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In propositional logic, how should the sentence "neither A nor B" be converted into a Well Formed Formula? Is it $\sim(A \lor B)$ or should it be $\sim(A \land B)$? Can it be interrupted both ways? I need a little help understanding this.

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3 Answers

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"Neither A nor B" is "not A and not B" and is formalized with Logical NOR i.e. $(¬A ∧ ¬B)$ that is equivalent to $¬(A ∨ B)$.

You can check with a simple truth table that $¬(A ∨ B)$ and $¬(A ∧ B)$ are not equivalent [when $A$ is TRUE and $B$ is FALSE, the first formula is FALSE, while the second one is TRUE].

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"Neither ...nor" is the negation of " OR" ; hence its name " NOR"

So neither A nor B means

• ~ (A OR B) [ read : it's not the case that (A is true OR B is true)]

which is equivalent, using DeMorgan's law, to

• ~ A & ~ B.

An Or statement is true as soon as at least one of the conjuncts is true; a NOR statement is false as soon as one of the 2 sentences is true.

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• in order to be true , the sentence ~ ( A&B) requires that one ( at least) out of the two propositions to be false; and the falsehood of only one proposition is sufficient for ~ (A & B) to be true

• the truth conditions of " neither A nor B" are different ; in order to be true, it requires BOTH propositions to be false.

• since " neither A nor B" and "~ (A&B)" do not have the same truth conditions, they are not equivalent.

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