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How do the statements if p then q and p only if q compare

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    anyway, both express that q is necessary if p, but they may not be semantically identical "According to the standard theory, there is a kind of reciprocity between necessary and sufficient conditions, and “if p, q” sentences can always be paraphrased by “p only if q” ones. However, as writers in linguistics have observed, neither of these claims matches either the most natural understanding of necessary (or sufficient) conditions, or the behaviour of “if” (and “only if”) in English. "
    – user64448
    Feb 1, 2023 at 0:02
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    See this very similar post. Feb 1, 2023 at 7:02
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    The relevant point of that answer is the fact that "if P, then Q" and "P only if Q" are the same. Feb 1, 2023 at 16:07
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    @Marsha - is this about mathematical logic? Maybe you're thinking that the use of "if" implies causality, and that's confusing you? Would it make more sense if phrased "whenever p, then also q" and "p only when also q"? Feb 1, 2023 at 16:17
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    Does this answer your question? Are "If P then Q" and "Q only if P" equivalent?
    – User
    Feb 6, 2023 at 9:45

7 Answers 7

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In simple cases at least, "if p, then q" and "p only if q" have the same truth conditions. But this is not the same as saying that they mean the same thing.

Typically with conditionals, the antecedent is logically, epistemologically or temporally prior to the consequent. David Sanford, in his book "If P then Q" gives an example of the difference as follows:

  1. If you learn to play the cello, I'll buy you a cello.
  2. You'll learn to play the cello only if I buy you a cello.

In both cases, what is ruled out is you learning to play cello and me not buying you one. But the first suggests by implicature that if you first learn the cello I'll buy you one, while the second suggests that me buying you a cello is a precondition of you learning it.

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    @άνθρωπος Those are different even as logical prepositions. "If you learn to play the cello, I'll buy you a cello" rules out the scenario where you learn to play the cello, but I don't buy you one; I can be generous and buy you one anyway even if you don't learn to play. Meanwhile, "I'll buy you a cello, only if you learn to play the cello" rules out the possibility of me buying you a cello if you don't learn to play; also, though the promise to buy a cello if you do learn to play may be suggested in colloquial English, it is not a logical consequence. Feb 1, 2023 at 2:37
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    Both rule out you learning to play the cello AND me not buying you one. Neither rule out other circumstances in which I buy you a cello.
    – Bumble
    Feb 1, 2023 at 6:51
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    I don't understand your comment, but I can assure you that sentences 1 and 2 have the same truth conditions and as such are very similar. The difference is mainly one of the implicatures involved in reversing the position of the antecedent and consequent.
    – Bumble
    Feb 1, 2023 at 9:32
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    <<Is this same "If you learn to play the cello, I'll buy you a cello" and "I'll buy you a cello, If you learn to play the cello"?>> Yes, those are the same. <<Is this similar "I'll buy you a cello, If you learn to play the cello" and "I'll buy you a cello, only if you learn to play the cello."?>> No. Those are quite different. But the second of those is not my sentence 2. The 'only if' is in a different place.
    – Bumble
    Feb 1, 2023 at 11:26
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    i said this is yours: "You'll learn to play the cello only if I buy you a cello." It is looks like " first you are learn to swim, and then i'll pour a water into a pool". But 1 one it not looks like this nonsense. So 2 is not similar or same as 1. Also you change time to the future = possibility in "you get", but in source "1" future time was in "I will do", so you broke all the causes, that is why yours 2 looks like nonsense ultimatum: if not I, you can't to do. And it may be not yours but Davids. Feb 1, 2023 at 11:48
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I'm not a logician, but there is a simple visual explanation. Consider Venn diagrams. Event p is a circle completely within the space of the larger circle q . "If p, then q" is satisfied as if you're in p, you're also in q . "p only if q" is satisfied as you can't be in p without being in q. But you can be in q without being in p.

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  • "A Venn diagram is a widely used diagram style that shows the logical relation between sets,"
    – user64448
    Feb 1, 2023 at 6:17
  • "If you're in p, you're also in q" just sounds like a rephrasing of "you can't be in p without being in q" (but then I guess the interpretation of the former relies on how one interprets "If p, then q"). Also, "you can be in q without being in p" applies to both (at least from the point of view of formal logic - look up any truth table of "if p then q" and you'll see that q can be true while p is false). Also also, visual explanations tend to work best when visualised.
    – NotThatGuy
    Feb 1, 2023 at 9:57
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    Actually I can't tell whether or not you're saying they are the same (but my comment above still applies). It sounds as if you're saying they're different, but you only presented one visual explanation. If they're different, you should present a visual explanation for each one, and if they're the same, you should probably say "they're the same", to make that clear.
    – NotThatGuy
    Feb 1, 2023 at 10:11
  • I read this answer as saying "I found one visual example that satisfies both statements". From which... We can't conclude much. Just because you found one example that satisfies both "if p then q" and "p only if q" we can't conclude that they're equivalent, nor that they're not equivalent. Just that they're both true on this one example.
    – Stef
    Feb 1, 2023 at 15:58
  • @Stef A Venn diagram is a commonly-used way to represent the overlap between different things. It's less an "example" and more one possible way to represent something, similar to how one could present a truth table, or just explain when overlap would occur: those could all represent the same data.
    – NotThatGuy
    Feb 2, 2023 at 10:39
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Since the question asker has not given much context for their question, I have chosen to interpret it through the lens of propositional logic in a rather narrow fashion. Other interpretations of the question, especially those which draw more from the "everyday" meanings of the words, are certainly plausible.

"if p, then q" and "p only if q" are the same thing

In propositional logic, "if p, then q" and "p only if q" are interpreted to mean the same thing. The idea being conveyed is that if p is true, and the proposed relationship between p and q is itself true, then q must also be true. This is called a "conditional", and the standard notation for it is "pq".

To translate this into more concrete terms:

  • "if p, then q" is like "If it is raining, then I carry an umbrella."
  • "p only if q" is like "It is raining only if I carry an umbrella."

This challenges our intuition, since those sentences don't seem to mean the same thing. Note that in pure conditional logic, we are not actually concerned with whether or not p causes q in any real-world sense, or even with the sequence of events. We are only concerned with what the truth or falsity of one proposition can tell us about the other propositions.

"If it is raining, then I will carry an umbrella" states that, at the time it is raining, I will have an umbrella with me. If I find myself in the rain, and I don't have an umbrella with me, that violates the conditional. But if it is not raining, I am free to either carry an umbrella or not carry an umbrella; neither state violates the conditional.

Here is a truth table for that:

Rain Umbrella "If it is raining, then I will carry an umbrella"
T T T
T F F
F T T
F F T

"It is raining only if I will carry an umbrella" states that the rain only occurs at the times that I carry an umbrella. If it rains at a time when I don't carry an umbrella, that violates the conditional. But if it does not rain at a time when I do carry an umbrella, that does not violate the conditional, because nothing in the conditional says that it will rain every time I will carry an umbrella. And of course, a sunny day with no umbrella doesn't violate the conditional either.

Here is a truth table for that:

Rain Umbrella "It is raining only if I will carry an umbrella"
T T T
T F F
F T T
F F T

Notice that the truth tables are the same in both cases.


"p if and only if q" is different

I do wonder if you meant to write "p if and only if q" (rather than "p only if q" or "p if only q"). This is called a biconditional, and it is the combination of "if p, then q" and "if q, then p". ("It is raining if and only if I will carry an umbrella.") The standard notation for this is "pq".

The main practical difference is this:

  • pq: if we hold this to be true, and p is false, then q could be true or false. "If it is raining, then I carry an umbrella" does not tell you whether or not I carry an umbrella on sunny days as well.
  • pq: If we hold this to be true, and p is false, then q must also be false. "It is raining if and only if I will carry an umbrella" tells you that I carry an umbrella on rainy days, and I don't carry an umbrella on rain-free days.

Here's an extended truth table:

p q pq pq
T T T T
T F F F
F T T F
F F T T

I consulted LibreTexts, Khan Academy, AlphaScore.com, and Wikipedia while writing this answer.

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  • Welcome to StackExchange, MJ713. Your answer is correct, but only up to a point. If you focus exclusively on the truth conditions of "if p then q" and "p only if q" then they are indeed the same. But meanings are more subtle things than mere truth conditions. The point I make in my answer is that even though these sentences have the same truth conditions, they have different implicatures, and so they tend to have different meanings when used in ordinary sentences. Still, your answer is correct, and several of the others here are not, so you get a +1 from me.
    – Bumble
    Feb 2, 2023 at 5:07
  • My comment to the answer by @Bumble may have some relevance here.
    – sdenham
    Feb 10, 2023 at 3:49
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How does "if p, then q" compare to "p only if q"?

"If p, then q" says that there is at least one event p that will result in event q. "p only if q" says that there is exactly one event q that will result in event p.

The difference is important to analyzing statements for fallacies. In the first example, "if not-p, then not-q" is fallacious. The problem is denying the antecedent; the terms of the if-then statement also allow for events x, y, or z to result in event q.

However, in the second example, "if not-q, then not-p" is valid; here, q is the only event in the world that will result in event p. Once q is negated, then so is p.

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    But you're looking at different if-then statements for the fallacious/valid distinction you're drawing. In both cases, "if not-q, then not-p" is a valid conclusion, while "if not-p, then not-q" is fallacious. Feb 1, 2023 at 2:39
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"P only if Q" seems to me to say "P happens, only if Q happens first." If that's the case, then "P only if Q" would be translated to "if Q then P" which is quite different than "if P then Q"

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How does "if p, then q" compare to "p only if q"?

The difference is the order of evaluation and assignment of the variables p and q.

  • If P then Q: Evaluate P then assign Q
  • P only if Q: Assign P then evaluate Q

A more common example of the difference is a list of steps to assemble an item (say Model X-100 for illustrative purposes). Assume the instructions are split between two pages and the page needs to be turned over to see the rest of the instructions:

  • Page 1. If you have Model X-201.
  • Page 2. Cut the provided Rod B into 5 even pieces.

This is quite different from

  • Page 1: Cut the provided Rod B into 5 even pieces.
  • Page 2 Only if you have Model X-201.

P only if Q leads to a situation where a task is performed before the required analysis: I cut Rod B into five pieces before confirming which model I had. This is particularly annoying if an uncut Rod B is required to complete assembly of Model X-100. ;-)

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Nelson Lande's Classical Logic and Its Rabbit-Holes (2013) expatiates on this difference between "P only if Q" vs. "P if Q" THE BEST! I forgo blockquotes, for the sake of readability.

Page 65/437 of my .epub file

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.

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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

Page 69/473 of my .epub file

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.

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  • This answer at least overlaps to a great extent with you answer to the very similar question about the difference between p if q and p only if q. (philosophy.stackexchange.com/questions/40878/…) So I'm afraid I have nothing to add to my comment there.
    – Ludwig V
    Feb 16, 2023 at 17:59

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