makes clear where the 'parentheses' are. Sep 20, 2013 at 16:21
• @Peter Would this be a good representation of the model that you gave?Let A: Amy is in the room and Let B be Bob is in the room. Then the logical forms...
– user93971
Sep 20, 2013 at 16:31

When translating these I suggest trying to find the principal connective first. This consists of the connective which holds the entire well-formed formula together. If you wrote in Polish/prefix notation you would also spot the principal connective as the first symbol of the well-formed formula (wff).

1. [¬(S∧E)]  The principal connective is ¬.


It is not the case that both sales will go up and expenses won't go up.

2. [¬S∧¬E]  The principal connective is ∧.


It is both the case that sales will not go up and expenses will not go up.

3. [¬(S∨E)]  The principal connective is ¬.


It is not the case that either sales will go up and expenses will go up.

4. [¬S∨¬E] The principal connective is ∨.


It is either the case that sales will not go up or expenses will not go up.

"Both sales and expenses will not go up."

In other words, not both of [expenses and sales] will go up. So at least one of them will fail to go up. So [expenses] will NOT go up or [sales] will NOT go up. $\require{cancel} \xcancel{S\lor E}$. This is $(\neg S \lor \neg E)$.

$\neg (S \lor E)$ represented in English is "Neither sales nor expenses will go up."

That English statement is more fitting for the (logically equivalent) statement $\neg S \land \neg E$.

Let's look at it this way: $(S \lor E)$ says that at least one of [sales, expenses] will go up.
While $\neg (S \lor E)$ says that it is not true that at least one of [sales, expenses] will go up.

And if "at least one" is false, then we have "less than one of [sales, expenses]..." (which is just zero/"none" in the case of counting numbers).

So none of [sales, expenses] will go up. Or EACH of [sales, expenses] will not go up.

• I don't think this interpretation of the English phrase "Both sales and expenses will not go up" is correct; see my comments above. Sep 20, 2013 at 16:26
• @Peter - wow. Yeah, I got switched around there! Will edit. Sep 20, 2013 at 20:07