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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.

mp27
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3 Answers3

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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].

Mauro ALLEGRANZA
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  • Dear sir, I know that it is true that if $\phi$ is a wff and $psi$ is a wff of length less than $phi$ ,then than the sum of the number of occurrences of atoms and connectives of $\psi$ will be less than that of $\phi$, but how to prove this fact-I tried induction but failed-, thank you – Vivaan Daga Jan 19 '22 at 14:50
  • @Logic - what is the connection with this 4-years old post? – Mauro ALLEGRANZA Jan 19 '22 at 14:53
  • The "length" is the number of symbols: a symbol is either an occurrence of an atom or a connective. Thus... – Mauro ALLEGRANZA Jan 19 '22 at 14:55
  • Sir the length also includes brackets, this problem is paining me and thought that you were the only one who could help me – Vivaan Daga Jan 19 '22 at 14:59
  • @Logic - by induction on the number of connectives. (i) Base case $n=1$; the formula must be either $(\lnot p)$ or $(p \lor q)$. The first one has $l=4$ while the second one has $l=5$; thus the condition of theorem applies. We have that $#(\lnot p)=2$ while $#(p \lor q)=3$. Thus the property holds for $n=1$ and the Base case is proved. Next (ii) Induction step... – Mauro ALLEGRANZA Jan 19 '22 at 15:04
  • Sorry sir I can’t figure out the induction step thanks a lot for your help. – Vivaan Daga Jan 19 '22 at 15:54
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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.