# The difference between "P if Q" and "P only if Q"?

Why "P only if Q" is different from "P if Q" in logic, though in English they have the same meaning?

• yes that's in logic but in the common language I can't see the difference between the too, the difference that motivate "P if Q" to mean P ⇒ Q and "P only if Q" to mean P ⇐ Q. Feb 12, 2017 at 15:41
• If your question is merely about how English grammar works, we're not the right SE. Feb 12, 2017 at 15:45
• The issue with this example is that we have a "causal connection" that is not formalized with the truth-functional conditional. The reading of the conditional P→Q: "if P, then Q", and "when P, then Q", and "Q when P" . Consider now "Q when P": if we have P, we are guaranteed that also Q holds. Thus, if we have "I give you a dollar if you eat this" and also "I give you a dollar when you eat this" the reading must be : if it is true that you eat, than we are licensed to infer that I give you a dollar is also true. Feb 12, 2017 at 16:03
• they certainly do not have the same meaning in English! "you may enter if you have a ticket" != "you may enter only if you have a ticket".
– user20153
Feb 13, 2017 at 20:30

Why "P only if Q" is different from "P if Q" in logic, though in English they have the same meaning?
In everyday English, the two are used interchangeably. Logically they are different.
In the first (only if), there exists exactly one condition, Q, that will produce P. If the antecedent Q is denied (not-Q), then not-P immediately follows.
In the second, the restriction on conditions is gone. The usual rules apply, and nothing follows from denying the antecedent Q.

• that sounds nice intuitively. but "only" is not a logical operator. nor is "produce". i think you and the op got things backwards. the rwo sentences do not have the same meaning in ordinary English. they do in logic, because "only" is logically redundant. "only if q then p" is just a restatement of "if q then p". actually that's not true, "only if q then p" is meaningless, logically. how would your write it symbolically? If you disagree, please show the truth table for "only if".
– user20153
Feb 13, 2017 at 20:23
• My answer assumes that the word "only" has a substantive meaning; the word adds something to the content of the if-then statement and so makes it different. Here, "only" means that there is no more than one condition Q that is followed by result P. Without the qualifier, the ordinary rule would apply that nothing follows from denying the antecedent; many conditions, not just Q, might result in P. But when there is only one such condition, Q, then negating that condition requires the conclusion not-P. Again, this answer assumes that the word "only" has substantive content. Feb 13, 2017 at 20:56
• sure, but that's all natural English. In logic, a word with substantive meaning is by definition extra-logical.
– user20153
Feb 13, 2017 at 21:04
• "no more than one condition Q that is followed by the result P". But material implication does not involve "followed by, for one. "2+2=4, therefore Paris is the capital of France" is a perfectly ok inolication, but the post-cedent does not "follow"from the antecedent.
– user20153
Feb 13, 2017 at 21:10
• plus, afaik there is no way in logic to express the idea that only one P can imply Q. i.e. P and nothing else -> Q.
– user20153
Feb 13, 2017 at 21:13

To see the difference, it can be useful to replace the "if..., then___" construction with a different one using "when".

So P → Q must be read as "Q if P" and as "Q when P".

The last version is more perspicuous : if we have P, we are guaranteed that also Q holds.

Example: "if n > 0, then n ≥ 0" (I've chosen it, because its converse : "if n ≥ 0, then n > 0", is not true).

According to the above proposal, we may read it with :

"n ≥ 0 when n > 0".

In terms of truth conditions, this is: when n > 0 holds, we are licensed to assert that also n ≥ 0 holds.

Consider now :

"n ≥ 0 only when n > 0".

It cannot be the same, because with n=0, the left clause is true while the right one is not. This means that it is not correct to assert that n ≥ 0 holds only in case (only when) n > 0 holds.

Thus, "n ≥ 0 when n > 0" must be rephrased with : "n > 0 only when n ≥ 0".

In conclusion, we have :

P → Q must be read as "if P, then Q", "Q if P", "P only if Q", "when P, then Q", "Q when P" and "P only when Q",

while :

Q → P must be read as "if Q, then P", "P if Q", "Q only if P", "when Q, then P", "P when Q" and "Q only when P".

Disclaim : what above does not mean that the truth functional conditional may correctly translate all usage of "if..., then___" (and related constructions) of natural language.

But in a context, like the mathematical discourse, where the "regimented" translation with the conditional is useful, we cannot conflate "P if Q" with "P oly if Q".

They do not mean the same thing in everyday English. One expresses a sufficient condition while the other expresses a necessary condition.

`````` You will pass the course if you "ace" the final

You will pass the course only if you "ace" the final
``````

give entirely different messages.

In the first sentence, "acing" the final is given as a sufficient condition for passing the course. No matter what your other grades may be, if you "ace the final, you will pass of course.

In the second sentence, "acing" the final is given as a necessary condition for passing the course. Other necessary conditions (attendance, passing grades in midterms, etc.) might also exist. This is left open.

Here are some other stark examples where you can see the difference clearly:

`````` I will die if I am beheaded (true)
I will die only if I am beheaded (false)

We can make bread if we have some kind of flour (false)
We can make bread only if we have some kind of flour (true)
``````

Such sentences are represented differently in symbolic logic because they mean different things in English.

See Nelson Lande's Classical Logic and Its Rabbit-Holes (2013). Lande showcases that "P only if Q" and "P if Q" differ in meaning, even in English!

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#### 2. Translating 'Only If'

Be prepared in this section to run up against your first conceptual speed bump. (If you aren't fully alert, don't read on; take a nap instead.) Consider the following sentence:

(1) Bobo is a widow only if Bobo is a woman.

Suppose that we dispense with propositional symbols just this once, for the sake of clarity. Interpret 'WIDOW to mean 'Bobo is a widow', and interpret 'WOMAN' to mean 'Bobo is a woman'. Translating (1) into Loglish--our halfway house (once again) between purely logical notation and English--yields the following:

(2) WIDOW only if WOMAN

Translating (1) and (2) into full-blown logical notation is a bit tricky. No doubt your first inclination may be to translate it as follows:

(3) WOMAN → WIDOW

But (3) can't be right; i.e., it can't possibly capture what (1) and (2) are claiming. Think of what (3) says: 'If Bobo is a woman then Bobo is a widow'. (1) and (2) are truths: if you know nothing about Bobo and I tell you that Bobo is a widow, you immediately know that Bobo is a woman. Why? Because being a woman is a necessary condition of being a widow: you can't be a widow unless you're a woman. (Once again: a man in a comparable position is a widower, not a widow.) (3), however, is a falsehood: if, once again, you know nothing about Bobo and I tell you that Bobo is a woman, you know nothing about her marital status. The claim that if she's a woman then she's a widow is simply false. So if (1) and (2) are truths and (3) is a falsehood, then (3) must be a mistranslation of (1) and (2). So then how do you translate (1) and (2)?

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It turns out that, your intuitions to the contrary, 'Bobo is a widow only if Bobo is a woman' has the very same meaning as 'If Bobo is a widow then Bobo is a woman'--in which case the correct translation of (1) and (2) is the converse (the very opposite) of (3):

(4) WIDOW → WOMAN

In the following paragraph, I shall write very loosely--so that you'll have an easier time grasping what's at issue. In the paragraph following the following paragraph, I'll express the same thought without the looseness.

The loose version first. All of the following statements have the exact same meaning; i.e., it should strike you that (6) has the same meaning as (5), that (7) has the same meaning as (6), etc.

(5) P only if Q.

(6) You have 'P' only if you have 'Q'.

(7) You have to have 'Q' in order to have 'P'.

(8) You can't have 'P' without 'Q'.

(9) It's not the case that you have 'P' without 'Q'; i.e.,--(P∧ -Q).

(10) If you have 'P' then you have 'Q'.

(11) If 'P' is true then 'Q' is true.

(12) If P then Q.

(13) P → Q.

So (5) has the same meaning as (12); i.e., 'P only if Q' has the same meaning as 'If P then Q'. Therefore because we translate 'If P then Q' as 'P→ Q', we translate 'P only if Q' as 'P→ Q'.

The non-loose version next. All of the following statements, (14) | through (19), have the exact same meaning; i.e., it should strike you that (15) has the same meaning as (14), that (16) has the same meaning as (15), etc.

(14) P only if Q.

(15) The truth of 'Q' is necessary for the truth of 'P'.

(16) It's not the case that 'P' is true and that 'Q' is not true; i.e., -(P∧ -Q).

(17) If 'P' is true then 'Q' is true.

(18) If P then Q.

(19) P → Q.

Once again, we see that 'P only if Q' has the same meaning as 'If P then Q'. And (once again) because we translate 'If P then Q' as 'P-→ Q', it follows that we translate 'P only if Q' as 'P-→ Q'.

If you're still unconvinced then it's time to beat a dead horse.... Notice first that the sentence 'If P then Q' amounts to the following two claims:

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(20) If 'P' is true then 'Q' is true; i.e., the truth of 'P' guarantees (i.e., is sufficient for) the truth of 'Q'.

(21) If 'Q' is not true then 'P' is not true; i.e., the truth of 'Q' is necessary for the truth of 'P'.

Notice next that the sentence 'P only if Q' amounts to the following two claims:

(22) If 'Q' is not true then 'P' is not true; i.e., the truth of 'Q' is necessary for the truth of 'P'.

(23) If 'P' is true then 'Q' is true; i.e., the truth of 'P' guarantees (i.e., is sufficient for) the truth of 'Q'.

Of course, (20) simply is (23), and (21) simply is (22). It should come as no surprise, therefore, that 'If P then Q' and 'P only if Q' have the same meaning. Accordingly, because the correct translation of 'If P then Q' is 'P→ Q', the correct translation of 'P only if Q' must also be 'P → Q'.

The obvious question: Why do our intuitions have to be dragged kicking and screaming before they will acknowledge that 'If P then Q' and 'P only if Q' have the exact same meaning?

The non-obvious answer: In the course of using conditionals in everyday conversation, we presume that the speaker (or writer) believes that there is a connection of some sort between the state of affairs to which the antecedent of the conditional refers and the state of the affairs to which the consequent refers. For example, the connection might be of the causal sort, of the definitional sort, or of the logical sort--and the context will normally make clear exactly which sort of conditional the speaker (or writer) is using.

Suppose that a dog-walker reprimands you: "If you pull Old Fido's tail one more time then he'll bite you." Your presumption is that the dog-walker believes that there is a causal connection between the former state of affairs--your pulling Old Fido's tail--and the latter state of affairs--Old Fido's biting you--such that the former will be the cause of the latter. But here's the crucial point: Your presumption, as well as the dog-walker's belief, are distinct from the meaning of the conditional sentence itself. The meaning of the conditional sentence itself is what it shares with all typical conditionals. Because a reference to causality doesn't characterize the other sorts of conditionals, a reference to causality can be no part of its own meaning.

## Page 68/437 of my .epub file

Its meaning--what it shares (once again) with all typical conditionals--is precisely this: It isn't the case that its antecedent is true and its consequent isn't true. Once you abandon the belief that there is a connection of some sort between the state of affairs to which the antecedent of the conditional refers and the state of affairs to which the consequent refers, then you should have no difficulty seeing that the meaning of a conditional consists exclusively in its not being the case that its antecedent is true and its consequent is not true.

Think of a conjunction. Suppose that on the first day of the semester, your instructor had walked into your class and said, "This is a course in formal logic and I shall now be taking roll." You would have found that entirely unsurprising. Suppose instead that on the first day of the semester your instructor had walked into your class and said, "This is a course in formal logic, and Lenin suffered his first stroke in May 1922." You would have found this more than a bit odd. Suppose that a short while later in the same class your instructor had then gone on to say, "There will be a quiz every other week, and Alexandria, Egypt, is named after Alexander the Great." At this point you would have begun to feel a bit uneasy and you would have looked around at the other students. Suppose finally that somewhat later, your instructor had then gone on to say, "The final exam will count for one-third of your course grade, and Euclid is credited with the proof that the square root of 2 is an irrational number." My guess is that at that point you and your fellow classmates would have started tiptoeing toward the exit. The collective bubble over all of your heads would have read: "What does the one thing have to do with the other? What's the connection between the first half of each of this instructor's sentences and the second half?" Or (if on that day you had known the terminology) your collective bubble would have read: "In each of the preceding conjunctions the two conjuncts have no relation to one another. Why, then, is this instructor conjoining such conjuncts?"

Your unease, however, would have concerned psychology (your instructor's) and not logic as such. Your confusion concerned not the meaning of your instructor's statements but rather your instructor's reasons for uttering them. The point is that you understood each of the sentences and you could have determined their truth-values without too much difficulty. Consider the sentence 'This is a course in formal logic, and Lenin suffered his first stroke in May 1922'. Had you known that each conjunct is true, you would have known in a jiffy that the entire conjunction is true. Whether the sentence is true or false is one thing; whether it's appropriate or inappropriate (i.e., bizarre) to utter it is another thing altogether. In logic our concern is exclusively with truth and falsehood, rather than with appropriateness and inappropriateness.

It simply doesn't matter--at least where the truth-value of the sentence is concerned--whether there's any connection between the left conjunct and the right conjunct in the conjunction 'This is a course in formal logic, and Lenin suffered his first stroke in May 1922'. By the same token, it simply doesn't matter--again, at least where the

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truth-value of the sentence is concerned--whether there's any connection between the antecedent and the consequent in the conditional 'If Fido wrote the Iliad then the Moon is made of pink fluff.

Once you divest yourself of the view that there has to be a connection, it should become somewhat easier to see that there's no difference either meaning-wise or truth-value-wise between the sentence 'If Fido wrote the Iliad then the Moon is made of pink fluff and the sentence 'Fido wrote the Iliad only if the Moon is made of pink fluff. Each of these sentences has the exact same meaning as the sentence 'It is not the case both that Fido wrote the Iliad and that the Moon is not made of pink fluff. Now, since this sentence is true--Fido did not write the Iliad--each of the two former sentences is true as well. And once you see that, it should become easy-ish to see that there's no difference truth-value-wise between the sentence 'If you work hard next semester then you'll pass' and the sentence 'You'll work hard next semester only if you'll pass'. They both mean that it's not the case that you'll work hard and yet that you won't pass.

• It is very important to read the question and tailor your answer to the question. There is an issue about the many ways that people try to explain informally what implication means and another about the many varieties of implication - logical, causal, etc. But you don't need to take all that on to answer this question. It is very tempting to paste in material that you have already written. But the guidelines point out that the community tends to vote down such answers because they tend to see it as self-promotion. I think you will find that the usual interpretation of "only if" is "if and only Feb 11, 2023 at 9:34