I cannot understand the meaning of if-and-only-if in definitions. For example, when we define a planar graph, we state that "A graph is G is planar, if-and-only-if it has no crossing edges". What does this mean? If we consider this whole proposition as a propositional variable x, then for x to be true, both the first and second parts must have the same truth values. But here when defining a planar graph, how can we find out the truth value of the first part , i.e. "A graph G is planar"? I mean how do we know whether this is true or false?
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While this is ultimately just an arbitrary convention, many style guides indeed recommend to use plain "if" for definitions. – Emil Jeřábek Mar 05 '18 at 08:13
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'if & only if ' aside, this is not a very good definition. For example $K_4$ is planar, but you can draw it with crossing edges – Max Punck-Institut Mar 05 '18 at 09:24
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You can see the related post: Are if and iff interchangeable in definitions ? – Mauro ALLEGRANZA Mar 05 '18 at 15:41
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Yeah got it. Thanks – sayantan dasgupta Mar 06 '18 at 03:02
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Formally, what is going on when you introduce such a definition is that you are adding a new predicate symbol, say $\mathsf{PG}$, and you are adding a new axiom $\forall g.\mathsf{PG}(g)\iff\dots$ where the "$\dots$" will contain the formal expression of the meaning, e.g. a formalization of "has no crossing edges" in this case. Using just "if", i.e. $\impliedby$, would allow the possibility that other things also give rise to planar graphs. That is, the condition in the "$\dots$" could fail to hold for $g$, yet $\mathsf{PG}(g)$ nevertheless held.
That said, for informal, natural language (or at least English) descriptions, when a definition is given the "only if" is often implied. This is just an aspect of normal speech patterns. People don't typically say "if and only if" in everyday conversation, even where technically it is appropriate or the intended meaning. In a more mathematical context, there is often an implicit assumption that a definition is exhaustive. If the author(s) had intended other possibilities, they would have mentioned them or at least mentioned their possibility.
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Does this mean that using "if and only if" in defining things is just the same as saying that the thing is defined to be holding those conditions which follow, and vice-versa? – sayantan dasgupta Mar 05 '18 at 09:43
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While it may be recommended to use 'if' in definitions, it is logically correct to use 'if and only if'. Because if a definition (using 'if' or 'if and only if') is accepted the two statements are equivalent.
Let's assume I do accept your definition "A graph is G is planar, if (an only if) it has no crossing edges". Then all the following statements are true as well:
"If $G$ is planar, $G$ has no crossing edges"
"If $G$ is not planar, $G$ has crossing edges"
"If $G$ has no crossing edges, $G$ is planar"
"If $G$ has crossing edges, $G$ is not planar"
how can we find out the truth value of the first part , i.e. "A graph G is planar"? I mean how do we know whether this is true or false?
I think this question yields the answer. You know it by defining it to be equivalent to the second part.
