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

"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 4

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 5

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) $B$.

Answer 6

$p$ 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 7

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 8

See 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 9

I have had to flesh out the logic of this myself. (after a late night of studying necessary and sufficient conditions for LSAT purposes, no less. I was a philosophy major and I couldn't just let it be a surface understanding). First, to all you legitimate technicians on this forum (whether by credential or study) if I have made some errors here please elucidate. (By necessity, this information is in the simplest terms possible)

It comes down to the obvious difference between the normal (read "logically unnecessary") way we express conditional statements in everyday arguments, and the INVERSE RELATIONSHIP between sufficiency and necessity.

We NORMALLY say "If _____, then ______ ." This has its analogue in propositional logic (which you should think of as necessarily different than the way we speak) as material implication and the rule of inference that makes the implication valid is Modus Ponens: P → Q (If P, then Q). Modus Ponens explicitly shows that P is sufficient for Q.

Ok, I know you understand that, but just bear with me.

NECESSITY AS A RULE OF INFERENCE FROM SUFFICIENT CONDITIONS:

What Modus Ponens does not explicitly show is that with P being sufficient with Q, we can infer that Q is necessary for P. We must use transposition and the second rule of inference, Modus Tollens, to explicitly state necessity. Now, we are taught that P implying Q is written or read as "If P, then Q", but we may miss the step of inference--which is also a rule of inference--that allows up to see that P implying Q can also be written or read so as to indicate the necessary relationship of Q to P (Q is necessary for P) as 'Only if Q, P' and 'P, only if Q'. When we write sufficient statements as P, if only Q' in order to show necessity we should understand that we are taking what modus tollens proves for granted. Moreover, seeing Modus Tollens as a rule of inference that explicitly states a backward inference to necessity will, I hope, clarify this. (for both of us!)

HERE'S THE TRICK Classically, in propositional logic, it seems we are taught necessity by seeing Modus Tollens as a valid deduction from the logic of Modus Ponens, and we are taught the rules of transposition, and 'the directions of inference' by seeing how sufficiency is the negation (read 'inverse') of necessity.

The 'first' step becomes Modus Ponens, it is a forward inference to obtain a conclusion; and, it is an explicit relationship of sufficiency that has a necessary relationship in terms of sufficiency, and this necessary relationship requires a deduction, or inference backwards to make necessity explicit: it requires another rule of inference, a deductive rule, to make the necessary relationship explicit.

You have to come to the explicit relationship of necessary statements through transposition and Modus Tollens (MT).

Definition of transposition: a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" to the truth of "Not-B implies not-A", and conversely.

Modus Tollens:

If P, then Q. (premise -- material implication) If not Q , then not P. (derived by transposition) Not Q . (premise) Therefore, not P. (derived by modus ponens)

[Note that even with Modus Tollens, it is never stated that P is necessary for Q (P, only if Q); rather, necessity is seen as a implication using transposition (inversion). I wonder about this, but I think it is because of the the nature of the subject: conditional statements]

Likewise, every use of modus ponens can be converted to a use of modus tollens and transposition.

Now we can see that Modus Tollens is the explicit statement of the necessary relationship implied by material implication and modus ponens.

Modus Tollens can be understood as the proof--and logical path--to these statements:

'Q is necessary for P' 'P only if Q' 'only if Q, P' 'the absence of Q implies the absence of P'

Notice that these are all equal statements: 'If P, then Q' = 'P, only if Q' = 'only if Q, P'. (you should see here that you need to be familiar with the rules more so than the order of the sufficient or necessary condition)

And we can also see from the conditionals AND the deductive reasoning from Modus Ponens to Modus Tollens how transposition works correctly and incorrectly:

Though Modus Tollens is an inference of transposition and a logical deduction from the rule of Modus Ponens that allows us to explicitly state necessity FROM sufficiency, it does not allow us to change the relationship of the statements, in their non-negated forms. To do so is to commit logical fallacies (denying the antecedent and affirming the consequent).

HOWEVER

We can transpose necessity onto sufficiency (denying the antecedent), only if we also transpose sufficiency onto necessity (affirming the consequent), which is to say that: 'If P, then Q; AND if Q, then P' = 'P, only if Q; AND Q, only if P' = 'Only if Q, P; AND only if P, Q'.

Importantly, 'if and only if' (the wording that contains the logic behind the transpostioning that leads to biconditionality) can be seen as a logical leap over the two fallacies.

For instance: Q = Madison will eat the fruit P = if it is an apple

1) "Madison will eat the fruit, if it is an apple." (equivalent to "Only if Madison will eat the fruit, is it an apple;" or "Madison will eat the fruit ← fruit is an apple")

2) "Madison will eat the fruit, only if it is an apple." (equivalent to "If Madison will eat the fruit, then it is an apple" or "Madison will eat the fruit → fruit is an apple")

3) "Madison will eat the fruit if and only if it is an apple" (equivalent to "Madison will eat the fruit ↔ fruit is an apple")

Note that If and only if statements can be viewed as the joining of non-negated, transposed statements that taken separately contradict each the other's necessary and sufficient conditions: in 2, by transposing P for Q (non-negated forms), we make both logical fallacies in one 'fell' swoop because this new information makes 1 a logical fallacy, too. From 2, the sufficient condition (if it is an apple) becomes the necessary condition (only if it is an apple); AND, from 1, vice versa.

So we see that (2), above, can be restated in the form of if...then as "If Madison will eat the fruit in question, then it is an apple"; taking this in conjunction with (1), we find that (3) can be stated as "If the fruit in question is an apple, then Madison will eat it; AND if Madison will eat the fruit, then it is an apple".

This exercise makes me remember the difference and similarities between mathematical logic and putting ideas into words. Both seek to be as efficient as possible: mathematical logic with truth, and the logic of our ideas with fallacies. Well, I guess it depends on the books we read.