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

Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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 "p → q".

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.

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"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 "p ↔ q".

The main practical difference is this:

• p → q: 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.
• p ↔ q: 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	p → q	p ↔ q
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.

Answer 5

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.

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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.

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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

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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.