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

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, ....

1. [(~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)]

• To avoid confusion I would not read them the same. My professors and other logicians I have heard have always said either "(stress the word verbally) NOT (pause) P or Q" or "It is not the case that P or Q." I do see the confusion with saying " (~P v Q) and ~(P v Q) out loud and I think disambiguation can come from where you put emphasis on the words. The first would be "not P (stress) OR (pause) Q" and and the second would be "(stress) NOT (pause) P or Q). I hope that is not too confusing. Jan 1 '17 at 12:14
• They are not the same : ~(p v q) is equivalent to (~p) ∧ (~q). Thus, it is necessary to read them differently... Jan 1 '17 at 13:17
• @MauroALLEGRANZA - De Morgan's Law they read (Wikipedia) -'the negation of conjunction is the disjunction of the negations Jan 1 '17 at 13:32
• but there is said also ' it is false, that either of A or B is true ' ~(p v q) Jan 1 '17 at 13:35
• and another one, [(~p) v (~q) ] - ' one (at least) or more of A or B must be false' Jan 1 '17 at 13:36

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.

~(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".

• I agree, but think the `should` in the second paragraph needs to be prefaced by a condition, "if we want to read things out loud in an non-confusing way" since it's neither a linguistic nor moral necessity that we do so. Jan 2 '17 at 10:15

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".

• There is more following after those expressions, I don't know how to type an arrow mark so I left those out (so it would be "if....., then ....;" or "if..... , then...implies..." and so on. Jan 1 '17 at 20:58
• can you give us a specific example of if...then that puzzles you? Remember, A -> B is not a synonym for "if A then B". It's quite counter-intuitive for beginners. If A is false, then A -> B is true no matter what B is. for example we can say "A is false, therefore A->B is true." crazy, but logically impeccable.
– user20153
Jan 1 '17 at 21:16
• your last quote: "if it is not the case that p or q" is again ambiguous. in speech you can use the words "disjunct" and "conjunct" to eliminate the ambiguity. so read "suppose [~(p v q)] as "suppose the disjunct of p, q is false". you could also say "suppose the disjunct of p and q is false", but then you've used informal "and", which could be confusing. but most people will get the point, I think.
– user20153
Jan 1 '17 at 21:34
• if you add brackets to you're readings you will see the problem. for example your 1st example is "[~(p v q)] -> .... If it is not p or q, ..." your gloss is ambiguous ; do you mean "If it is not [p or q], ..." or "If it is [not p] or [q], ..."?
– user20153
Jan 1 '17 at 21:53
• of course we know you should mean "if it is not [p or q], but we have no way of knowing that that is what you do intend.
– user20153
Jan 1 '17 at 21:55