Conditional Statements: "only if"

Question

For some reason, be it some bad habit or something else, I can not understand why the statement "p only if q" would translate into p implies q. For instance, I have the statement "Samir will attend the party only if Kanti will be there." The way I interpret this is, "It is true that Samir will attend the party only if it is true that Kanti will be at the party;" which, in my mind, becomes "If Kanti will be at the party, then Samir will be there."

Can someone convince me of the right way?

EDIT:

I have read them carefully, and probably have done so for over a year. I understand what sufficient conditions and necessary conditions are. I understand the conditional relationship in almost all of its forms, except the form "q only if p." What I do not understand is, why is p the necessary condition and q the sufficient condition. I am not asking, what are the sufficient and necessary conditions, rather, I am asking why.

Answer 1

Think about it: "$p$ only if $q$" means that $q$ is a necessary condition for $p$. It means that $p$ can occur only when $q$ has occurred. This means that whenever we have $p$, it must also be that we have $q$, as $p$ can happen only if we have $q$: that is to say, that $p$ cannot happen if we do not have $q$.

The critical line is whenever we have $p$, we must also have $q$: this allows us to say that $p \Rightarrow q$, or $p$ implies $q$.

To use this on your example: we have the statement "Samir will attend the party only if Kanti attends the party." So if Samir attends the party, then Kanti must be at the party, because Samir will attend the party only if Kanti attends the party.

EDIT: It is a common mistake to read only if as a stronger form of if. It is important to emphasize that $q$ if $p$ means that $p$ is a sufficient condition for $q$, and that $q$ only if $p$ means that $p$ is a necessary condition for $q$.

Furthermore, we can supply more intuition on this fact: Consider $q$ only if $p$. It means that $q$ can occur only when $p$ has occurred: so if we don't have $p$, we can't have $q$, because $p$ is necessary for $q$. We note that if we don't have $p$, then we can't have $q$ is a logical statement in itself: $\lnot p \Rightarrow \lnot q$. We know that all logical statements of this form are equivalent to their contrapositives. Take the contrapositive of $\lnot p \Rightarrow \lnot q$: it is $\lnot \lnot q \Rightarrow \lnot \lnot p$, which is equivalent to $q \Rightarrow p$.

Answer 2

I don't think there's really anything to understand here. One simply has to learn as a fact that in mathematics jargon the words "only if" invariably encode that particular meaning. It is not really forced by the everyday meanings of "only" and "if" in isolation; it's just how it is.

By this I mean that the mathematical meaning is certainly a possible meaning of the English phrase "only if", the mathematical meaning is not the only possible way "only if" can be used in everyday English, and it just needs to be memorized as a fact that the meaning in mathematics is less flexible than in ordinary conversation.

To see that the mathematical meaning is at least possible for ordinary language, consider the sentence

John smokes only on Saturdays.

From this we can conclude that if we see John pulsing on a cigarette, then today must be a Saturday. We cannot, out of ordinary common sense, conclude that if we look at the calendar and it says today is Saturday, then John must currently be lighting up -- because the claim doesn't say that John smokes continously for the entire Saturday, or even every Saturday.

Now, if we can agree that there's no essential difference between "if" and "when" in this context, this might as well he phrased as

John is smoking now only if today is a Saturday.

which (according to the above analysis) ought to mean, mathematically, $$ \mathit{smokes}(\mathit{John}) \implies \mathit{today}=\mathit{Saturday} $$

Answer 3

I see it this way:

"If Kanti will not be at the party, then neither will Samir", which translates to $\neg q \to \neg p$ which is logically equivalent to $p \to q$.

Answer 4

"P only if Q" means, as it says, that P will happen ONLY if Q happens. That is, P cannot happen without Q happening also, which means that if P is happening, then Q must be happening -- if P, then Q, or $P \rightarrow Q$, not $Q \rightarrow P$.

Answer 5

The mathematician R.L. Moore, who was very careful with his language, interpreted "only if" to mean "if and only if". In his mind, "A only if B" was a stronger statement than "A if B". In other words, "A only if B" tells us that "A if B", but also gives us a little extra information: "A only if B". He would have insisted that any other interpretation of "only if" is inconsistent with the standard use of the English language.

(However, Moore did frequently say "if and only if" for the sake of clarity.)

Therefore, if you think that "only if" should mean "if and only if", you are not alone. But, mathematicians have established the convention that "A only if B" means "if A then B", and so now we must follow this convention.

Answer 6

I think that is not easy to find a good "explanation".

The propositional connectives are a (very simple) mathematical model of natural language, suited for modelling very simple arguments.

Their definition is through truth-table; after you have defined them, you will check how they are "proxing" natural language mechanism.

Someone better (negation and conjunction), someone with some arbitrariness (disjunction, inclusive : vel instead of aut); someone with a "big" approximation : "implies".

I have found useful the discussion in Stephen Cole Kleene, Mathematical Logic (1967), pag.9 and pag.58-on.

As Kleene says, a lot of controversies aroused around truth-functional definition of "implies".

For me, the traditional locutions : "necessary ... " and "sufficient condition" are a little bit misleading, because they are suggesting a sort of "causal" link between the two statement.

The mathematical model of "if $A$ then $B$" represented by truth-tables does not require any sort of "link" between them.

Assuming now my personal "quasi-conventionalist" reading of the truth-functional connectives, I will try a sort of "reverse engineering" to answer your question.

1) Starting from $A \equiv B$ and agreeing on its "natural" translation as "$A$ if and only if $B$", we have that :

$A \equiv B$ is $A \rightarrow B$ and $B \rightarrow A$.

This is translatable into : "if $A$ then $B$" and "if $B$ then $A$".

But unpacking "if and only if" we have that "$A$ if $B$" and "$A$ only if $B$".

At this point, the "wisdom of the ancients" (see Kleene, pag.63) says that :

"if $A$ then $B$" is "$A$ only if $B$" and that "if $B$ then $A$" is "$A$ if $B$".

The second pair sound more natural to me : into "$A$, if $B$", the "if" is attached to $B$, so it becomes : "if $B$, then $A$".

Then ... les jeux sont fait !

2) And now, what about "sufficient" and "necessary" ?

Let us agree on avoiding the discussion (started in modern times at least from C.I.Lewis, A survey of symbolic logic (1918)) that the truth-fuctional reading of "implies" is not correct, and it is necessary to involve modal concepts in order to correctly explain it.

I think that we must take into account the "isomorphism" between the truth-functional connective "if ... then" and the inference rule of

modus ponens that allows us to infer from the premises $A$ and $A \rightarrow B$, the conclusion $B$.

We must read it as Gottlob Frege did in his Begriffsschrift (1879) :

assuming as true both the premises, the assumption that $A \rightarrow B$ is true, rule-out the row $T-F$ in the truth-table for implies, while the assumption that also $A$ is true rule out two other rows ($F-F$ and $F-T$, respectively). Then, the conclusion that $B$ is true is licensed.

So, assuming the truth-functional definition of "$A$ implies $B$", we have that (the truth of) $A$ is a sufficient condition for (that of) $B$.

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See also Jan von Plato, Elements of Logical Reasoning (Cambridge UP, 2013 - just printed), page 11:

The two sentences if A, then B and B if A seem to express the same thing. Natural language seems to have a host of ways of expressing a conditional sentence that is written $A \rightarrow B$ in the logical notation. Consider the following list :

From A, B follows, A is a sufficient condition for B [...], B is a necessary condition for A, A only if B.

The last two require some thought. The equivalence of $A$ and $B$, $A \leftrightarrow B$ in logical notation, can be read as A if and only if B, also A is a necessary and sufficient condition for B. Sufficiency of a condition as well as the 'if' direction being clear, the remaining direction is the opposite one. So A only if B means $A \rightarrow B$ and so does B is a necessary condition for A.

It sound a bit strange to say that B is a necessary condition for A means $A \rightarrow B$. [...] A necessary condition is instead something that necessary follows, therefore not a condition in the causal sense.

Answer 7

$p$ only if $q$ is saying if $q$ were false then $p$ wouldn't have been true, i.e., $$\neg q \implies \neg p$$which is equivalent to $$p \implies q$$

Answer 8

I first summarise Professor Scott's helpful comment, and then exemplify it.

$A$ only if $B$
= $A$ is the case/can only happen only if $B$ is the case/has happened.
= $B$ is a necessary (pre)condition for $A$.
= $A \Longrightarrow B$.

I had been confused why $ \quad [A$ only if $B] \quad \neq \quad [B \Longrightarrow A] \quad $,
but I created the following aquatic example which should aid, with these abbreviations:
$F$ := There is freshwater fish.
$W$ := There is water.

$\color{green}{\text{Common sense regarding 'fish' enables us to presume: $F$ only if $W$.}}$

Notice that I purposely did not specify the type of water defined in $W$, because I was constructing $W$ NOT to be a sufficient condition. $W$ is not a sufficient condition, because it says nothing about ALL other conditions necessary for freshwater fish, such as the salinity of the water
(if $W$ is saltwater, then $F$ is false and the fish will perish).

Thus, because we know nothing about the water in $W$ and because $F$ may require other conditions, we do not know that $W \; \overset{?}{{\Longrightarrow}} \; F$. $\color{green}{\text{All we know is the green: $F \implies W$.}}$

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User NikolajK's comment dated 2013 Aug 9:

+1 for the explanation. Though I think your answer exaggerates the way in which the formal logic captures the truth of the statements. If you don't provide a rule which lets one deduce $F\Rightarrow W$, then I don't think it's part of "All we know". What if the fish is dead, etc.

Answer 9

Take your example."Samir will attend the party only if Kanti will be there". I think this as following. Assume I saw Samir at a party. Then can safely bet on the fact that Kanti is there at the party. Because he will attend only if Kanti is there. But presence of Kanti doesn't mean that Samir is there. Samir may be not there, because he may be sick or something. So I can safely say "If Samir attend to a party Kanti is there", which explains p → q.

Answer 10

Looking at Venn diagrams for these statements

1. "If P then Q" means that that there are no Ps which are not Qs but there could be Qs which are not Ps.
2. "Q only if P" means that there are no Qs that are not Ps, but there could be Ps which are not Qs.

Or given the Kanti/Samir example:

1. "If Kanti will be there, then Samir will attend the party" means that Samir will definitely go to the party if Kanti goes, but Samir may decide to attend the party even if Kanti doesn’t go. So in this case, if Kanti is at the party we know Samir is there. Therefore, Kanti’s presence is the fact from which we can draw a conclusion.

2. "Samir will attend the party only if Kanti will be there" mean that the only way Samir will attend the party is if Kanti will be there, but if Kanti attends, Samir may still not go. If Kanti is at the party we don’t know if Samir is there, but if Samir is at the party we know Kanti is there. Therefore, Samir's presence is the fact from which we can draw a conclusion.