# Logical meaning of “vice versa”?

I was wondering what "vice versa" means using the language of logic?

1. For example,

If P is unbounded, D is infeasible, and vice versa.

Does the vice versa part mean that "If D is infeasible, P is unbounded", "If D is unbounded, P is infeasible", or something else?

2. How to understand "vice versa" generally?

Thanks!

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saying "If D is unbounded then..." doesn't add any info (the proof will be the same with a change of letters) and the vice versa part is redundant. In generally when a claim "p->q" is presented, the vice versa part means "and also p<-q", i.e the two claims are equivalent. – kneidell Apr 3 '11 at 5:06
@kneidell: Looks like the author mean the contrary to your view. By "vice versa", he meant "If D is unbounded, P is infeasible", because in the next page there is another corollary regarding "If D is infeasible, ...". So is his usage wrong? – Tim Apr 3 '11 at 5:12
You're right, and the author's usage is not wrong, though ambiguous. @kneidell falsely assumed that $P$ and $D$ stand for exactly the same kind of thing; in that case indeed it would be redundant to say "and vice versa" (though people sometimes write redundant things). But $P$ and $D$ were introduced as specific things with different specific properties, and in that case it's not redundant. Both meanings of "vice versa" that you listed are in use, and as this discussion demonstrates, it's not necessarily always clear which one is intended. – joriki Apr 3 '11 at 5:22
@Joriki: Thanks! (1) I would had agreed with kneidell's explanation, had I not seen the corollary on the next page. I still feel the author's usage of vice versa seems much more unusual than what I seen so far. (2) Are these two cases all the possible meanings of vice versa? Just off the top of your head, have you seen it has other meanings? – Tim Apr 3 '11 at 5:25
My impression is the opposite; I think if I'd had to choose without seeing the next page, I would have chosen the intended meaning -- but that just goes to show how ambiguous it is :-). No, I can't think of any systematically different meanings, I think much like in everyday usage it generally means "the other way around", "with things swapped", and it depends on the circumstances what's meant to get swapped -- in this case, the entire statements or just their subjects. – joriki Apr 3 '11 at 5:34

NOTE: p & q here are logical statements (propositions, having fixed truth values).So in this example of yours

• p: P is unbounded
• q: D is infeasible

'and vice versa' in the language of logic means p <--> q is true, which is actually the case, when both p-->q & q-->p are true. i.e. (truth of p implies truth of q) AND (truth of q implies truth of p). Hence in your example,

• If P is unbounded, D is infeasible AND
• If D is infeasible, P is unbounded

There're some cases, where 'vice versa' may mean what you doubt it means, like in the following example:

• If a proposition is false, it's negation is true & vice versa.

where 'vice versa' seem to imply :-

• If proposition is true, it's negation is false OR

• If negation of proposition is true, proposition is false.

So, it pretty much depends on the context in which this's said, since both appear to be logically (!) correct. In your context, you need to mention what P & D actually are (what mathematical structures are they?) , so as to check if D can ever be unbounded or P can ever be infeasible.

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Thanks! (1) You have two kinds of interpretation: switching predicates, or switching the clauses. The former is new to our discussion so far in my post and the comments following. And your two kinds of interpretation are not what the author meant. He meant switching the subjects. (2) D and P are from the book I linked. P means the primal problem, and D its dual problem, under the context of optimization. – Tim Apr 3 '11 at 6:10
Could people please explain why they upvoted this answer? It's in direct contradiction to what had already been discussed in the comments. It seems unambiguous to me from the following page that the authors meant none of this logical stuff and merely intended the subjects of the statements to be swapped. – joriki Apr 3 '11 at 6:17
'Switching the clauses' or 'Switching the predicates' seemed so natural a possibility in the example that I cited, but I've to go study about P & D (that you mentioned what they're) since it's really important to see what can be switched. Excuse me since I don't know about Primal & Dual problems. It must be easy for you, though, to get the things the way that seems most intuitive, given you're sure of whether D can be unbounded or P can be infeasible (which seems to be the possible case when I skimmed the part of the book you linked). – Amit L Apr 3 '11 at 6:22
"vice versa" is not a matter of logic, it's a matter of English (or Latin) usage. In this case it is ambiguous, and the author should have been more explicit. The correct statement is: if P is unbounded, then D is infeasible, and if D is unbounded, then P is infeasible. However, it is not true that if D is infeasible then P is unbounded: it is possible to have both P and D infeasible. For example, consider the problem maximize $x - 2 y$ subject to $x - y \le -1$ $-x + y \le -2$ – Robert Israel Apr 3 '11 at 6:39