Translating “neither…nor” into a mathematical logical expression

Question

Having some difficulty doing translations for complicated neither...nor sentences.

With these characters:

~  Negation
V  Disjunction
&  Conjunction

I'm trying to translate and understand, for example:

"Neither John nor Mary are standing in front of either Jim or Cary"

I have been told that a successful translation of "Neither e nor a is to the right of c" is translated as follows: ~(RightOf(e, c) V RightOf(e, c))

What about just doing a translation on: "I like neither chocolate nor vanilla"

~(Like(chocolate) V Like(Vanilla))

Any food for thought would be appreciated.

Answer 1

As @Nickolodeon said, De Morgan's laws are the key to understanding "neither/nor" statements. The laws might look a little scary, but they have a quite natural interpretation. Statements of the form "Neither P nor Q" can be a little tricky to work with, because natural sentences aren't formed just like that. However, "Neither P nor Q" can be rephrased as "It is not the case that P, and it is not the case that Q". If we have a natural sentence such as "I like neither chocolate nor vanilla", we could rewrite it into that form: "It is not the case that I like chocolate, and it is not the case that I like vanilla". Then, we see that the statement "I like chocolate" plays the role of P, and that "I like vanilla" plays the role of Q, and that our sentence is indeed of the form "Neither P nor Q". But let's stick with the "It is not the case that P, and it is not the case that Q" formulation, which can be written with symbols as "~P & ~Q". Claiming that both of P and Q are false is the same as claiming that none of them are true. This can be reformulated as "It is not the case that at least one of P and Q are true", which is the negation of "At least one of P and Q are true" - in symbols, "~(P V Q)". This is one of De Morgan's laws, and it can also be verified with a truth table. There is a similar reasoning behind the other law, which states that "~P V ~Q" is equivalent to "~(P & Q)".

Many logical sentences can be formulated in terms of predicates, which help us make a clear distinction between the individual statements we make (which we now call predicates) and the objects we make statements about. For instance, an alternative way of translating "It is not the case that I like chocolate, and it is not the case that I like vanilla" would be "~L(chocolate) & ~L(vanilla)", where "L(x)" means "I like x". Now, the structure of the sentence is clearer: we are making the same assertion, but about two different objects. When using predicates, we gain more flexibility to manipulate our statements, but the old rules (such as De Morgan's) still apply, so it is still valid to rewrite that sentence to "~(L(chocolate) V L(vanilla))".

Now, let's first consider "Neither John nor Mary are standing in front of either Jim or Cary" as a statement about John and Mary. The predicate is then F(X): "X is standing in front of either Jim or Cary", and we can first reformulate the sentence to "It is not the case that John is standing in front of either Jim or Cary, and it is not the case that Mary is standing in front of either Jim or Cary", which in symbols becomes "~F(John) & ~F(Mary)". If we want to, we can instead consider the sentence as a statement about the relative positions of all four people, using the predicate G(X, Y): "X is standing in front of Y". Then, "X is standing in front of either Jim or Cary", which we can rewrite to "X is standing in front of Jim, or X is standing in front of Cary" becomes "G(X, Jim) V G(X, Cary)", and the entire sentence becomes "~(G(John, Jim) V G(John, Cary)) & ~(G(Mary, Jim) V G(Mary, Cary))". Now, try using DeMorgan's laws (first on each of the innermost statements, then on the outermost statement) and see the results - and try to "see" that the resulting statement expresses the same thing.