Does the idiom "nothing implies everything" make sense?

Question

Does the idiom

Nothing implies everything.

make sense?

If here the term nothing means empty set $\emptyset$ of premises, and imply means "semantically implication, $\vDash$"(or syntactically implication, $\vdash$; doesn't matter), then it is not true; since, for example, $\emptyset\not\vDash P$. Emptyset only implies tautologies, e.g. $\emptyset\vDash P\vee\neg P$. So, why the idiom "nothing implies everything" occurs or arises from? It does not make sense right?

Let $A, B$ be two proposition. I think the meaning of this statement is $A \implies B$ is always true when $A$ is false — ictibones, Jul 15 '17 at 12:09
See also Why in an inconsistent axiom system every statement is true? (For Dummies) — Bill Dubuque, Jul 15 '17 at 14:43
Depending on the context nothing can be nothing or everything. E.g. A teacher asks "Any questions?" The students "No". Then the teacher states "no questions implies all questions are covered". — farruhota, Jul 15 '17 at 17:39

score 5 · Accepted Answer · answered Jul 15 '17 at 12:41

5

I have never heard of 'nothing implies everything' ... are you sure it's idiom?

And I'm with you, it doesn't make sense; either it says that everything can be implied from the empty set, which you are are correct to say is clearly false, or it says that there isn't anything from which everything can be inferred, but that is false too, because there is something from which everything can be inferred, which is a contradiction.

Indeed, 'a contradiction implies everything' is true and does make sense. Maybe that is what someone tried to tell you but they didn't say it right?

answered Jul 15 '17 at 12:41

Bram28

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So complete answer! By the way, it was my teacher that told me so. I just googled it, not finding any result, so it appears not an idiom! lol – Eric Jul 15 '17 at 16:58
@Eric Ah! I thought so ... :) – Bram28 Jul 15 '17 at 22:39

score 3 · Answer 2 · answered Jul 15 '17 at 12:09

One possible interpretation of the implication $P\vDash Q$ is that $S_P\subseteq S_Q$ where $S_P$ and $S_Q$ are the set of "objects" with property $P$ and property $Q$ respectively. So the idiom "nothing implies everything" is like saying that the empty set $\emptyset$ is a subset of any set.

Does the idiom "nothing implies everything" make sense?

2 Answers2