# How can I teach intuition why ‘If P then Q’ and ‘P only if Q’ mean the same, to first year undergraduates?

In every of my ten years as course instructor introducing first year undergraduates to proofs, I always project this quotation on the lecture screen, then dictate the whole caboodle word for word! But for the tenth straight year, this quotation failed to teach intuition why $$\color{mediumseagreen}{\text{If P then Q}}$$ and $$\color{darkGoldenrod}{\text{P only if Q}}$$ mean the same! On the midterm that I am marking now, some students always answer false to this question

True or false ? $$\color{mediumseagreen}{\text{'If P then Q'}}$$ is logically equivalent to $$\color{darkGoldenrod}{\text{'P only if Q'}}$$.

#### How else can I teach intuition for $$\color{MediumAquamarine}{\text{If P then Q}}$$ = $$\color{ Goldenrod}{\text{P only if Q}}$$? How can I better assist students to intuit this equivalence? I don't want them to memorize anything. Truth Tables and proofs don't teach intuition.

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.
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.
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 truthvalue 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.

Lande NP, Classical logic and its rabbit-holes: A first course (Hackett Publishing 2013), pages 53-5.

• Is this long quotation from Lande the same one you make your students read every year? If so, there's your problem. Besides being very long, it overcomplicates the whole issue, in my opinion. Nov 6, 2023 at 0:43
• @user1110341 People don't learn by having a long quote read to them either. Here's a frame challenge: don't use the phrasing "only if" in class ever! As a professional mathematician, I would never use this phrasing when communicating mathematics (and indeed every time I see it, I have to reconstruct what it's equvialent to anew—it is a very unnatural phrasing). Why teach students something that isn't actually used in practice and is just confusing? Nov 6, 2023 at 8:36
• I have a degree in mathematics (perhaps a bit dated), and considerable experience with formal logic. The terminology "P only if Q" is unfamiliar to me in formal context, however, and I understand the sense you are using only from your assertion that it is equivalent to "if P then Q". My natural English reading of the former comes out closer to "P if and only if Q", which of course is not equivalent to "if P then Q". Nov 7, 2023 at 17:51
• It’s really good that you want to improve how you teach this topic. You might look into the Socratic method for a challenging topic like this. And use copious examples. Avoid reading long quotes or simple dictating. Intuition comes from seeing patterns, building hypotheses, testing and refining them. You need to give your students the opportunity to do this with this material (and if you aren’t already, with other material too).
– bob
Nov 8, 2023 at 0:18
• @user1110341 You say it "thrives" on Math SE, but many of those search results are people who are confused about what "only if" means, and only have to learn it because they are studying propositional logic. So their problem would go away if propositional logic classes ceased to use the confusing term. It's bad enough that the English phrasing "if ... then ..." already doesn't really correspond with the propositional $\Rightarrow$. Nov 8, 2023 at 19:41

I would totally tune out if you dictated that massive quote to me. I can't even bring myself to properly skim it. You say teaches intuition, but for me, at least, the whole point of intuition is to make something memorable with a concise obvious-sounding explanation (or possibly just an example that extrapolates well). And an 8-paragraph, 1000+ word wall of text is definitely not that.

Here's what feels intuitive to me:

Suppose we're sitting next to each other in class. If you say "hello" then I hear "hello." You say "hello" only if I hear "hello" (because if I didn't hear it then you didn't say it). But other people can also say hello to me so it's not true that I hear "hello" only if you say "hello."

But personally, that intuition isn't what I use in practice. In practice, I just think about this visually with inference arrows:

• the phrase "if X" tells you the arrow starts at X,

• but when the word "only" precedes it, that indicates to flip the arrow.

For the phrases you mentioned:

• "if P then Q" starts at P: $$P \rightarrow Q$$

• "P if Q" starts at Q: $$P \leftarrow Q$$

• "P only if Q" is like the above but flipped: $$P \rightarrow Q$$

Conveniently, these inference arrows can be replaced with implications:

• "if P then Q" means $$P \Rightarrow Q$$

• "P if Q" means $$P \Leftarrow Q$$

• "P only if Q" means $$P \Rightarrow Q$$

• +1 Arrows are the way to think about implications. After almost a decade in math research I still shiver when a proof of the theorem "Foo is equivalent to Shmoo" has two paragraphs that start with the words "Necessity" and "Sufficiency" rather than with the neat arrows "$\Leftarrow$" and "$\Rightarrow$". "Only if" is the little brother of this kind of misguided language acrobatics - a bit easier to parse, but still immediately turned into an arrow the moment it enters my head. Nov 6, 2023 at 1:12
• You should probably avoid suggesting that P implies Q somehow means that P causes Q. Scientists say, for example, that smoking CAUSES cancer. We cannot infer from this that smoking IMPLIES cancer. Nov 6, 2023 at 17:33
• I strongly agree with @DanChristensen . As another example: my friend Alice always stays at home on rainy sundays. From this we can conclude: "if Alice goes outside on sunday, then it is sunny" or equivalently "Alice only goes outside on sunday if it is sunny". However, Alice certainly has no power over the weather.
– Stef
Nov 7, 2023 at 21:42
• Personally I would need examples to develop intuition. Arrows are helpful (very helpful; they’re how I learned logical implication) but not enough here. Students need to understand what implication actually means in practical terms.
– bob
Nov 8, 2023 at 0:49
• Basically to me this answer feels more like providing the student with useful tools for manipulating logical implications rather than a gut understanding of logical implications. They’re orthogonal to each other even though both are important.
– bob
Nov 8, 2023 at 0:52

I'll do a frame challenge answer that is closely related to the others: the phrase "only if" is generally only useful inside the larger phrase "if and only if". It's not hard to intuit the idea that "if and only if" means "is logically equivalent to". Once students are okay with that, since "if" is clearly one of the implications, "only if" must be the other one. This doesn't need to cause a ton of confusion.

Aside from that, I agree with the people who say there's not much benefit to using the phrase "only if" at all.

• "Aside from that, I agree with the people who say there's not much benefit to using the phrase "only if" at all." See my comment overhead. Nov 8, 2023 at 9:25

To avoid confusion, in examples of implication, it may help to avoid any suggestion of causality, and to have both antecedent and consequent expressed in the present tense.

EXAMPLE

Consider the implication: If it is raining (R), then it is cloudy (C).

R => C

Clearly, rain does not cause cloudiness. And it is raining ONLY IF it is cloudy.

• [1.] Sorry, but your answer feels like a comment? It's too snippy to answer my question. Pls elaborate? [2.] And can you use a more self-evident truism? Your example isn't obvious at all. It's unnecessarily esoteric to presume knowledge of Meteorology. Too many adults misbelieve that rain causes cloudiness. Nov 8, 2023 at 9:18
• Your question was about teaching university undergrads, folks with supposedly with high enough grades in school to get into a university program where propositional logic is taught. I am almost certain they will know that rain does not cause cloudiness. Nov 8, 2023 at 14:09
• @DanChristensen - The vast majority of US undergrads are at open admissions colleges/universities, where effectively anyone with a high school diploma or GED can attend. (My university technically has minimum high school GPAs to attend, but grade inflation in high school has made these effectively meaningless.) The first course in which propositional logic is taught usually has no prerequisites. We do in fact have a substantial minority of students who believe rain causes cloudiness. Nov 9, 2023 at 22:45
• @AlexanderWoo If so, the instructor has only to point out that rain does not actually cause cloudiness. Just like they might have to point out that pigs cannot actually fly. Neither is a difficult concept. This is not rocket science. One sentence, a hint, ought to do it. (Hint: Rain does not cause cloudiness. And pigs cannot fly.) Nov 10, 2023 at 4:41

Two expressions are equivalent if they produce the same true/false result under the same circumstances.

For "if" sentences, to match intuition it's easier to concentrate on counter-examples, because the mathematical definition of "if P then Q" being true for P=false does not match with intuition so well.

Let the students create counter-examples:

• Under which circumstances can "if P then Q" be falsified?
• Under which circumstances can "P only if Q" be falsified?

They'll (hopefully) come to the very same situation for both cases: both sentences can be falsified only with P=true and Q=false.

(1) Maybe, you want to start with real-world sentences instead of the symbols P and Q, depending on the expected abstraction capabilities of your students.

An example that comes to my mind:

• If the lamp shines (P), the switch is on (Q). [Not the other way round, there might be a power outage, the fuse might be blown etc.]
• The lamp shines (P) only if the switch is on (Q).

True or false ? 'If P then Q' is logically equivalent to 'P only if Q'

You then provide the example for if P then Q - if you pull Fido's tail (P) then he'll bite you (Q). The 'P only if Q' version would be 'Only if Fido bites you (Q) you'll pull his tail (P)' which to a "normal" person (ie me) isn't the same thing at all - perhaps a better example would help

• I like your approach, but be careful for the formulation: $P$ meaning "You pull his tail", not "You will pull his tail". (Undergraduates can be very picky on details like this) Nov 6, 2023 at 10:10

I want to challenge the frame:

• if you want people to answer this question correctly on a test, repeat the correct answer 5 times and insist that it will be on a test.
• if you rather want to check whether people really understand this, do it in a different way. For example, give them two actual mathematical statements and ask whether they have the same meaning. (As an aside, I would struggle to find and example for the "Q only if P" since people rarely phrase math like this.)

There are multiple problems with the question as it is asked. First one is that it is posed in an abstract form, with $$P$$ and $$Q$$, which adds an additional layer of difficulty for the students. More importantly, it is likely that you never explained to your students properly what questions like this even mean. For example, what are $$P,Q$$ and "If $$P$$, then $$Q$$" - are they sentences or predicates? In the long text you've quoted, the two are badly conflated, without ever acknowledging the difference: there are examples that are sentences, but old Fido one is secretly a predicate - truth/or falsity of statements depend on an additional input (your behavior).

Now, suppose it is presumed they are sentences. I think it is safe to assume you never explained what "sentence" means (Is "this sentence is false" a sentence? Is a sentence like "Fido will bite you" a sentence, and if it is, does it (yet?) have a true/false value?). Maybe you have at least explained what it means for two sentences to be "logically equivalent". Did you tell them that it means that either both are true, or both are false? Did you go through the examples like "2+2=4" and "Joe Biden is the POTUS", insisting they are "logically equivalent"?

But even assuming you did, the question as you asked still does not fully make sense, since it contains placeholders for unknown statements $$P,Q$$. Is "If $$P$$, then $$Q$$" still a statement, or is it now a predicate taking two statements as arguments?

How are students suppose to "intuit" a jargon that you never give a precise meaning to?

• Only a mathematician would think people can't intuit a jargon they haven't been given a precise meaning for. A practiced musician looks at their conductor's subtle gestures, which have never been defined for them, and they almost always do what the conductor intended. (The meaning of the big gestures - e.g. the beat patterns, are explicitly taught.) Everywhere else in the humanities, scholars learn the meanings of words by seeing how they are used in the writings of previous scholars. There are good reasons mathematics works differently, but we shouldn't pretend it's the only way. Nov 7, 2023 at 18:35
• "if you rather want to check whether people really understand this" This is my goal, yes. "if you want people to answer this question correctly on a test, " No, this isn't my goal. It is advisable to delete this bullet point. Nov 8, 2023 at 9:26
• @AlexanderWoo, we are speaking about a course introducing undergraduates to proofs. Nov 8, 2023 at 9:41

If all you are trying to convey is an intuition, could a Venn diagram work, with a region P completely inside a region Q?

This would intuitively show that if P is true, then Q is true (if P, then Q). It would also show that if Q is not true, then P cannot be true (P only if Q).

If you wanted to make this less abstract (more intuitive) than a Venn diagram, you could express regions P and Q as locations on a map, for example Kansas and the USA (or wherever you may be located): If you are in Kansas, then you are in the USA. Also, if you are not in the USA, then you are not in Kansas.

• Nov 9, 2023 at 2:13
• Thanks, David. That is the type of Venn diagram I had in mind. Nov 9, 2023 at 15:14
• I edited my answer to include an example about using locations on a map, in case Venn diagrams are still too abstract / not intuitive enough. Nov 9, 2023 at 18:33

Point out that the question is designed to be deceptive by employing a sense of causality that doesn't exist in formal logic, which describes relationships but not necessarily causality.

It's fine to say "If light P is on then switch Q must be closed," even though it's the state of the switch which controls the state of the light. One could also say "light P is on only if switch Q is closed." You haven't affected the causality, just looked at the relationship another way.

The expressions “if $$P$$, then $$Q$$” and “$$P$$ only if $$Q$$” have the same literal meaning, but they carry different connotations—different enough that in many situations, only one of them sounds natural. To internalize their shared literal meaning, it might help to practice using them in situations where they both sound natural.

The king frowned. “The witch? She’s dangerous—too dangerous to ever set foot on this island again. I’m calling on her only if I have no one else.”

“And you’re going? After what he did to you? Let someone else clean up his mess.”

The witch shook her head. “If he’s calling on me, he has no one else.”

“Don’t worry about the gumshoe,” the DA chuckled. “If she’s as smart as they say, then she drops that ring down a storm drain.”

He tossed the ring on my desk, followed by a stack of twenties. “I hope you’re as smart as they say.”

I shrugged. “I’m as smart as they say only if I drop this ring down a storm drain.”

The geologist broke the silence. “He’s alive only if there’s oxygen down there somewhere. Oxygen we’ve seen no trace of.”

The captain’s voice crackled in her ear. “Tjio, continue to base. Your air reserves are gone. If you stay out, you’ll never make it back.”

Tjio was already wheeling around toward the signal. “If he’s alive, there’s oxygen down there somewhere.”

“Can’t you”—she sobbed—“can’t you see? You’ll never be safe from your past, until—” She whirled, eyes blazing. “The wedding goes ahead only if—only if you never see him again!”

He gazed across the table. The firelight vanished into the depths of his brown eyes. “The day you met her,” he said quietly, “was the first day you looked happy.”

“If the wedding goes ahead... I never see you again.”

Following on from @Vectornaut's answer and the comment by @John Bollinger: I think the issue is that 'only if' can be interpreted differently in everyday language. For instance, suppose I said the following to my housemate:

I will do the washing-up only if you clean the living room.

Upon hearing this utterance, my housemate thinks:

If I clean the living room, then he will do the washing-up.

In other words, the everyday interpretation of 'only if' in the above utterance is the opposite to its interpretation in logic. Now, there are plenty of linguistic and logical nuances around the meaning and interpretation of the above utterance that one could discuss, but I think the immediate thought of my hypothetical housemate alone explains why students find the logical equivalence of 'if P, then Q' and 'P only if' confusing. As @Greg Martin, @Jochen Glueck and others have commented, using 'only if' outside of the phrase 'if and only if' is generally avoided, even by professional mathematicians.

An example would be helpful. As someone that might have answered 'false' on that question, I will tell you what I am thinking...

If P then Q

This says to me that if proposition P is true, then Q must also be true, but there is no reason to say that P is true.

P only if Q

This seems to start with the assumption that given no other information, P would be true, but only if Q is true, and Q is the only thing required to make P true.

The second is much less common in normal English usage. Adding a comma when written or only a short pause when spoken changes the meaning. "P, only if Q" makes the statement that P is definitely true as long as Q is true. If spoken out loud, the comma would only be a short pause.

Say you are looking at a chicken and make the statement "That animal is a mammal only if it is warm blooded". Now add a short pause "That animal is a mammal, only if it is warm blooded". Maybe that's part of it as well, the statement doesn't seem so strange to me when talking in abstracts: "An animal is a mammal only if it is warm blooded".