How is the correct way to read out negation in symbolic expression?

Question

I am not sure does the following parts of symbolic expressions read the same way or not when being the first part of the expression:

1. [~(p v q)] -> .... If it is not p or q, ...

or perhaps

If it is not the case that p or q, ....

2. [(~p) v (~q)] -> .... - Should read I believe "if it is not p or not q ,... "? Or if not, how is the correct way?

(There was an example given in my book " if it is not p or not q, then it is not the case that p or q" - so this as a function would look I think like this:

[(~p) v (~q)] -> [~ (p v q)]

Answer 1

For myself, I would take a much more radical approach and avoid reading expressions such as you have there using the English words 'and' 'or' 'not' and 'if'. When we use the propositional calculus, we are creating an artificial and formal language with well defined symbols. This calculus includes the symbols ∧∨ ¬ and → whose meanings can be specified proof-theoretically by introduction and elimination rules, or model-theoretically by truth tables. These symbols are not representations of the English words; English words don't have formal stipulated definitions. ∧ is not shorthand for 'and' - it is a symbol in a formal language.

If we ask, how closely do the symbols approximate the corresponding English words, the answer is roughly, but not all that accurately. ∧ is quite close in use to 'and' though it lacks the pragmatic force of suggesting sequence; ∨ is rather less similar to 'or' since the English word typically suggests ignorance or a choice. ¬ is a classical negation and a lot of ink has been spilled in arguing over whether 'not' in English has this sense. → is a crude approximation to 'if/then' and only works in simple contexts where a truth functional conditional is being expressed.

So to take your symbolic expressions, I would read [¬(p ∨ q)] → ... as "the negation of the disjunction of p with q materially implies..." and [(¬p) ∨ (¬q)] → ... as "the disjunction of the negation of p with the negation of q materially implies...". I am especially averse to reading → as if/then because material implication is not usually what we mean by 'if' in English.

This approach may seem clumsy, but it has the added benefit of helping to maintain a clear separation between object language and metalanguage, which is important when studying logic.

Answer 2

~(p v q) is not the same as [(~p) v (~q)]; it is however the same as [(~p) ^ (~q)].

~(p v q) probably should be read aloud as "it is not the case of either p or q"; [(~p) v (~q)] would possibly be "it is not the case of p, or it is not the case of q".

Answer 3

negation only applies to propositions.

(p v q) is a proposition, call it r, so read ~(p v q) as "it is not the case that the proposition r is true".

p and q are also propositions, so e.g. ~p is the proposition "it is not the case that p".

Read [(~p) v (~q)] as "it is the case that either (it is not the case that p) or (it is not the case that q).

~(p v q) is thus true only if both p and q are false.

[(~p) v (~q)] is true only if at least one of p, q is false.

there is no "if" involved in either case.

hth

postscript ok, I used "if" in a confusing way. to quote myself:

"~(p v q) is thus true only if both p and q are false." I used "if" here. my bad. there is really no getting around this circularity, but it's a circularity of informal English, not logic.

a better quasi-formal reading would be something like "~(p v q) is true" just means "it is not the case that [(it is the case that p is true) OR (it is the case that q is true)]".

in other words, even though we use "if" informally to explain these things, their meanings do not involve "if" -there is no contingency there.

Addendum if you need to convey these ideas in speech you can use pauses, as @Not_Here suggests, but to be really clear you need to name your conjunctions and disjunctions, e.g. "the disjunction of A and B is true".