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This is of the form P -> Q, which is equivalent to ~P \/ Q. The equivalent statement would be the following: "Either you don't study hard, or you'll pass the test." My question is, why does this make sense? Could someone please elaborate on what it means intuitively? I get how translating from ~P -> Q to P \/ Q makes sense, but for P -> Q, it is rather confusing.

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    The truth of this implication is only consistent with at least one of the following two events happening: you do not study hard; you pass the test. If neither of them happens, and only then, the implication is false. The if-then phrasing may suggest a causal connection often associated with colloquial conditionals, which is missing in the disjunction, but the material conditional that this is does not express any such thing either. Natural language phrasing tempts people to read more into it "intuitively" than is meant to be there.
    – Conifold
    Commented Aug 11, 2020 at 9:24

5 Answers 5

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The Law of Excluded Middle is the reason.

Either you do not study hard, or you do. ~P v P

If you do study hard, then you pass the test. P -> Q

Therefore... You do not study hard, or you pass the test. ~P v Q

[Note: this is an "inclusive or". You might pass the test and not study hard; or you might do only one from these; however, it must be at least one from these.]

A quick proof by cases

|_ P -> Q
|  ~P v P
|  |_ ~P
|  |  ~P v Q
|  +
|  |_ P
|  |  Q
|  |  ~P v Q
|  ~P v Q

You do not study hard, or you pass the test. ~P v Q

(It must be at least one of these. If one is denied, then the other must be affirmed.)

So, if you do study hard, then you pass the test. P -> Q

|_ ~P v Q
|  |_ ~P
|  |  |_ P
|  |  |  Q      explosion
|  |  P -> Q
|  +
|  |_ Q
|  |  |_ P
|  |  |  Q      
|  |  P -> Q
|  P -> Q
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IF statements and OR statements don't translate cleanly between English and logic.

Your English phrasing of "If you study hard, then you'll pass the test." actually implies if and only if (i.e. that you'll also fail the test if you don't study hard), in contrast to the one-directional IF you're trying to express.

To be clearer in your statement, you might say something like "You may or may not pass the test now, but if you study, you will pass for sure."

Next, your statement "Either you don't study hard, or you'll pass the test." ambiguously implies an exclusive OR, which is incorrect. You'd need to say "One or more of the following is true: You didn't study, or you passed."

With these clarifications, you can see the implication intuitively is that when you studied, you passed.

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  • "actually implies if and only if (i.e. that you'll also fail the test if you don't study hard)" This is misleading. An implication φ → ψ implies the transposed implication ¬ψ → ¬φ but "If you study hard, then you'll pass the test" does not imply "If and only if you study hard, you'll pass the test". In other words, it is possible you'll pass the test even if you don't study hard. Commented Aug 11, 2020 at 16:22
  • @Speakpigeon Yes, I understand what the logic is. I'm saying the way that English sentence is phrased often implies IFF, and I went on to suggest how to clarify the phrasing in the very next paragraph. I've edited my answer to be clearer on this. Commented Aug 11, 2020 at 16:29
  • But "If A,then B" never suggests "If A and only if A, then B". In the situation considered here, we all know that one can pass a test without studying hard. Further, while "If A and only if A, then B" can be suggested by particular A's and particular B's, this has nothing to do with the "If-then" form in itself. The "If-then" form is perfectly unambiguous. Commented Aug 11, 2020 at 16:45
  • "We all know that one can pass a test without studying hard" I'd have to disagree with you there. As another example, "If I water this plant, it will live" heavily implies that the plant will die if I don't water it. If you're used to communicating with people who are very precise and logical I understand if they avoid this kind of ambiguity, but in my experience non-logical/philosophical Anglophones in day-to-day English often use "If" as an IFF (and indeed rarely use the phrase "only if"). Commented Aug 11, 2020 at 16:55
  • The conditional "If I water this plant, it will live" is obviously false. It is either very bad English or wilfully misleading. The reality that there is a large majority of English people who have a poor command of English, who equivocate all the time, who dissemble, who obfuscate, who lie etc. is completely irrelevant to what a conditional means. It is just an irritant and a hindrance. And apparently does confuse philosophers no end. Commented Aug 11, 2020 at 17:47
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Think about it this way: one way to read the implication is as follows.

"If you study hard, then you'll pass the test. If you don't study hard, maybe you'll still get lucky, so you may or may not pass the test, I just don't know [since it's not if and only if]. Equivalently, the only thing that I'm sure will not happen is that you study hard and don't pass the test."

One way to read the disjunction is as follows.

"You will either not study hard, or pass the test (or both [since it's not exclusive or]). Equivalently, the only thing that I'm sure will not happen is that you study hard and don't pass the test."

Therefore they are equivalent.

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    Not equivalent, no. Your reasoning is a logical fallacy. You have argued first (φ → ψ) → ¬(φ ∧ ¬ψ), and second (¬φ ∨ ψ) → ¬(φ ∧ ¬ψ), both true, but from that you infer (φ → ψ) ⇔ (¬φ ∨ ψ)., which is a non sequitur (undistributed middle). In a nutshell, you have just argued ((A → C) ∧ (B → C)) → (A ⇔ B), which is of course a logical fallacy. Thank you to correct. Commented Aug 11, 2020 at 16:36
  • I have replaced "Therefore" by "Equivalently" to make this clear.
    – present
    Commented Aug 11, 2020 at 17:13
  • This is still fallacious. You now argue first (φ → ψ) ⇔ ¬(φ ∧ ¬ψ), but this would require that you prove first ¬(φ ∧ ¬ψ) → (φ → ψ), which is the same as (¬φ ∨ ψ) → (φ → ψ), which is in fact what you are trying to prove, so now you are committing the fallacy of assuming the consequent. Commented Aug 11, 2020 at 17:56
  • I don't think the argument is fallacious at this point. It is a verbal argument for why they are equivalent that can be read in both directions and therefore also argues for what you want to be proven. Of course it is not a formal proof. But that is not what the person asking the question is asking for -- the question is explicitly just about getting some intuition. The formal proof would not be useful to the person.
    – present
    Commented Aug 11, 2020 at 18:52
  • Assuming the consequent is a logical fallacy. In order to prove your conclusion, you start by assuming this very same conclusion: ((φ → ψ) ∧ (ψ → φ)) → (ψ → φ). This is logically "valid", but nonetheless fallacious. It is a "circular argument". You first "Equivalently" simply assumes (¬φ ∨ ψ) → (φ → ψ), which is the very conclusion you pretend to be proving. Commented Aug 12, 2020 at 7:11
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The neurology of human logic and the logic of an arithmetic-logic unit of a microprocessor architecture built on formal algebras aren't the same. Once you phrase a question in the language of intuition, properly speaking, you're more in the territory of the pscyhology of logic rather the technical formalisms of it. So we have to build an intuition for it.

Human intuition is a powerful tool, and is absolutely necessary in defeasible reasoning and informal logic. In computer science, much effort has gone into building systems that use common sense and intuition, and progress has been scant. The human brain is a massively parallel computer complicated by emotions. If you want a great introduction, Thinking, Fast and Slow by Kahneman is worth the read, and explains human intuition in judgement. He won the Nobel prize for his research on the topic.

As for your example, let's try the conditional with a slightly different example since space is more intuitive than causality. Let's say, if Socrates is in the kitchen, then he is in the house. (P->Q)

  • Then we have by logical equivalence the contraposition. (~Q->~P) If Socrates is not in the house, he is not in the kitchen. Easy enough.
  • But what about the inclusive disjunct (~P OR Q)? Does it make sense intuitively to say Either 'He's not in the kitchen' or 'He's in the house' or both? Certainly not like expressed like that! We don't communicate positions of people in such convoluted language. It's like using a double-negative in English; it can get confusing because of how our language works. It's not intuitive to use (~P OR Q) in natural language. But it can be explained more intuitively.

So, let's start with (P->Q) being true. Is it possible that he's not in the kitchen? Sure, remember that (P->Q) is identical with (~Q->~P), but it makes no claims about the inverse and converse. (~P->~Q, Q->P) may be true or false. That is, in the case that Socrates is not in the kitchen, he may be in the living room or outside. So, let's re-examine what is being said with (~P OR Q) to build an intuition:

'He's not in the kitchen' or 'He's in the house' or both? can be revised as: 'He's must be somewhere else in the house or outside since he's not in the kitchen' or 'he's in the house' or both. And that's why it's confusing in natural language. The natural language expression is redundant! Not too common to communicate a positions by saying 'If he's in the house but not in the kitchen, then he still can be in the house or not'. So, just to be thorough, the cases are as follows:

  • 'He's in the house outside of the kitchen' AND 'he's in the house' are logically consistent.
  • 'He's outside the house' OR 'he's inside' BUT NOT BOTH is consistent.
  • NEITHER 'He's in the kitchen' NOR 'He's outside' is consistent (because he could be in the living room if both are false, right?

That's a real headspinner! That's because our intuition (in this case our ability to convey concepts through syntax) doesn't process these redundancies naturally. Like those of us who are native English speakers who don't use the double-negative, we have to build an intuition through practice.

Lastly, we can use our visual smarts a la John Venn:

Venn diagram

That leaves us with the question of why isn't it intuitive not fully addressed. There's another reason we tend to think P->Q -> Q->P. It's the way the brain builds connections through association, if we constantly are exposed to P->Q thinking, we tend to associate Q with P such that Q->P. Human associations tend psychologically to go in both directions. If I say, 'animal with a trunk', you say elephant, and if I say name the part of an elephant, likely you'll say trunk since it differentiates the species from the genus of mammals.

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    I like your answer, though I think there is more than just association going on when we hear a suggestion of Q->P. We need to take into account the pragmatics of the utterance. "If you study you'll pass" is intended in the circumstances to provide an incentive to study. If you believe you'll pass anyway, there is no incentive. So, "if you don't study you won't pass" is not implied by the statement, but it is a conversational implicature. It would be irrelevant to tack on the "if you study" unless the speaker believes the studying is relevant to the passing.
    – Bumble
    Commented Aug 11, 2020 at 18:27
  • @Bumble I hadn't considered implicature. I think it is cert ainly defensible to tender the notion that goal-oriented factors may entice a person to think speciously generally. My experience is that people tend to believe words coming from the hand that feeds them. In this particular example, though, how should read implicature without context?
    – J D
    Commented Aug 11, 2020 at 20:21
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The conditional is the canonical form we can used to express logical implications in everyday language.

The so-called material implication used in mathematical logic is not an implication in any meaningful sense, and so it is not possible to interpret correctly a conditional statement expressing a logical implication in terms of the material implication.

Clearly, the conditional "If you study hard, then you'll pass the test" does not express a true logical implication, since it is very easy to see that there is a logical possibility that "you study hard" be true and "you'll pass the test" be false. Presumably, we all understand that the relation is only probable, not absolute.

Still, the statement "If you study hard, then you'll pass the test" does express a logical implication, if only a false one. As such, it can be adequately expressed in the formal way we all understand, for example Sx → Px, where x is the subject, S stands for "study hard" and P stands for "pass the exam".

However, Sx → Px here, true or false, would not be logically equivalent to Sx ⊃ Px, even if in some instances of the two expressions they could happen to have the same truth values.

There is nonetheless a straightforward interpretation of the conditional "If you study hard, then you'll pass the test" as a true logical implication, but it wouldn't apply in this context.

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