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If I have $A \iff B,$ does that means $A$ is sufficient for $B$ to occurs and $B$ is necessary for $A$ to occurs?

I am confused about the direction of meaning sufficient and meaning necessary, I usually say $\implies$ this direction means the premises is sufficient for the result i.e. $A \implies B$ means $A$ is sufficient for $B$ to occur. And $\Longleftarrow $ means the premises is sufficient for the result i.e. $B \impliedby A$ means $B$ is necessary for $A$ to occur.

But I have seen some people not sticking to this and taking reverse directions for necessary and sufficient. Could anyone explains this for me please?

Xander Henderson
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    If something is needed for a truth, and that something is produced by that truth, those things are logically equivalent. "Sufficient for" means implied by. – David P Apr 08 '20 at 00:06
  • my mother tongue is not English so what is the difference between implied by and is implied by?@DavidPeterson –  Apr 08 '20 at 00:10

4 Answers4

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If $A \implies B$ then $B$ is necessarily true when $A$ is true. As well, $A$ is a sufficient condition for $B$ to be true.

If $A$ is true then it must be the case that $B$ is true (necessary).

If we want to imply $B$ it is sufficient to determine that $A$ is true.

Imply is a synonym for deduce.

palindromicprime
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  • sometimes in the proof they just say the necessary condition and the sufficient condition as if it is always known and standard. –  Apr 08 '20 at 00:13
  • what about the necessary one? –  Apr 08 '20 at 00:14
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    It is known and standard. Necessary and sufficient conditions are abstract concepts pertaining to the direction of deduction. – palindromicprime Apr 08 '20 at 00:15
  • Material implication does not mean that " if A is true, B is necessarily true" it only mean " if A is true, B actually IS true". I can correctly say that " If the Earth has only one natural satellite, then there are 5 continents on Earth". The reason is that factually, the 2 sentences are true, so the material implication is true. But that does not mean that " the Eart has only one natural satellite" necessarily implies " there are 5 continents". $(A\rightarrow B) $ does not mean $\square (A\rightarrow B) $ nor $ (A\rightarrow \square B) $ . Material implication is not logical implication. –  Apr 09 '20 at 11:45
  • Reference: Lipschutz, Outline Of Set Theory ( at archive.org), chapter about the Algebra Of Propositions. –  Apr 09 '20 at 11:52
  • Another way to say this is that some conditions are " contingently sufficient" and that others are " necessarily sufficient". –  Apr 09 '20 at 11:53
  • As far as I know, when one talks about sufficient and necessary conditions, one talks about the meta relationship between the truth values of $A$ and $B$. This is because the words "sufficient" and "necessary" pertain to a description of the morphism between the objects, and does not speak of the actual truth values in the proposition. – palindromicprime Apr 09 '20 at 12:04
  • Being a sufficient or a necessary relation ( in the sense of material implication) is not a relation pertaining to the meta-level . The relation that belongs to meta-logic is *logical implication*. –  Apr 09 '20 at 13:07
  • You may have a look at : Mummert's answer to https://math.stackexchange.com/questions/68932/whats-the-difference-between-material-implication-and-logical-implication –  Apr 09 '20 at 13:16
  • Yes, I am aware of the difference, it is what I hinted at with the morphism abstraction. What I am further claiming is that necessity and sufficiency are abstract concepts that arise naturally from morphism and not the objects themselves. We then use this concept to “come back to earth” and materialize the objects while the morphism between them survives. Hence necessity and sufficiency are concepts that arise from logical implication and not material implication. We see necessity and sufficiency in material implication only in the cases in which we could also see it in the logical/abstract. – palindromicprime Apr 09 '20 at 16:03
  • Hence, explaining why your example of earth does not acquire the usual use of the term necessity and sufficiency, and the abstract concept they necessarily entail; that is of a logical implication. – palindromicprime Apr 09 '20 at 16:10
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$A$ is a sufficient condition for $B$ is symbolised as $$A \to B$$

$A$ is a necessary condition for $B$ is symbolised as $$B \to A$$

Example (taken from the book "forallX" by P.D. Magnus):

Symbolization key:

  • $P$: Jean is in Paris.

  • $F$: Jean is in France.

We can symbolise this sentence

If Jean is in Paris, then Jean is in France.

as $P \to F$.

An intuitive way of understanding the concept of sufficient condition, is: $P$ is a sufficient condition for $F$ means $P$ being true guarantees the truth of $F$. In this example, if Jean is in Paris, I know he is definitely in France.

Along these lines, $F$ is a necessary condition for $P$ means $P$ would not have happened without $F$. In this example, if Jean is not in France, I can be sure he is not in Paris.

$A$ if and only if $B$ is simbolised as $$A \leftrightarrow B$$

It means $A$ is a sufficient and necessary condition for $B$.

F. Zer
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  • The fact that "the Earth has only one natural satellite" is a sufficient condition for " there are 5 continents on Earth" to be true, for the material implication relation holds between these two sentences. But it is not the case that the first one guarantees the truth of the second: no-one could ever deduce the second from the first. Material implication is not the same thing as logical implication. There is no modal component in material implication. See : Lipschutz, Outline Of Set Theory, Chapter on the Algebra of proposiitons ( at archive.org). –  Apr 09 '20 at 12:06
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    Good point, @Ray LittleRock. If $A \to B$ is true, $A$ being true guarantees the truth of $B$. Do you think this wording is incorrect? How would you express this subtlety? – F. Zer Apr 09 '20 at 12:26
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    Thanks for your answer on hypothetical scenarios @Ray LittleRock. It really helped. I was going to accept it but then the answer disappeared. – F. Zer Apr 09 '20 at 12:27
  • Shortly : $(A \rightarrow B) $ is not the same thing as $\square (A\rightarrow B)$. What is true, nevertheless, is that : "if A implies B and if A is true, it is guarantied ( logically) that B is true". –  Apr 09 '20 at 12:35
  • " Guarantees" means that " it is impossible for B to be false when A is true". , or, if you prefer, that there is no possible world in which the first sentence is true and the second false. But certainly the Earth could have had only one natural satellite and , though, have more ( or less) than 5 continents. Simply, in the actual world ( the world as it is) the 2 sentences are true, and therefore, the material implication relation holds ( due to the truth table of " if...then"). That does not mean , notwithstanding, that the first sentence " guarantees" or " logically implies" the second. –  Apr 09 '20 at 12:39
  • Are you saying that, “A being true guarantees the truth of B”, does not necessarily guarantees what is actually being symbolized by B ? I see where you are heading. That concept is clear for me, but I wonder if we are talking about wording, here. – F. Zer Apr 09 '20 at 12:46
  • There is actually a little " subtlety" here. You may have a look at Wolfram's Philosophical logic (Introduction, § 1.2.2) , in which the author explains the distinction between a sufficient condition that is only " contingently sufficient" ( as a matter of fact) and a sufficient condition that is " necessarily sufficient" ( " as a point of logic").- Also note that : though material implicaton is a logical operator, *material implication is not the same thing as logical implication*. –  Apr 09 '20 at 12:49
  • Sentence A logically implies ( " entails") sentence B iff the material implication $(A\rightarrow B)$ is a tautology ( i.e. is true in all possible cases, all possible " worlds".) –  Apr 09 '20 at 12:50
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$A\implies B$ reads that "$B$ is true whenever $A$ is true".

However this does not promise anything about $B$ when $A$ is false. It is possible that $B$ might be true while $A$ is false, and if so, then $A\implies B$ will still be satisfied.

So $A\implies B$ claims that the truth of $A$ is sufficient but not necessary for the truth of $B$.

Likewise, $A\impliedby B$, also written as $B\implies A$, claims that the truth of $A$ is necessary but not sufficient for the truth of $B$.

Finally $A\iff B$, being equivalent to $(A\implies B)\wedge(A\impliedby B)$, claims that the truth of each is sufficient and necessary for the truth of the other.

Graham Kemp
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If I have A⟺B, does that means A is sufficient for B to occurs and B is necessary for A to occurs?

Yes it means this, but it means more than this.

If it only meant this, there would be no difference between bi-implication and simple implication.

Simple implication ( A $\rightarrow$ B) already means by itself that A is sufficient for B and B is necessary for A.

  • Saying that A is sufficient for B to be true means than $ A \rightarrow B$ is true. ( Which implies, by itself, that B is necessary for A to be true).

  • Saying that A is necessary for B to be true means that if A is not true, B is not true either , that is : $\neg A \rightarrow \neg B$.

  • Now, suppose you want the expression $(A \iff B)$ to mean : " A is both sufficient and necessary for B to be true", you will define this expression as follows :

$$(A \iff B)\equiv_{Df_1} [( A \rightarrow B) \land (\neg A \rightarrow \neg B)]$$.

  • But, contraposition law ( with double negation) allows you to rephrase the second part of the conjunction as : $(\neg\neg B \rightarrow \neg\neg A) \equiv (B\rightarrow A)$.

So your first definition is equivalent to :

$$(A \iff B)\equiv_{Df_{2}} [( A \rightarrow B) \land (B \rightarrow A)]$$.

This shows that : saying "A is both sufficient and necessary for B" is equivalent to saying that "(1) A is sufficient for B and (2) B is sufficient for A".

  • In fact , with contraposition, one can show that $(A\iff B)$ means equivalently :

(1) each sentence is sufficient for the other

OR

(2) each sentence is necessary for the other

OR

(3) each is both necessary and sufficient for the other.