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Why is $\neg (p \land q)$ different than $(\neg p \land \neg q)$?

If we let:

p: Blair is a liar

q: Bush is a liar

Then:

¬(p & q) is "Neither Bush nor Blair are liars"

which seems to be the same as: 

(¬p & ¬q) "Bush is not a liar and Blair is not a liar"

But the logic tables for them are different.

$\neg$(p & q)

($\neg$p & $\neg$q)

Andy
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    The key thing here is that the first one says that "Bush and Blair are not both liars". Thus if Bush is a liar, but Blair is not, the first sentence remains true, while the second is not true. – Jsevillamol Oct 01 '16 at 10:47
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    Ah I see, because ¬(p & q) is logically equivalent to (¬p v ¬q)? – Andy Oct 01 '16 at 10:49
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    @Andy and that would be a yes. This is one part of De Morgan's laws. – Parcly Taxel Oct 01 '16 at 10:50
  • So then for the sentence: "If either the Butler or the Chambermaid was guilty, then neither the Scullery-maid nor the Chauffeur was guilty." then would that be (p v q) -> (¬r & ¬s) or (p v q) -> ¬(r & s)? I understand why they're different, just finding it hard to establish the difference between them in written sentences. I'm struggling to see any difference in "neither the Scullery-maid nor the Chauffeur was guilty." and "the Scullery-maid was not guilty and the Chauffeur was not guilty." – Andy Oct 01 '16 at 10:57
  • Also is there any difference between: "Bush and Blair are not both liars" and "Neither Bush nor Blair are liars"? Thanks @ParclyTaxel and Jsevillamol for your earlier help! – Andy Oct 01 '16 at 11:04
  • In your example above in your comment $(B\lor C)\rightarrow (\lnot S \land \lnot \text{chauff})$, is correct. – amWhy Oct 01 '16 at 11:33

1 Answers1

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"Neither Bush nor Blair are liars" can be written in two equivalent ways:

$$\lnot (p\lor q) \equiv \lnot p \land \lnot q$$

"Not both Bush and Blair are liars"

$$\lnot (p\land q) = \lnot p \lor \lnot q$$

$\lnot(p\land q)$ means "Not both (p and q)." This means that that either $\lnot p$ or $\lnot$ q.

$\lnot(p \lor q)$ means it's not the case that (either p holds or q holds), i.e. "neither p nor q," This is equivalent, as noted above, to $\lnot$ p

Andy
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amWhy
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  • What helps me to differentiate between "either/or" and "neither/nor" is to remember that neither/nor is the negation of either/or. $(p\lor q)$ is either p or q. $\lnot (p\lor q) $ means neither p nor q, i.e. $\lnot (p\lor q)\equiv \lnot p \land \lnot q$ – amWhy Oct 01 '16 at 11:52
  • What did you click? – amWhy Oct 01 '16 at 12:25